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algebraic approaches to differential calculus | https://ncatlab.org/nlab/source/algebraic+approaches+to+differential+calculus | Derivatives and differentials are usually expressed in terms of limits in the sense of analysis. However it became clear in about the last half century that much of the knowledge on usual [[differential calculus]] can be inferred from using just algebraic properties of differentials and derivatives, most notably the Leibniz rule for differentiating products ($D (f g) = (D f) g + f (D g)$); an alternative [[synthetic differential geometry|synthetic]] formalism also appeared which did not use limiting procedures as well.
A [[derivation]] of an [[associative algebra]] is simply a linear [[endomorphism]] satisfying the Leibniz rule. Then for example the [[tangent vector fields]] on a [[smooth manifold]] are obtained as derivations of the algebra of $C^\infty$-functions on the manifold. The differential of a map is a linearized approximation. This is clear in various non-classical analytic setups, for example for maps between [[Banach spaces]] and for differentiable [[manifolds]].
This linearization idea has been obtained at the level of [[sheaves]] of $\mathcal{O}$-modules by [[Grothendieck]] for algebraic varieties a the view toward the differential calculus for varieties in prime characteristics. It is interesting that he related differential calculus to resolutions of the diagonal, where he considered the sheaves of modules supported on [[infinitesimal neighborhood]]s of diagonal. Indeed, to define a derivative in analysis, one needs to start with consideration of differences of values of a function at points which are close to one to another, $x$ and $x + \Delta x$, and this means that means that the pair $(x, x + \Delta x)$ is close to the diagonal of the cartesian square $X \times X$. In [[numerical analysis]], various approximation schemas for higher order differential operators are involved which obviously live around higher diagonals in $X^n = X \times X \times \ldots \times X$ ($n$ times). One of the products of that thinking is Grothendieck's notion of a [[regular differential operator]]. This led later to the creation of the theory of [[D-modules]] which are sheaves of modules over the [[sheaf]] of rings of regular differential operators over a scheme, or a complex analytic manifold.
We plan in the nLab to cover many aspects of the interaction between geometry and differential calculi of various sorts including [[synthetic differential geometry]] and algebraic counterparts of notions from differential calculi. It is hard to say, however, where [[homological algebra]] belongs: the differential in the sense of homological algebra is rather a notion which can be systematized into the more general subject of [[homotopical algebra]], but in some cases it is related to analogues of [[exterior differentiation]] for the de Rham [[complex]] of [[differential forms]] (say on manifolds). But there is also an analogue of the [[Taylor series]] for functors in some homotopical contexts (say [[Goodwillie calculus]]).
One should also point out that an elaborate schema for
differential calculus in
noncommutative geometry has been proposed by Tsygan
in terms of algebras with higher brackets. In one version,
a differential calculus is given there by a Gerstenhaber
algebra and a Batalin-Vilkovisky module over it.
See [[derivation]], [[regular differential operator]], [[differential form]], [[differential bimodule]], [[universal differential envelope]], [[differential forms in synthetic differential geometry]], [[connection]], [[connection for a differential graded algebra]], [[D-module]], [[Fox derivative]], [[crystal]], [[Fermat theory]]...
[[!redirects algebraic approach to differential calculus]] |
algebraic category | https://ncatlab.org/nlab/source/algebraic+category |
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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
# Algebraic categories
* table of contents
{: toc}
## Idea
An algebraic category is a [[concrete category]] which behaves very much like the categories familiar from [[algebra]], such as [[Grp]], [[Ring]], and [[Vect]], but characterised in category-theoretic terms. But many other categories are also algebraic, most famously [[compact Hausdorff space|CompHausTop]]; one can describe these in purely algebraic terms, but only using infinitary (perhaps even largely many) operations.
There are several definitions of 'algebraic' in the literature. Here, we will follow AHS (see references) in using a generous interpretation, but other authors follow Johnstone in using 'algebraic' to mean monadic (a stricter requirement), while some authors add finiteness conditions that remove examples such as $Comp Haus Top$. However, all of these notions are related, and we will discuss them here.
The definitions in AHS also includes a requirement violating the [[principle of equivalence]], which we omit: that of unique strict lifts of isomorphisms, which serves to fix algebraic categories up to [[isomorphism of categories|isomorphism]] (instead of mere [[equivalence of categories|equivalence]]).
## Definitions
Let $A$ be a [[concrete category]]; that is, $A$ is equipped with a [[forgetful functor]] $U\colon A \to Set$ to the [[Set|category of sets]]. For some authors, such a category is called 'concrete' only if $U$ is [[representable functor|representable]], but that follows in all the cases considered below; in particular, if $A$ has free objects (that is, if $U$ has a [[left adjoint]] $F$), then $U$ is representable by $F(1)$, where $1$ is a [[singleton]].
+-- {: .un_defn}
###### Definition (based on AHS 23.38)
The concrete category $A$ is __algebraic__ if the following conditions hold:
* $A$ has [[free objects]].
* The category $A$ has all binary [[coequalizers]].
* The forgetful functor $U$ preserves and reflects [[extremal epimorphisms]].
=--
+-- {: .un_defn}
###### Definition (based on AHS ...)
The concrete category $A$ is __monadic__ if the following conditions hold:
* $A$ has [[free objects]].
* The [[adjunction]] $F \dashv U$ is [[monadic adjunction|monadic]].
=--
+-- {: .un_defn}
###### Definition (based on AHS 24.11)
An algebraic (or monadic) category is __bounded__ if the following condition holds:
* For some [[cardinal number]] $\kappa$ and every $\kappa$-[[directed colimit]] in $A$, the universal [[cocone]] is jointly [[surjection|surjective]] in $Set$.
=--
+-- {: .un_defn}
###### Definition (based on AHS 24.4)
An algebraic (or monadic) category is __finitary__ if the following condition holds:
* For every finitely [[directed colimit]] in $A$, the universal [[cocone]] is jointly [[surjection|surjective]] in $Set$.
=--
Note that this is a weakening of the condition that the forgetful functor $U$ is [[finitary functor|finitary]] (that is, that $U$ preserves directed colimits); every universal cocone in $Set$ is jointly surjective, but not conversely.
## Properties
Every monadic category is algebraic; an algebraic category is monadic if and only if the forgetful functor $U$ preserves [[congruences]]. (AHS 23.41)
A category is algebraic if and only if it is a [[reflective subcategory]] of a monadic category with [[regular epimorphism|regular epic]] reflector; given an algebraic category, this monadic category is the [[Eilenberg–Moore category]] of the monad $U \circ F$. (AHS 24.3)
Every monadic category is the category of algebras for some [[variety of algebras]], although we must allow potentially a [[proper class]] of infinitary axioms; that is, every monadic category is [[equationally presentable category|equationally presentable]]. Similarly, every algebraic category is the category of algebras for some [[quasivariety of algebras]]; that is, we allow [[conditional statements]] of equations among the axioms. (AHS 24.11)
As special cases of the last item:
* A concrete category is bounded monadic if and only if it is equationally presentable (presented by a variety) with a small set of operations (and hence equations).
* A concrete category is bounded algebraic if and only if it is presented by a quasivariety with a small set of operations.
* A concrete category is finitary monadic if and only if it is the category of algebras for some finitary variety; that is, we have only a small set of finitary operations.
* A concrete category is finitary algebraic if and only if it is the category of algebras for some finitary quasivariety.
Also, every algebraic category whose [[forgetful functor]] preserves [[filtered colimits]] is the category of [[models]] for some [[first-order theory]]. The converse is false.
## Examples
The typical categories studied in [[algebra]], such as [[Grp]], [[Ring]], [[Vect]], etc, are all finitary monadic categories. The [[monad]] $U \circ F$ may be thought of as mapping a set $x$ to the set of words with alphabet taken from $x$ and the connections between letters taken from the appropriate algebraic operations, with two words identified if they can be proved equal by the appropriate algebraic axioms.
The category of cancellative [[monoids]] is finitary algebraic but not monadic. The category [[Field]] of [[fields]] is not even algebraic.
Assuming the [[ultrafilter principle]], the category of [[compact Hausdorff spaces]] is monadic, but not bounded algebraic. The monad in question takes a set $x$ to the set of [[ultrafilters]] on $x$. (Without the ultrafilter principle, this monad still exists, but it may be quite small, possibly even the [[identity monad]]; passing to [[locales]] does not help.)
Similarly, the category of [[Stone space]]s is algebraic, but not monadic or bounded algebraic.
## References
The original definitions can be found in
* [[F. William Lawvere]], _Algebraic theories, algebraic categories, and algebraic functors_. 1965 Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), 413–418. North-Holland, Amsterdam.
Our definitions are taken from
* **AHS**: [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]]; _Abstract and Concrete Categories: [[The Joy of Cats]]_, Sections 23 & 24; [web](http://katmat.math.uni-bremen.de/acc).
Actually, AHS discusses the more general concept of algebraic (etc) *functors*, generalising from $U\colon A \to Set$ to arbitrary functors (not necessarily faithful, not necessarily to $Set$). We actually take our definitions from AHS\'s characterisation theorems in the case of faithful functors to $Set$. We probably should discuss the more general concept, perhaps at [[algebraic functor]]; we already have [[monadic functor]].
* [[Peter Johnstone]]; _[[Stone Spaces]]_, Section 3.8
For Johnstone, a concrete category is 'algebraic' if and only if it is monadic. However, Johnstone also discusses [[equationally presentable category|equationally presentable categories]].
Another modern reference is
* [[Jiří Adámek]], [[Jiří Rosický]], [[Enrico M. Vitale]],
_Algebraic theories_,
Cambridge Tracts in Mathematics 184 (2011), Cambridge University Press.
doi:10.1017/cbo9780511760754.
[[!redirects algebraic category]]
[[!redirects algebraic categories]]
[[!redirects bounded algebraic category]]
[[!redirects bounded algebraic categories]]
[[!redirects finitary algebraic category]]
[[!redirects finitary algebraic categories]]
[[!redirects monadic category]]
[[!redirects monadic categories]]
[[!redirects bounded monadic category]]
[[!redirects bounded monadic categories]]
[[!redirects finitary monadic category]]
[[!redirects finitary monadic categories]] |
algebraic cobordism | https://ncatlab.org/nlab/source/algebraic+cobordism |
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###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Algebraic cobordism_ is the bigraded [[generalized cohomology theory]] represented by the [[motivic Thom spectrum]] $MGL$. Hence it is the algebraic or [[motives|motivic]] analogue of [[complex cobordism]]. The $(2n,n)$-graded part has a geometric description via [[cobordism]] classes, at least over fields of characteristic zero.
## Definition
Let $S$ be a [[scheme]] and $MGL_S$ the [[motivic Thom spectrum]] over $S$. **Algebraic cobordism** is the [[generalized motivic cohomology theory]] $MGL_S^{*,*}$ represented by $MGL_S$:
... formula here ...
## Properties
### The (2n,n)-graded part
Let $S = Spec(k)$ where $k$ is a field of characteristic zero.
A geometric description of the $(2n,n)$-graded part of algebraic cobordism was given by [[Marc Levine]] and [[Fabien Morel]]. More precisely, [Levine-Morel](#LevineMorel) constructed the universal _oriented cohomology theory_ $\Omega^* : \Sm_k \to CRing^*$. Here _oriented_ signifies the existence of [[direct image]] or [[Gysin homomorphisms]] for [[proper morphisms of schemes]]. This implies the existence of[[Chern classes]] for [[vector bundles]].
+-- {: .num_theorem}
###### Theorem
**([Levine-Morel](#LevineMorel))**.
There is a canonical isomorphism of graded rings
$$ \mathbf{L}^* \stackrel{\sim}{\longrightarrow} \Omega^*(\Spec(k)) $$
where $\mathbf{L}^*$ denotes the [[Lazard ring]] with an appropriate grading.
=--
+-- {: .num_theorem}
###### Theorem
**([Levine-Morel](#LevineMorel))**.
Let $i : Z \hookrightarrow X$ be a [[closed immersion]] of smooth $k$-schemes and $j : U \hookrightarrow X$ the complementary [[open immersion]]. There is a canonical exact sequence of graded abelian groups
$$ \Omega^{*-d}(Z) \stackrel{i_*}{\to} \Omega^*(X) \stackrel{j^*}{\to} \Omega^*(U) \to 0, $$
where $d = \codim(Z, X)$.
=--
+-- {: .num_theorem}
###### Theorem
**([Levine-Morel](#LevineMorel))**.
Given an embedding $k \hookrightarrow \mathbf{C}$, the canonical homomorphism of graded rings
$$ \Omega^*(k) \longrightarrow MU^{2*}(pt) $$
is invertible.
=--
+-- {: .num_theorem}
###### Theorem
**([Levine 2008](#Levine2008))**.
The canonical homomorphisms of graded rings
$$ \Omega^*(X) \longrightarrow MGL^{2*,*}(X) $$
are invertible for all $X \in \Sm_k$.
=--
## Related concepts
* [[motivic Thom spectrum]]
* [[complex cobordism]]
* [[motivic homotopy theory]]
* [[motivic cohomology]]
* [[algebraic K-theory]]
\linebreak
[[!include flavours of cobordism cohomology theories -- table]]
## References
There are two notions of "algebraic cobordism", not closely related, one due to [Snaith 77](#Snaith77), and one due to [Levine-Morel 01](#LevineMorel01).
### Snaith's construction
{#SnaithConstructiom}
* {#Snaith77} [[Victor Snaith]], _Towards algebraic cobordism_, Bull. Amer. Math. Soc. 83 (1977), 384-385 ([doi:10.1090/S0002-9904-1977-14281-X](https://doi.org/10.1090/S0002-9904-1977-14281-X))
* {#Snaith79} [[Victor Snaith]], _Algebraic Cobordism and K-theory_, Mem. Amer. Math. Soc. no 221 (1979)
* {#GepnerSnaith08} [[David Gepner]], [[Victor Snaith]], _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Documenta Math. 2008 ([arXiv:0712.2817](http://arxiv.org/abs/0712.2817))
The construction in [Snaith 77](#Snaith77), motivated from the [[Conner-Floyd isomorphism]], uses a variant of his general construction ([[Snaith's theorem]]) of a periodic [[multiplicative cohomology theory]] $X(b)^*(-)$ out of a pair consisting of a homotopy commutative [[H-monoid]] $X$ and a class $b\in \pi_n(X)$:
When $X = B S^1$ (the [[classifying space]] of the [[circle group]]) and $b$ is a generator of $\pi_2(BS^1)\cong\mathbb{Z}$ then $X(b)^*(-)$ is isomorphic with 2-periodic [[complex K-theory]].
When $X = B U$ and $b$ a generator of $\pi_2(BU)\cong\mathbb{Z}$ one obtains $MU^*[u_2,u_2^{-1}]$ where [[MU]] is the (topological) [[complex cobordism cohomology]] and $u_2$ is the periodicity element.
Then Snaith introduces a variant of such constructions with a more general ring $A$ replacing the complex numbers; and uses the Quillen's description of [[algebraic K-theory]] of a ring $A$ in terms of the [[classifying space]] $B GL(A)$; this way he obtains an algebraic cobordism theory.
Later, [Gepner-Snaith 08](#GepnerSnaith08) returned to the question of algebraic cobordism this time using the motivic version of algebraic cobordism of Voevodsky, namely the [[motivic spectrum]] $M GL$ representing universal oriented motivic cohomology theory (which is different from Morel-Voevodsky algebraic cobordism), and to the motivic version of [[Conner-Floyd isomorphism]] for which they give a comparably short proof.
### Morel-Levine's construction
* {#LevineMorel01} [[Marc Levine]], [[Fabien Morel]], _Cobordisme algébrique I_, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 723--728, 2001 (<a href="http://dx.doi.org/10.1016/S0764-4442(01)01833-X">doi:10.1016/S0764-4442(01)01833-X</a>); _Cobordisme algébrique II_, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 815--820, 2001 (<a href="http://dx.doi.org/10.1016/S0764-4442(01)01832-8">doi:10.1016/S0764-4442(01)01832-8</a>).
* [[Marc Levine]], _Algebraic cobordism_, Proceedings of the ICM, Beijing 2002, vol. 2, 57--66, [math.KT/0304206](http://arxiv.org/abs/math/0304206)
* {#LevineMorel} [[Marc Levine]], [[Fabien Morel]], _Algebraic cobordism_, Springer 2007, [pdf](https://www.uni-due.de/~bm0032/publ/AlgCobordBook4.pdf).
* [[Marc Levine]], _A survey of algebraic cobordism_ ([pdf](http://www.uni-due.de/~bm0032/publ/SurveyAlgCobord.pdf‎))
* [[Marc Levine]], _Three lectures on algebraic cobordism_, University of Western Ontario Mathematics Department, 2005, [Lecture I](http://www.uni-due.de/~bm0032/publ/CobordismLec1LS.pdf), [Lecture II](http://www.uni-due.de/~bm0032/publ/CobordismLec2LS.pdf), [Lecture III](http://www.uni-due.de/~bm0032/publ/CobordismLec4LS.pdf).
* [[Marc Levine]], [[Fabien Morel]], [[Oberwolfach]] Arbeitsgemeinschaft mit aktuellem Thema, April 2005 [report](http://www.ems-ph.org/journals/show_abstract.php?issn=1660-8933&vol=2&iss=2&rank=2), [notes](http://www.ems-ph.org/journals/show_pdf.php?issn=1660-8933&vol=2&iss=2&rank=2)
* [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], _A universality theorem for Voevodsky's algebraic cobordism spectrum_, Homology, Homotopy and Applications, 2008, 10(2), 211-226 ([arXiv:0709.4116](http://arxiv.org/abs/0709.4116))
* [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], _On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory_, Inventiones mathematicae, 2009, 175(2), 435-451, [doi:10.1.1.244.7301](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.244.7301), [arXiv:0709.4124](http://arxiv.org/abs/0709.4124).
* [[Marc Hoyois]], _From algebraic cobordism to motivic cohomology_, [pdf](http://math.mit.edu/~hoyois/papers/hopkinsmorel.pdf), [arXiv](http://arxiv.org/abs/1210.7182).
* [[Markus Spitzweck]], _Algebraic cobordism in mixed characteristic_ ([arXiv:1404.2542](http://arxiv.org/abs/1404.2542))
* [[Marc Levine]], [[Girja Shanker Tripathi]], _Quotients of MGL, their slices and their geometric parts_, [arXiv:1501.02436](http://arxiv.org/abs/1501.02436).
More chat about the relation to [[motivic homotopy theory]]:
* _Interdependence between A^1-homotopy theory and algebraic cobordism_, [MO/36659](http://mathoverflow.net/questions/36659/interdependence-between-a1-homotopy-theory-and-algebraic-cobordism/36698#36698).
A simpler construction was given in
* M. Levine, R. Pandharipande, _Algebraic cobordism revisited_ ([math.AG/0605196](http://arxiv.org/abs/math/0605196))
A [[Borel-Moore homology]] version of $MGL^{*,*}$ is considered in
* [[Marc Levine]], _Oriented cohomology, Borel-Moore homology and algebraic cobordism_, [arXiv](http://arxiv.org/abs/0807.2257).
The comparison with $MGL^{2*,*}$ is in
* {#Levine2008} [[Marc Levine]], _Comparison of cobordism theories_, Journal of Algebra, 322(9), 3291-3317, 2009, [arXiv](http://arxiv.org/abs/0807.2238).
The construction was extended to [[derived algebraic geometry|derived schemes]] in the paper
* Parker Lowrey, [[Timo Schuerg]]. _Derived algebraic bordism_, 2012, [arXiv:1211.7023](http://arxiv.org/abs/1211.7023).
The close connection of algebraic cobordism with [[K-theory]] is discussed in
* José Luis González, Kalle Karu. _Universality of K-theory_. 2013. [arXiv:1301.3815](http://arxiv.org/abs/1301.3815).
An algebraic analogue of h-cobordism:
* Aravind Asok, Fabien Morel, _Smooth varieties up to $\mathbb{A}^1$-homotopy and algebraic h-cobordisms_ ([arXiv:0810.0324](http://arxiv.org/abs/0810.0324)).
A construction of algebraic cobordism as a non-[[A1-homotopy theory|$\mathbb{A}^1$-invariant]] [[cohomology theory]] on [[derived schemes]] and the resulting [[Conner-Floyd isomorphism]]:
* Toni Annala, [[Marc Hoyois]], Ryomei Iwasa, _Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory_ ([arXiv:2303.02051](https://arxiv.org/abs/2303.02051)).
[[!redirects algebraic cobordism spectrum]]
[[!redirects algebraic cobordism spectra]]
|
algebraic correspondences | https://ncatlab.org/nlab/source/algebraic+correspondences | See [[pure motives]] for now. |
algebraic curve | https://ncatlab.org/nlab/source/algebraic+curve |
#Contents#
* table of contents
{:toc}
## Idea
An algebraic curve is an [[algebraic variety]] of dimension $1$. Typically
one restricts considerations to either affine or projective algebraic curves. Most often one treats the plane algebraic curves, i.e. curves
with an embedding into $\mathbf{A}^2$ or $\mathbf{P}^2$; they are the locus of solutions of a single algebraic equation.
An _algebraic curve_ over a [[field]] $F$ is the locus of solutions of $(n-1)$-[[polynomials]] in $n$-[[variables]] of [[type]] $F$, provided the
Krull dimension of the ring is $1$.
## Properties
### General
* Every projective algebraic curve is [[birationally equivalent]] to a plane algebraic curve
* [[Mordell conjecture]]: every algebraic curve of genus $g\geq 2$ defined over rationals has at least one point over rationals
* To a nonsingular curve $C$ over the field of complex numbers one associates an [[abelian variety]], namely its [[Jacobian variety]] together with the period map or Abel-Jacobi map $C\to J(C)$.
### Function field analogy
[[!include function field analogy -- table]]
## Related concepts
* [[complex curve]], [[Riemann surface]]
* [[arithmetic curve]]
* [[elliptic curve]]
* [[stable curve]],
* [[moduli stack of curves]]
* [[Jacobian variety]],
* [[Mordell conjecture]],
* [[Bezout's theorem]]
## References
* [[William Fulton]], _Algebraic curves_ ([pdf](http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf))
* Wikipedia, _[Algebraic curve](http://en.wikipedia.org/wiki/Algebraic_curve)_
* Patrick J. Morandi, _Error Correcting Codes and Algebraic Curves_ , lecture notes New Mexico State University 2001. ([pdf](http://www.math.nmsu.edu/~pmorandi/math601f01/LectureNotes.pdf))
[[!redirects algebraic curves]]
|
algebraic cycle | https://ncatlab.org/nlab/source/algebraic+cycle |
#Contents#
* table of contents
{:toc}
## Definition
Let $X$ be a [[noetherian scheme]]. One defines a **$k$-dimensional algebraic [[cycle]]** as an element of the [[free abelian group]] $Z_k(X)$ generated by the [[closed subspace|closed]] [[integral scheme|integral]] subschemes of [[dimension]] $k$, and dually a $k$-[[codimension|codimensional]] cycle is an element of the [[free group]] $Z^k(X)$ generated by the closed integral subschemes of [[codimension]] $k$ in $X$. (An important special case is the group $Z^1(X)$ of 1-codimensional cycles, better known as the group of **[[Weil divisors]]**.) One usually writes a cycle as a formal sum
$$ C = \sum_{Z \subset X} n_Z.[Z] $$
## Direct image and inverse image of cycles
For [[proper morphisms]] $f : X \to Y$, one defines the [[direct image]] of a $k$-cycle by assigning
$$ f_*([Z]) = \deg_{R(f(Z))}(R(Z)) . [f(Z)] $$
when $\dim(f(Z)) = \dim(Z)$ and 0 otherwise. Here $f(Z)$ is considered as an integral subscheme of $Y$ with the [[reduced scheme|reduced]] subscheme structure induced from $Y$. $R(Z)$ denotes the field of [[rational functions]] on $Z$ and $\deg$ denotes the degree of the [[field extension]]. One gets homomorphisms $f_* : Z_k(X) \to Z_k(Y)$ for each $k$.
For [[flat morphisms]] of [[relative dimension]] $n$, one defines the inverse image of a $k$-cycle by assigning, for a closed integral subscheme $Z \subset Y$ of dimension $k$,
$$ f^*([Z]) = \sum_{Z_\alpha \subset f^{-1}(Z)} \length_{O_{X,z_\alpha}}(O_{f^{-1}(Z),z_\alpha}) [Z_\alpha] $$
where the sum is taken over the [[irreducible components]] $Z_\alpha$ of $f^{-1}(Z)$, $\length$ denotes length of [[modules]], and $z_\alpha$ are the [[generic points]] of $Z_\alpha$. Hence one gets homomorphisms $f^* : Z_k(Y) \to Z_{k+n}(X)$.
## Weil divisors and rational functions
A **[[Weil divisor]]** on $X$ is a 1-codimensional cycle.
A [[rational function]] $r \in R(X)$ on an [[integral scheme]] $X$ corresponds via canonical isomorphisms $R(X) \to \Frac(O_{X,x})$, for every $x \in X$, to elements $a_x/b_x \in \Frac(O_{X,x})$, and one defines the **order of vanishing** of $r$ at $x$ as
$$ \ord_x(r) = \length_{O_{X,x}}(O_{X,x}/(a_x)) - \length_{O_{X,x}}(O_{X,x}/(b_x)) $$
where $\length$ denotes length of [[modules]].
Then one defines the **[[Weil divisor]] associated to the rational function $r$** as
$$ div(r) = \sum_{Z \subset X} \ord_z(r).[Z] $$
where the sum goes over closed integral subschemes $Z$ of codimension 1 and with [[generic point]] $z \in Z$.
## Related concepts
* [[Chow groups]]
* [[Cartier divisor]]
* [[Abel-Jacobi map]]
* [[standard conjectures]]
## References
Standard references are
* [[EGA IV]], section 21
* [[Stacks Project]], [02QQ](http://stacks.math.columbia.edu/tag/02QQ)
On the relation with [[Weil cohomology theories]], [[algebraic K-theory]], [[Beilinson-Lichtenbaum conjectures]], and [[motivic cohomology]]:
* [[Marc Levine]], _Algebraic cycle complexes_, talk notes, 2008, [pdf](https://www.uni-due.de/~bm0032/publ/CycleComplexes.pdf).
A relation to [[iterated integrals]] and [[diffeological spaces]] is discussed in
* [[Richard Hain]], _Iterated Integrals and Algebraic Cycles: Examples and Prospects_ ([arXiv:math/0109204](http://arxiv.org/abs/math/0109204))
[[!redirects algebraic cycles]] |
algebraic definition of higher categories | https://ncatlab.org/nlab/source/algebraic+definition+of+higher+categories | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
* [[geometric definition of higher categories]]
* **algebraic definition of higher categories**
***
# Contents
* automatic table of contents goes here
{:toc}
## Idea
In an _algebraic definition_ of [[(n,r)-categories]] [[composition]] of [[higher morphism]]s is a genuine algebraic operation. As opposed to a _geometric model_ where composites are only guaranteed to exist, in an algebraic model there is prescription for finding these composites.
These choices may be given in an elementary way as binary composition operations as in an ordinary [[category]], or the composition operation may be controled by sophisticated algebraic structures such as [[operad]]s that, while keeping track of all specified composites, may allow large combinations of possible composition operations.
The composition operation in an algebraic model for an $(n,r)$-category are subject to [[associativity]] [[coherence law]]s. On the geometric side these reflect the fact that the spaces of possible choices of composites are contractible.
Typically an algebraic model for higher categories admits a [[nerve]] operation that turns it into an equivalent geometric model. Conversely, typically one can obtain an algebraic model from a geometric model by making _choices_ of composites.
## Examples
* The series of notions
* [[category]], [[groupoid]]
* [[bicategory]]
* [[tricategory]]
* [[tetracategory]]
are algebraic models for [[n-categories]] with $1 \leq n \leq 4$ given in terms of direct explicit operations: no [[operad]]s or other tools are used but all the possible composition operations are defined elementarily and all the relevant [[coherence law]]s are demanded explicitly. This makes these models very concrete and hands-on. But it also has the disadvantage that beginning with tricategories these definitions become quite unwieldy.
* For [[strict n-categories]] all subtleties with [[associator]]s and [[coherence law]]s are absent (by definition) and therefore there are straightforward algebraic models for these See [[strict omega-category]], [[strict omega-groupoid]] and [[n-fold category]]. The drawback is of course that these strict models capture only a very restricted part of higher category theory.
* The [[Michael Batanin|Batanin]]/[[Tom Leinster|Leinster]] approach to higher categories involves algebraic structure imposed all at once (using higher [[operad|operads]]) on a globular set. See [[Batanin omega-category]].
* The [[Todd Trimble|Trimble]]/[[Peter May|May]] approach to higher categories involves algebraic structure imposed in stages by a process of iterative enrichment. See also [[Trimble n-category]].
* These models turn out to be closely related to an original idea by [[Alexandre Grothendieck]] that was resurrected and formalized by [[Georges Maltsiniotis]]: [[Grothendieck-Maltsiniotis ∞-categories]]
* Using the tool of [[model structure on algebraic fibrant objects]] many geometric models for higher categories may be realized equivalently as algebraic models. This is notably true for [[∞-groupoid]]s and [[(∞,1)-categories]]. See [Algebraic fibrant models for higher categories](http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects#AlgebaicHigherCategories).
## Properties
When an [[(∞,1)-category]] of [[(n,r)-categories]] is presented by a [[model category]], then algebraic models tend to be fibrant objects (while geometric models tend to be cofibrant objects). For instance in all the [[folk model structure]] on algebraic higher categories and groupoids, all objects are fibrant.
In a strict version of the [[homotopy hypothesis]], one may make this 'algebraicity' of the model structure (not to be confused with the notion at [[algebraic model structure]]) a requirement.
[[!redirects algebraic definition of higher categories]]
[[!redirects algebraic definitions of higher categories]]
[[!redirects algebraic definition of higher category]]
[[!redirects algebraic model of higher categories]]
[[!redirects algebraic model of higher category]]
|
algebraic dynamics | https://ncatlab.org/nlab/source/algebraic+dynamics | While standard dynamics studies processes in real time, sometimes it also considers discrete dynamical systems. The step of discrete dynamics is in the appropriate category.
In __algebraic dynamics__ one typically studies discrete dynamical systems on algebraic varieties. Such a system is given by a regular endomorphism $D: X\to X$ of a variety $X$.
* L. Szpiro, _Algebraic dynamics_, [pdf](http://wfs.gc.cuny.edu/lszpiro/www/papers/AlgebraicDynamics2.pdf)
The case over [[number field]]s is also called arithmetic dynamics,
see
* wikipedia [arithmetic dynamics](http://en.wikipedia.org/wiki/Arithmetic_dynamics) |
algebraic effect | https://ncatlab.org/nlab/source/algebraic+effect | [[!redirects algebraic side effect]]
[[!redirects algebraic side effects]]
[[!redirects algebraic effects]]
#Contents#
* table of contents
{:toc}
## Idea
**Algebraic effects** are computational [[side effect|effects]] that can be represented by an equational theory, or [[algebraic theory]], whose operations produce the effects at hand.
They provide an alternative to using [[monads (in computer science)]] for modelling _effects_ with a [[functional programming language]]. This approach has been realized in the programming language [[Eff]].
[[continuation monad|Continuations]] are not algebraic effects.
## References
* [[Andrej Bauer]], _[Programming with algebraic effects and handlers](http://math.andrej.com/2012/03/08/programming-with-algebraic-effects-and-handlers/)_
|
algebraic element | https://ncatlab.org/nlab/source/algebraic+element |
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## Definition
Given a [[field]] $K$ and a field extension $K \subseteq F$, an element $\alpha \in F$ is **algebraic** if the subfield $K(\alpha) \subseteq F$ is a finite-degree [[field extension]] of $K$.
## Related concepts
* [[algebraic number]]
* [[algebraic extension]]
* [[transcendental element]]
## References
See also
* Wikipedia, _[Algebraic element](https://en.wikipedia.org/wiki/Algebraic_element)_
[[!redirects algebraic element]]
[[!redirects algebraic elements]] |
algebraic equation | https://ncatlab.org/nlab/source/algebraic+equation |
#Contents#
* table of contents
{:toc}
## Idea
An [[algebraic equation]] is an [[equation]] of the form
$$
P(x_1, \cdots, x_n) = 0
$$
where $P(x_1, \cdots, x_n)$ is a [[polynomial]] with [[coefficients]] in some [[field]].
## Related concepts
* [[linear equation]]
* [[Diophantine equation]]
## References
* Wikipedia, _[Algebraic equation](http://en.wikipedia.org/wiki/Algebraic_equation)_
[[!redirects algebraic equations]]
|
algebraic extension | https://ncatlab.org/nlab/source/algebraic+extension |
#Contents#
* table of contents
{:toc}
## Idea
A kind of [[field extension]]
## Related concepts
* [[perfect field]]
* [[transcendental extension]]
## References
* Wikipedia, _[Algebraic extension](http://en.wikipedia.org/wiki/Algebraic_extension)_
[[!redirects algebraic extensions]]
|
algebraic function | https://ncatlab.org/nlab/source/algebraic+function |
## References
* Wikipedia, _[Algebraic function](http://en.wikipedia.org/wiki/Algebraic_function)_
[[!redirects algebraic functions]] |
algebraic fundamental group | https://ncatlab.org/nlab/source/algebraic+fundamental+group |
#Contents#
* table of contents
{:toc}
## Idea
The _algebraic fundamental group_ is the [[fundamental group]] of a [[scheme]], as defined by [[Grothendieck]] in [[SGA1]]. It is essentially the fundamental group as seen by [[étale homotopy]], the _[[étale fundamental group]]_.
For [[fields]] this is essentially the [[Galois group]]. For [[smooth varieties]] it is the [[Galois group]] of the maximal [[unramified extension]] of the [[function field of a variety|function field]] of the variety (e.g. [MK 09, p. 3](#Kim09)).
In [[arithmetic geometry]] one also speaks of the _arithmetic fundamental group_.
##Definition
Let $S$ be a connected [[scheme]]. Recall that a finite [[étale cover]] of $S$ is a finite flat
surjection $X\to S$ such that each fibre at a point $s \in S$ is the spectrum of a finite [[étale algebra]] over the local ring at $s$. Fix a [[geometric point]] $\overline{s} : Spec(\Omega) \to \Omega$.
For a finite étale cover, $X\to S$, we consider the [[geometric fibre]], $X\times_S Spec (\Omega)$, over $\overline{s}$, and denote by $Fib_\overline{s} (X)$ its
underlying set. This gives a set-valued functor on the category of finite étale covers
of $X$.
The *algebraic fundamental group*, $\pi_1(S, \overline{s})$ is defined to be the automorphism group
of this functor.
For more on this area, see at _[[étale homotopy]]_.
(This entry is a stub and needs more work, including the linked entries that do not yet exist! Also explanation of $\Omega$. It is adapted from the first reference below.)
##References
* [[Tamás Szamuely ]], [Heidelberg Lectures on Fundamental Groups](http://pagine.dm.unipi.it/tamas/pia.pdf), 2010.
or in a lengthier form:
* [[Tamás Szamuely ]], Galois Groups and Fundamental Groups, Cambridge Studies in Advanced Mathematics, vol. 117, Cambridge University Press, 2009.
An earlier version is to be found [here](http://math.uchicago.edu/~aanders/books/szamuely_-_galois_groups_and_fundamental_groups.pdf).
A paper on a closely related subject is
* Feng-Wen An, _On the arithmetic fundamental groups_ ([arXiv:0910.0605](http://arxiv.org/abs/0910.0605))
See also
* {#Kim08} [[Minhyong Kim]], _Fundamental groups and Diophantine geometry_, 2008 ([pdf](http://www.ucl.ac.uk/~ucahmki/leeds.pdf))
* {#Kim09} [[Minhyong Kim]], _Galois theory and Diophantine geometry_, 2009 ([pdf](http://www.ucl.ac.uk/~ucahmki/cambridgews.pdf))
[[!redirects algebraic fundamental groups]]
[[!redirects arithmetic fundamental group]]
[[!redirects arithmetic fundamental groups]] |
algebraic geometry | https://ncatlab.org/nlab/source/algebraic+geometry |
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=--
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#Contents#
* table of contents
{:toc}
## Overview
__Algebraic geometry__ is, in origin, a [[geometry|geometric study]] of solutions of systems of [[polynomial]] equations and generalizations.
The set of zeros of a set of polynomial equations in finitely many variables over a [[field]] is called an __affine [[algebraic variety|variety]]__ and it is equipped with a particular topology called [[Zariski topology]], whose closed sets are subvarieties. The system of polynomial equations defines an [[ideal]] in the ring of polynomials over the [[ground field]]; one of the first insights of algebraic geometry is that the ideal is a more invariant notion than the original set of equations. The [[quotient object|quotient]] of the ring of polynomials by the defining ideal is the ring of coordinate functions of the affine variety; a basic theorem asserts that this ring is [[noetherian ring|Noetherian]] algebra over the ground field. This ring determines the variety up to a natural notion of [[isomorphism]] of varieties. __Projective varieties__ are zeroes of systems of homogeneous polynomial equations in projective $n$-dimensional space; if the coordinates are rescaled by a common nonzero multiple, then by definition they still define the same point in projective space. Zariski open subsets of affine (or projective) varieties are called __quasiprojective varieties__.
It is often useful to take a [[pullback]] along a morphism of fields, usually called [[base change]]. Since [[Alexander Grothendieck|Grothendieck]], one generalizes the coordinate rings of affine varieties to arbitrary commutative unital rings, not necessarily Noetherian nor finitely generated; and interprets the [[opposite category]] of the category of commutative rings as a category of affine [[scheme]]s $\mathrm{Aff}$; affine schemes are traditionally constructed by the affine [[spectrum]] functor $\mathrm{Spec}:\mathrm{CommRing}^{\mathrm{op}}\to\mathrm{lrSp}$ into the category of locally [[ringed space|ringed]] topological spaces. Points of the affine spectrum are the prime ideals of the ring. Assuming the [[axiom of choice]], every affine scheme corresponding to a ring with $0 \neq 1$ has therefore at least one point. The slice category $\mathrm{Aff}/(\mathrm{Spec} F)$ over a spectrum of a fixed field $F$ contains the category of varieties over $F$ as a full subcategory. However the points of an affine variety and of the corresponding affine spectrum do not coincide: only *maximal* ideals are points of usual varieties. Affine schemes have a natural topology, also called a Zariski topology; the ringed spaces locally isomorphic to affine schemes are called __schemes__. Schemes include projective schemes and more generally quasiprojective schemes; if they are relative over a fixed [[ground field]], then they contain the subcategories of projective (resp. quasiprojective) varieties over the same field.
Although [[Grothendieck]] in the late 1950s envisioned many generalizations of [[scheme]] theory, his coauthor Dieudonné wrote later in EGA that algebraic geometry is the study of algebraic and [[formal schemes]], which is clearly a too dogmatic definition. Grothendieck's school studied in addition locally affine spaces in various [[Grothendieck topology|Grothendieck topologies]] on $\mathrm{Aff}$ (including [[algebraic space]]s), [[algebraic stack]]s ([[Deligne-Mumford stack]]s and [[Artin stack]]s), [[ind-object|ind-schemes]] and so on; in SGA the study of ringed spaces is replaced by more general [[ringed site]]s and [[ringed topos|ringed topoi]]. Modern generalizations include derived schemes, almost schemes (with the theory of almost rings of Gabber developing after some ideas of [Faltings](http://de.wikipedia.org/wiki/Gerd_Faltings)), generalized schemes of [[Nikolai Durov]], so-called schemes over the '[[field of one element]]' $F_1$ of various authors, dg-schemes, slightly noncommutative [[D-scheme]]s etc. Many ideas of scheme theory and the spectral theory of rings have influenced parallel developments in the analytic setup (Stein manifolds ([see the English Wikipedia](http://en.wikipedia.org/wiki/Stein_manifold)), rigid [[analytic geometry|analytic spaces]], etc.) and in the noncommutative setup give rise to [[noncommutative algebraic geometry]]. Deligne has also suggested how to do algebraic geometry in an arbitrary symmetric monoidal category.
## Perspective of structured $(\infty,1)$-toposes
Grothendieck took the viewpoint that the schemes, algebraic spaces etc. are sheaves on $Aff$ in some subcanonical Grothendieck topology (functor of points point of view).
Algebraic geometry starts with study of [[space]]s that are locally modeled on ([[object]]s in the [[category]]) Aff = [[CRing]]${}^{op}$ -- main categories being of algebraic [[scheme]]s and of [[algebraic space]]s; one also allows infinitesimal thickenings leading to formal schemes and other ind-objects in schemes. Hakim, [[Deligne]] and [[Ofer Gabber|Gabber]] extend the setup internally to a symmetric monoidal category, where $Aff$ is replaced by the opposite to the category of monoids in that category; [[Nikolai Durov|Durov]] on the other side takes monoids in the category of endofunctors in [[Set]], i.e. [[monad]]s as the opposite to the local objects in a generalized scheme theory.
This is to be compared to and contrasted with for instance [[differential geometry]], that studies [[space]]s locally modeled on ([[object]]s in the [[category]]) [[CartSp]] -- called (smooth) [[manifold]]s.
The general formalization of the notion of [[space]] and hence, by the general lore of [[space and quantity]], that of [[sheaf]], [[stack]], [[∞-stack]] has originally been crucially inspired by [[Grothendieck]]'s work on algebraic geometry and is more recently being greatly revived and further extended by the developments in [[derived algebraic geometry]] by people like [[Bertrand Toen]] and [[Jacob Lurie]].
In particular in [[Structured Spaces]] Lurie presents a general formalism of [[generalized scheme]]s that encompasses the [[space]]s studied in algebraic geometry and [[derived algebraic geometry]] just as well as ordinary [[smooth manifold]]s, [[derived smooth manifold]] and harmonizes with other axiomatizations such as in [[synthetic differential geometry]]: all of these [[space]] object are realized as special cases of [[structured (∞,1)-topos]]es, differing only in the choice of [[geometry (for structured (∞,1)-toposes)]] that they are modeled on.
From this perspective,
* ordinary algebraic geometry is the study of [[structured (∞,1)-topos]]es for the [[geometry (for structured (∞,1)-toposes)|Zariski or etale geometry]] $\mathcal{G}_{Zar}$, $\mathcal{G}_{et}$ on [[CRing]]${}^{op}$. In fact one has as series of geometries for every integer $n\geq 0$, where classical case is at level $0$ and derived at level $\infty$. Cf. 4.2.9 in [[Structured Spaces]]. The fully faithful embedding of schemes into derived schemes does not commute with limits, what is relevant e.g. for intersection theory.
* [[derived algebraic geometry]] is the study of [[structured (∞,1)-topos]]es for the [[geometry (for structured (∞,1)-toposes)|Zariski or etale pre-geometry]] $\mathcal{T}_{Zar}$, $\mathcal{T}_{et}$ on [[CRing]]${}^{op}$.
Despite of this, an axiomatic formulation of algebraic geometry along the lines of [[synthetic differential geometry]], that would de-emphasize the peculiarities of $CRing^{op}$ and emphasize structural aspects such as to facilitate for instance the transportation or interpretation of results in algebraic geometry to other geometries, is currently hardly to be found in the elementary literature. [[SGA]], specially SGA IV was written however to reflect "algebraic" geometry over any topos.
>Maybe we could talk more about [[synthetic differential geometry applied to algebraic geometry]] to unify perspective of algebraic with differential geometry.
## Related pages
* [[synthetic algebraic geometry]]
* [[contributors to algebraic geometry]]
* [[books in algebraic geometry]],
* [pages in the category "algebraic geometry"](http://www.ncatlab.org/nlab/list/algebraic+geometry)
* [[functorial geometry]]
* [[derived algebraic geometry]]
* [[noncommutative geometry]]
## References
{#References}
Original references: [[EGA]] and [[SGA]].
Textbook accounts:
* {#ShafarevichVol1} [[Igor Shafarevich]], *Basic Algebraic Geometry 1 -- Varieties in Projective Space*, Springer (1977, 1994, 2013) [[pdf](http://userpage.fu-berlin.de/aconstant/Alg2/Bib/Shafarevich.pdf), [doi:10.1007/978-3-642-57908-0](https://link.springer.com/book/10.1007/978-3-642-57908-0)]
* [[Siegfried Bosch]], *Algebraic Geometry and Commutative Algebra*, Universitext, Springer (2017) [[doi:10.1007/978-1-4471-4829-6](https://doi.org/10.1007/978-1-4471-4829-6)]
* Ulrich Görtz, Torsten Wedhorn, *Algebraic Geometry I: Schemes*, Springer (2020) [[doi:10.1007/978-3-658-30733-2](https://doi.org/10.1007/978-3-658-30733-2)]
* * [[Robin Hartshorne]], Algebraic Geometry, Graduate Texts in Mathematics volume 52, Springer 1977. [ [doi link](https://doi.org/10.1007/978-1-4757-3849-0) ]
Lecture notes:
* [[Ravi Vakil]], _Foundations Of Algebraic Geometry_, Course notes ([web](http://math.stanford.edu/~vakil/216blog/))
See also the references at _[[functorial geometry]]_.
and see
* _[[The Stacks Project]]_
On [[synthetic algebraic geometry]], i.e. via the [[internal logic]] of the [[sheaf topos]] over a [[scheme]] ([[Zariski topos]]/[[étale topos]]):
* {#Blechschmidt15} [[Ingo Blechschmidt]], _Using the internal language of toposes in algebraic geometry_, talk at [Toposes at IHES](https://indico.math.cnrs.fr/event/747/), November 2015 ([pdf](https://github.com/iblech/internal-methods/blob/master/slides-ihes2015.pdf), [recording](https://www.youtube.com/watch?v=7S8--bIKaWQ))
* [[Ingo Blechschmidt]], *Using the internal language of toposes in algebraic geometry*, PhD thesis (2017) [[pdf](https://rawgit.com/iblech/internal-methods/master/notes.pdf), [[Blechschmidt-InternalLanguage.pdf:file]]]
* [[Felix Cherubini]], [[Thierry Coquand]], [[Matthias Hutzler]], *A Foundation for Synthetic Algebraic Geometry* (2023) [[arXiv:2307.00073](https://arxiv.org/abs/2307.00073)]
* [[Felix Cherubini]], *A Foundation for Synthetic Algebraic Geometry*, talk at *[Homotopy Type Theory Electronic Seminar Talks](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html)* (Oct 2023) [slides:[pdf](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Cherubini-2023-10-18-HoTTEST.pdf), video:[YT](https://www.youtube.com/watch?v=lp4kcmQ0ueY)]
For more see:
* [[Ingo Blechschmidt]], _Internal methods_ [[github](https://github.com/iblech/internal-methods)]
category: algebraic geometry
[[!redirects algebraic geometer]] |
algebraic group | https://ncatlab.org/nlab/source/algebraic+group |
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#Contents#
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## Definition
Given a (typically [[algebraically closed field|algebraically closed]]) field $k$, an __algebraic $k$-group__ is a [[group object]] in the category of $k$-[[algebraic variety|varieties]].
## Linear algebraic groups and abelian varieties
There are two important classes of algebraic groups whose intersection is trivial (the identity group): Linear algebraic groups and abelian varieties.
Any algebraic group contains a unique normal linear algebraic subgroup $H$ such that their quotient $G/H$ is an abelian variety.
### Linear algebraic group
An algebraic $k$-group is [[linear algebraic group|linear]] if it is a [[Zariski topology|Zariski]]-closed subgroup of the [[general linear group]] $GL(n,k)$ for some $n$.
An algebraic group is linear iff it is affine.
An algebraic group scheme is *affine* if the underlying scheme is [[affine scheme|affine]].
The category of affine group schemes is the [[opposite category|opposite]] of the category of commutative [[Hopf algebras]].
### Abelian variety
Another important class are connected algebraic $k$-groups whose underlying variety is [[projective variety|projective]]; these are automatically commutative so they are called *abelian varieties*. In dimension $1$ these are precisely the [[elliptic curve]]s. If $k$ is a [[perfect field]] and $G$ an algebraic $k$-group, the theorem of Chevalley says that there is a unique linear subgroup $H\subset G$ such that $G/H$ is an abelian variety.
#### Elliptic curve
An abelian variety of dimension $1$ is called an *[[elliptic curve]]*.
## Other prominent classes of algebraic groups
Some of the definitions of the following classes exist more generally for [[group schemes]].
### Jacobian
(...)
### Unipotent algebraic groups
(See also more generally [[unipotent group scheme]].)
+-- {: .num_defn}
###### Definition
An element $x$ of an affine algebraic group is called *unipotent* if its associated right translation operator $r_x$ on the affine [[coordinate ring]] $A[G]$ of $G$ is locally unipotent as an element of the ring of linear endomorphism of $A[G]$ where ''locally unipotent'' means that its restriction to any finite dimensional stable subspace of $A[G]$ is unipotent as a ring object.
=--
+-- {: .num_theorem}
###### Theorem
([[Jordan-Chevalley decomposition]])
Any commutative linear algebraic group over a perfect field is the product of a unipotent and a [[semisimple object|semisimple algebraic group]].
=--
## Properties
The group objects in the category of [[algebraic schemes]] and [[formal scheme]]s are called (algebraic) [[group schemes]] and [[formal groups]], respectively.
Among group schemes are 'the infinite-dimensional algebraic groups' of Shafarevich.
Algebraic analogues of [[loop group]]s are in the category of [[ind-scheme]]s. All linear algebraic $k$-groups are affine.
## Examples
The [[affine line]] $\mathbb{A}^1$ comes canonically with the structure of a group under addition: the [[additive group]] $\mathbb{G}_a$.
The affine line without its origin, $\mathbb{A}^1 - \{0\}$ comes canonically with the structure of a group under multiplication: the [[multiplicative group]] $\mathbb{G}_m$.
## Generalizations
* [[group scheme]]
## Related concepts
* [[isogeny]]
* [[form of an algebraic group]]
## References
* [[Gerhard P. Hochschild]], *Basic Theory of Algebraic Groups and Lie Algebras*, Graduate Texts in Mathematics **75**, Springer (1981) [[doi:10.1007/978-1-4613-8114-3_16](https://doi.org/10.1007/978-1-4613-8114-3_16)]
* M. Demazure, P. Gabriel, _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970; English edition is _Introduction to algebraic geometry and algebraic groups_, North-Holland, Amsterdam 1980 (North-Holland)
* M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, _Schemas en groupes_, [[SGA3]]
* {#Milne17} [[James Milne]], *Algebraic Groups -- The theory of group schemes of finite type over a field*, Cambridge University Press 2017 ([doi:10.1017/9781316711736](https://doi.org/10.1017/9781316711736), [webpage](http://www.jmilne.org/math/Books/iag.html), [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf))
* A. Borel, _Linear algebraic groups_, Springer (2nd edition much expanded)
* W. Waterhouse, _Introduction to affine group schemes_, GTM 66, Springer 1979.
* S. Lang, _Abelian varieties_, Springer 1983.
* D. Mumford, _Abelian varieties_, 1970, 1985.
* J. C. Jantzen, _Representations of algebraic groups_, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)
* T. Springer, _Linear algebraic groups_, Progress in Mathematics 9, Birkhäuser Boston (2nd ed. 1998, reprinted 2008)
* Roe Goodman, Nolan R. Wallach, Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255. Springer, Dordrecht, 2009.
* J. Milne, _Algebraic groups, Lie Groups, and their arithmetic subgroups_, [pdf](http://www.jmilne.org/math/CourseNotes/ALA.pdf)
* [[!redirects algebraic groups]]
|
Algebraic Homotopy | https://ncatlab.org/nlab/source/Algebraic+Homotopy |
The book
* [[Hans Baues]],
_Algebraic Homotopy_,
Cambridge studies in advanced mathematics 15,
Cambridge University Press, (1989)
[doi:10.1017/CBO9780511662522](https://doi.org/10.1017/CBO9780511662522)
gives an account of the author's interpretation of [[Henry Whitehead]]'s [[Algebraic Homotopy Theory]] as described in his ICM talk (1950) and his famous papers, _[[Combinatorial homotopy I]]_, (1949), and _[[Combinatorial homotopy II]]_, again (1949). Although the material contained in the first of these papers became central to the development of homotopy theory (CW complexes etc.) soon after its publication, the second paper, treating harder ideas including those of [[crossed complexes]], was relatively 'unstudied' until much more recently. This book gives one interpretation of the ideas it developed from a modern point of view. That development continued in [[Combinatorial Homotopy and 4-Dimensional Complexes]].
#Contents#
* table of contents
{:toc}
## Preface
### Introduction
###I Axioms for homotopy theory and examples of cofibration categories
###II Homotopy theory in a cofibration category
###III The homotopy spectral sequences in a cofibration category
###IV Extensions, coverings and cohomology groups of a category
###V Maps between mapping cones
###VI Homotopy theory of CW-complexes
###VII Homotopy theory of complexes in a cofibration category
### VIII Homotopy theory of Postnikov towers and the Sullivan-de Rham equivalence of rational homotopy categories
###IX Homotopy theory of reduced complexes
###Bibliography
###Index
## References
See also:
* [[Tim Porter]], _Review of "Algebraic Homotopy'' by H.J.Baues_, in Bull. London Math. Soc. 22 (1990) 196-197.
category: reference |
algebraic homotopy | https://ncatlab.org/nlab/source/algebraic+homotopy |
#Contents#
* automatic table of contents goes here
{:toc}
## Whitehead's algebraic homotopy programme
In his talk at the 1950 ICM in Harvard, [[Henry Whitehead]] introduced the idea of *algebraic homotopy theory* and said
_"The ultimate aim of algebraic homotopy is to construct a purely
algebraic theory, which is equivalent to homotopy theory in the same sort of
way that 'analytic' is equivalent to 'pure' projective geometry."_
A statement of the aims of 'algebraic homotopy' might thus include the
following homotopy classification problem (from the same source,
J.H.C.Whitehead, (ICM, 1950)):
_Classify the homotopy types of polyhedra, $X$, $Y$, $\ldots$ , by algebraic data._
_Compute the set of homotopy classes of maps, $[X,Y]$, in terms of the
classifying data for $X$, $Y$._
These aims are still valid, but, within the context of these webpages, with the enlargement of the class of objects of study to
include many other types of spaces, and ultimately $\infty$-[[infinity-groupoid|groupoids]].
One may summarise them, optimistically,
by saying that one searches for a nice "algebraic" category $\mathbf{A}$ together
with a functor or functors
$$\mathbf{F} : \mathbf{Spaces }\rightarrow \mathbf{A}$$
and an algebraically defined notion of 'homotopy' in $\mathbf{A}$ such that
a) if $X\simeq Y$ in $\mathbf{Spaces}$, then $F(X) \simeq F(Y)$ in $\mathbf
A$;
b) if $f \simeq g$ in $\mathbf{Spaces}$, then $F(f)\simeq F(g)$ in $\mathbf
A$,
and $F$ induces an equivalence of [[homotopy category|homotopy categories]]
$$Ho(\mathbf{Spaces}) \simeq Ho(\mathbf{A}).$$
(Here $\mathbf{Spaces}$ is a category, perhaps of topological spaces such as
polyhedra or CW-complexes, but it may be larger than this and may contain the
sort of 'generalised space', [[topos]], etc., used in other contexts such as algebraic
geometry, and, of course, $\infty$-[[infinity-groupoid|groupoids]].)
## More recent developments
* [[Baues]] has developed an approach to Whitehead's basic programme using a mix of [[cofibration category|cofibration categories]] and categories with a particular type of [[cylinder functor]], that he calls [[I-category|I-categories]]. These are treated in separate entries. Cofibration categories are very similar to the dual of K.S. Brown's abstract homotopy theory, as discussed in [[category of fibrant objects]] and [[BrownAHT]].
* [[Ronnie Brown]]'s [[nonabelian algebraic topology]] has developed Whitehead's theory of crossed complexes along the lines suggested by the original papers of Whitehead, but extending that, in particular, using generalisations of van Kampen's theorem. (This is discussed in detail in the entry: [[nonabelian algebraic topology]].)
##Is Algebraic Homotopy 'the same as' Homotopical Algebra?
Often the objects of study are the same, and there is an enormous interaction between the two areas, but the aims and objectives seem to be different. Perhaps, tentatively, one could say that 'algebraic homotopy' is 'combinatorial homotopical algebra'.
##References
* [[Hans-Joachim Baues]], _[[Algebraic Homotopy]]_, Cambridge studies in advanced mathematics 15, Cambridge University Press, (1989);
* [[Hans-Joachim Baues]], _[[Combinatorial Homotopy and 4-Dimensional Complexes]]_, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991);
* [[Hans-Joachim Baues]], _[[Homotopy Types]]_, in I.M.James, ed., _Handbook of Algebraic Topology_, 1--72, Elsevier, (1995).
[[!redirects Algebraic Homotopy Theory]]
|
algebraic integer | https://ncatlab.org/nlab/source/algebraic+integer | +-- {: .rightHandSide}
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* table of contents
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## Idea
In [[number theory]], the concept of _algebraic integer_ is a generalization of that of [[integer]] to more general base-[[number fields]]. These algebraic integers form what is called the _[[ring of integers]]_ and so in order to distinguish that from the standard integers $\mathbb{Z}$ these are sometimes called _rational integers_, since they are the algebraic integers in the ring of [[rational numbers]].
## Definition
Colloquially, an **algebraic integer** is a solution to an equation
$$x^n + a_1 x^{n-1} + \ldots + a_n = 0 \qquad (1)$$
where each $a_i$ is an [[integer]] (hence a [[root]] of the [[polynomial]] on the left). More precisely, an element $x$ belonging to an [[algebraic extension]] of the [[rational numbers]] $\mathbb{Q}$ is an (algebraic) integer, or more briefly is _integral_, if it satisfies an equation of the form (1). Equivalently, if $k$ is an algebraic extension of $\mathbb{Q}$ (e.g., if $k$ is a number field), an element $\alpha \in k$ is integral if the subring $\mathbb{Z}[\alpha] \subseteq k$ is finitely generated as a $\mathbb{Z}$-module.
This notion may be relativized as follows: given an [[integral domain]] in its [[field of fractions]] $A \subseteq E$ and a finite [[field extension]] $E \subseteq F$, an element $\alpha \in F$ is **integral** over $A$ if $A[\alpha] \subseteq F$ is finitely generated as an $A$-module.
If $\alpha, \beta$ are integral over $\mathbb{Z}$ (say), then $\alpha + \beta$ and $\alpha \cdot \beta$ are integral over $\mathbb{Z}$. For, if $\beta$ is integral over $\mathbb{Z}$, it is _a fortiori_ integral over $\mathbb{Z}[\alpha]$, hence
$$(\mathbb{Z}[\alpha])[\beta] = \mathbb{Z}[\alpha, \beta]$$
is finitely generated over $\mathbb{Z}[\alpha]$ and therefore, since $\alpha$ is integral, also finitely generated over $\mathbb{Z}$. It follows that the submodules $\mathbb{Z}[\alpha + \beta]$ and $\mathbb{Z}[\alpha \cdot \beta]$ are therefore also finitely generated over $\mathbb{Z}$ (since $\mathbb{Z}$ is a [[Noetherian ring]]). Thus the integral elements form a [[ring]]. In particular, the integral elements in a [[number field]] $k$ form a ring often denoted by $\mathcal{O}_k$, usually called the **[[ring of integers]]** in $k$. This ring is a [[Dedekind domain]].
## Examples
* The algebraic integers in the [[rational numbers]] are the ordinary [[integers]].
* The algebraic integers in the [[Gaussian numbers]] are the [[Gaussian integers]].
* [[golden ratio]]
## Related concepts
* [[ring of integers]]
## References
Textbook account:
* {#Cassels86} [[J. W. S. Cassels]], Section 10.3 of: *Local Fields*, Cambridge University Press, 1986 (ISBN:9781139171885, [doi:10.1017/CBO9781139171885](https://doi.org/10.1017/CBO9781139171885))
Lecture notes:
* [[James Milne]], Chapter 2 of: *Algebraic number theory*, 2020 ([pdf](https://www.jmilne.org/math/CourseNotes/ANT.pdf))
[[!redirects algebraic integer]]
[[!redirects algebraic integers]]
[[!redirects rational integer]]
[[!redirects rational integers]]
|
algebraic K-theory | https://ncatlab.org/nlab/source/algebraic+K-theory |
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#### Cohomology
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#Contents#
* table of contents
{:toc}
## Idea
Algebraic K-theory is about natural constructions of [[cohomology theories]]/[[spectra]] from [[algebra|algebraic]] data such as [[commutative rings]], [[symmetric monoidal categories]] and various [[homotopy theory|homotopy theoretic]] refinements of these.
From a modern perspective, the algebraic K-theory [[spectrum]] $\mathbf{K}(R)$ of a [[commutative ring]] is simply the [[K-theory of a symmetric monoidal (∞,1)-category|∞-group completion]] of [[algebraic vector bundles]] on $Spec(R)$; this will be discussed in more detail [below](#AsTheKTheoryOfAlgebraicVectorBundles). In particular there is a natural concept of algebraic K-theory of "[[brave new rings]]", i.e. of [[ring spectra]]/[[E-∞ rings]].
Historically, the _algebraic K-theory_ of a [[commutative ring]] $R$ (what today is the "0th" algebraic K-theory group) was originally defined to be the [[Grothendieck group]] of its [[symmetric monoidal category]] of [[projective modules]] (under [[tensor product]] of modules). Under the [relation between modules and vector bundles](module#RelationToVectorBundlesInIntroduction), this is directly analogous to the basic definition of [[topological K-theory]], whence the common term. (In fact when applied to the _[[stack]]_ of [[vector bundles]] then algebraic K-theory subsumes [[topological K-theory]] and also [[differential K-theory]], see [below](#OnMonoidalStacks)).
There are canonical maps $K_0(R)\to Pic(R)$ from the 0th algebraic K-theory of a ring to its [[Picard group]] and $K_1(R)\to GL_1(R)$ from the first algebraic K-theory group of $R$ to its [[group of units]] which are given in components by the [[determinant]] functor. This fact is sometimes used to motivate algebraic K-theory as a "generalization of [[linear algebra]]" (see e.g. [this MO discussion](http://mathoverflow.net/a/171369/381)). This is also how the traditional [[regulator of a number field]] relates to [[Beilinson regulators]] of algebraic K-theory.
More generally, following the axiomatics of _[[Whitehead-generalized cohomology]]_ any _algebraic K-theory_ should be given by a sequence of [[functors]] $K_i$ from some suitable class of [[categories]] of "algebraic nature" to [[abelian groups]], satisfying some natural conditions. Moreover, following the [[Brown representability theorem]] these groups should arise as the [[homotopy groups]] of a [[spectrum]], the algebraic _[[K-theory spectrum]]_. Classical constructions producing this by combinatorial means are known as the _[[Quillen Q-construction]]_ defined on _[[Quillen exact categories]]_ and more generally the _[[Waldhausen S-construction]]_ defined on _[[Waldhausen categories]]_.
For more on the history of the subject see ([Arlettaz 04](#Arlettaz04), [Grayson 13](#Grayson13)) and see at at _[[Algebraic K-theory, a historical perspective]]_.
There are two ways to think of the traditional algebraic K-theory of a commutative ring more conceptually: on the one hand this construction is the [[group completion]] of the [[direct sum]] [[symmetric monoidal category|symmetric monoidal]]-structure on the [[category of modules]], on the other hand it is the group completion of the addition operation expressed by [[short exact sequences]] in that category. This leads to the two modern ways of expressing and viewing algebraic K-theory:
1. **monoidal.** The [[core]] of a [[symmetric monoidal category]] or more generally of a [[symmetric monoidal (∞,1)-category]] has a [[universal construction|universal]] completion to an [[abelian ∞-group]]/[[connective spectrum]] optained by universally adjoining [[inverses]] to the symmetric monoidal operation -- the [[K-theory of a symmetric monoidal (∞,1)-category|∞-group completion]]. This yields the concept of [[K-theory of a permutative category|algebraic K-theory of a symmetric monoidal category]] and more generally that of [[algebraic K-theory of a symmetric monoidal (∞,1)-category]];
1. **exact/stable.** Analogously, inverting the addition operation expressed by the [[exact sequences]] in an [[abelian category]] or more generally in a [[stable (∞,1)-category]] yields the [[algebraic K-theory of a stable (∞,1)-category]]. Explicit ways to express this are known as the _[[Quillen Q-construction]]_ and the _[[Waldhausen S-construction]]_. This turns out to be a universal construction in the context of [[non-commutative motives]].
Here the second construction may be understood as first [[split exact sequence|splitting]] the [[exact sequences]] and then applying the first construction to the resulting [[direct sum]] monoidal structure. Typically the first construction here contains more information but is harder to compute, and vice versa (see also MO-discussion [here](http://mathoverflow.net/a/98602/381) and [here](http://mathoverflow.net/a/102583/381)).
Both of these constructions produce a [[spectrum]] (hence [[Brown representability theorem|representing]] a [[Whitehead-generalized cohomology theory]]) -- called the _[[K-theory spectrum]]_ -- and the algebraic K-theory groups are the [[homotopy groups]] of that spectrum.
The classical case of the algebraic K-theory of a commutative ring $R$ is a special case of this general concept of algebraic K-theory by either forming the [[symmetric monoidal category]] $(Mod(R), \oplus)$ and applying the [[abelian ∞-group]]-completion to that, or else forming the [[stable (∞,1)-category of chain complexes]] of $R$-modules and applyong the [[Waldhausen S-construction]] to that. In both cases the result is a [[spectrum]] whose degree-0 [[homotopy group]] is the ordinary algebraic K-theory of $R$ as given by the [[Grothendieck group]] and whose higher homotopy groups are its _higher algebraic K-theory_ groups.
## Constructions
### Symmetric monoidal K-theory
For a [[symmetric monoidal category]] $C$, K-theory may be defined by taking
* the [[maximal subgroupoid]] $i C \subset C$,
* the [[classifying space]] $B(i C)$ (a [[topological monoid]]),
* the [[group completion]] $\Omega B B (i C)$.
See at
* [[K-theory of a symmetric monoidal (∞,1)-category]]
* [[K-theory of a permutative category]]
* [[K-theory of a bipermutative category]]
### Quillen Q-construction (for exact categories)
Given an [[Quillen exact category]] $E$, one defines $K(E)$ by applying
* the [[Quillen Q-construction]] $Q(E)$,
* the [[group completion]] $\Omega B Q(E)$.
See at
* [[Quillen Q-construction]]
### Waldhausen $S_\bullet$-construction (for Waldhausen categories)
Given a [[Waldhausen category]] $(C, w C)$, one defines its $K$-theory by applying
* the [[Waldhausen S-construction]] $w S_\bullet C$ (a [[simplicial category]]),
* the [[nerve]] (a [[bisimplicial set]]),
* the [[colimit]] (a [[simplicial set]]),
* the [[geometric realization]] (a [[space]]).
There is also a [[Waldhausen S-construction]] for [[stable (infinity,1)-categories]] and, most generally, for [[Waldhausen (infinity,1)-categories]].
See at
* [[Waldhausen S-construction]]
* [[K-theory of a stable (infinity,1)-category]]
## Examples
### For rings
We recall several constructions of the [[algebraic K-theory]] of a [[ring]].
See ([Weibel, IV.4.8, IV.4.11.1](#Weibel)) for details.
#### Plus construction
Given an associative unital [[ring]] $R$, one may define the algebraic K-theory [[space]] $K(R) = BGL(R)^+$ by taking
* the [[general linear groups]] $GL_n(R)$ for $n \ge 0$,
* their [[classifying spaces]],
* the [[colimit]] $BGL(R) = colim_n BGL_n(R)$,
* the [[Quillen plus construction]].
#### Direct sum K-theory
Consider the category $P(X)$ of [[finitely generated]] [[projective]] (right) $R$-modules.
It has a [[symmetric monoidal structure]] given by [[direct sum]].
The algebraic K-theory $K(R)$ may be described as the [[K-theory of a symmetric monoidal (infinity,1)-category]] of $P(R)$.
That is, it is the [[group completion]]
$K(R) = \Omega B B (i P(X))$
where $i P(X)$ denotes the [[maximal subgroupoid]].
See ([Weibel, IV.4.8, IV.4.11.1](#Weibel)).
#### Exact K-theory
Consider the category $P(X)$ of [[finitely generated]] [[projective module|projective]] (right) $R$-modules.
This is an [[exact category]] and the K-theory $K(R)$ may be described via the [[Quillen Q-construction]]:
$$ K(R) = \Omega B (Q(P(R)). $$
### For schemes
For [[schemes]], there are two constructions which do not agree in full generality.
See [Thomason-Trobaugh 90](#ThomasonTrobaugh90).
#### Quillen K-theory
The Quillen K-theory of a [[scheme]] $X$ is defined as the algebraic K-theory of the [[exact category]] $Vect(X)$ of [[vector bundles]] on $X$ (using the [[Quillen Q-construction]]).
#### Thomason-Trobaugh K-theory
Let $Perf(X)$ be the category of [[perfect complexes]] on $X$.
This admits the structure of a [[Waldhausen category]], and the Thomason-Trobaugh K-theory of $X$ is defined via the [[Waldhausen S-construction]].
It may also be defined as the [[K-theory of a stable (infinity,1)-category]] of $Perf(X)$ viewed as a [[stable (infinity,1)-category]].
Thomason-Trobaugh K-theory coincides with Quillen K-theory for schemes that admit an ample family of [[line bundles]], but has the advantage of better global descent properties.
### For smooth manifolds
Discussion of algebraic K-theory as a [[smooth spectrum]] $SmoothMfd^{op} \longrightarrow Spectra$ via $X \mapsto K(C^\infty(X))$ is in ([Bunke-Nikolaus-Voelkl 13](#BunkeNikolausVoelkl13), [Bunke 14](#Bunke14)).
For more on this see at
* [[algebraic K-theory of smooth manifolds]]
* [[differential cohomology hexagon]], section _[Algebraic K-theory of smooth manifolds and the e-invariant](differential+cohomology+diagram#SoothVectorBundlesWithConnectionAndEInvariant)_
## Properties
### Chern characters
#### Regulators and relation to ordinary cohomology
See at _[[Beilinson regulator]]_.
#### Cyclotomic trace and relation to topological Hochschild homology
Given a ring $R$, then there is a natural morphism of [[spectra]]
\begin{tikzcd}
& TC(R) \arrow[d]\\
K(R) \arrow[ru] \arrow[r] & THH(R)\\
\end{tikzcd}
from the algebraic K-theory spectrum to the [[topological Hochschild homology]] spectrum and factoring through the [[topological cyclic homology]] spectrum called the _[[cyclotomic trace]]_ which much like a [[Chern character]] map for algebraic K-theory.
### Comparison map and Relation to topological K-theory
* [[comparison map between algebraic and topological K-theory]]
### Descent
{#Descent}
See also
* Moritz Kerz, Florian Strunk, [[Georg Tamme]], _Algebraic K-theory and descent for blow-ups_ ([arXiv:1611.08466](https://arxiv.org/abs/1611.08466))
#### Zariski and Nisnevich descent
The algebraic K-theory spectrum $\mathbf{K}$ satisfies [[descent]] to give a [[sheaf]] of [[connective spectra]] on the [[Zariski site]]. For regular noetherian schemes this statement is due to ([Brown Gersten 73](#BrownGersten73)). The generalization to finite dimensional noetherian schemes is due to ([Thomason-Trobaugh 90](#ThomasonTrobaugh90)).
Moreover, $\mathbf{K}$ satisfies descent with respect to the [[Nisnevich site|Nisnevich topology]] (which lies between Zariski and étale). This is due to ([Nisnevich 89](#Nisnevich89)) and was generalized in turn to finite dimensional noetherian schemes in the same paper of Thomason.
Further generalization of the descent result to finite dimensional quasi-compact quasi-separated schemes is due to ([Rosenschon 06](#Rosenschon06)).
#### Etale descent
The question of descent of $\mathbf{K}$ over the [[étale site]] is closely related to the [[Lichtenbaum-Quillen conjecture]], see also ([Thomason 85](#Thomason85)). This is now a theorem of Rost and Voevodsky and it implies that K-theory does satisfy etale descent in sufficient large degrees.
> [MO comment](http://mathoverflow.net/a/180265/381)
#### Description of the K-theory sheaf via algebraic vector bundles
{#AsTheKTheoryOfAlgebraicVectorBundles}
Let $Sch$ denote the gros [[Zariski site]] of regular, separated, noetherian schemes.
It is explained in ([Bunke-Tamme 12, section 3.3](#BunkeTamme12) that the presheaf of [[spectra]] on $Sch$ defined by algebraic K-theory admits the following description.
Regard the [[stack]] $\mathbf{Vect}^\oplus$ of [[algebraic vector bundles]]
on $Sch$ as taking values in [[symmetric monoidal (∞,1)-categories]], via the [[direct sum]] of vector bundles. Then apply the [[K-theory of a symmetric monoidal (∞,1)-category]]-construction $\mathcal{K}$ to this, yielding a [[sheaf of spectra]].
This identifies with the usual Thomason-Trobaugh K-theory sheaf,a fact that follows from
1. Zariski descent for Thomason-Trobaugh K-theory,
1. the Zariski-local equivalence between Thomason-Trobaugh K-theory, Quillen K-theory, and direct sum K-theory.
### Relation to non-commutative topology and non-commutative motives
{#RelationToKKAndMotives}
[[!include noncommutative motives - table]]
### Red-shift conjecture
* [[red-shift conjecture]]
[[!include chromatic tower examples - table]]
## Examples
### On monoidal stacks
{#OnMonoidalStacks}
Algebraic K-theory is traditionally applied to single [[symmetric monoidal (∞,1)-categories|symmetric monoidal]]/[[stable (∞,1)-category|stable]] [[(∞,1)-categories]], but to the extent that it is [[(∞,1)-functor|functorial]] it may just as well be applied to [[(∞,1)-sheaves]] with values in these.
Notably, applied to the monoidal [[stack]] of [[vector bundles]] (with [[connection on a bundle|connection]]) on the [[site]] of [[smooth manifolds]], the [[K-theory of a symmetric monoidal (∞,1)-category|K-theory of a monoidal category]]-functor produces a [[sheaf of spectra]] which is a form of [[differential K-theory]] and whose [[geometric realization]] is the [[topological K-theory]] spectrum. For more on this see at _[differential cohomology hexagon -- Differential K-theory](differential%20cohomology%20diagram#DifferentialKTheory)_.
## Related concepts
* [[Eilenberg swindle]]
* [[equivariant algebraic K-theory]]
* [[iterated algebraic K-theory]]
Types of categories for which a theory of algebraic K-theory exist include notably the notions
* [[Quillen exact category]],
* [[abelian category]],
* [[Waldhausen category]],
* [[triangulated category]].
Concrete examples of interest include for instance
* the category of finitely generated [[projective object]]s over a unital $k$-[[associative unital algebra|algebra]],
* the category of [[coherent sheaf|coherent sheaves]] over a [[noetherian ring|noetherian]] [[scheme]],
* the category of locally free sheaves over a scheme,
* [[Gersten resolution]]
* [[Milnor's K2]] ([[Steinberg group]], [[universal central extension]])
* [[higher algebraic K-theory]] [[Quillen exact category]], [[Quillen's Q-construction]], [[Waldhausen S-construction]], [[Volodin spaces]];
* [[topological cyclic homology]], [[algebraic K-theory of operator algebras]]
* [[Beilinson regulator]]
* [[differential algebraic K-theory]]
* [[non-connective algebraic K-theory]]
* [[real algebraic K-theory]]
* [[bivariant algebraic K-theory]]
* [[K-motive]]
* [[red-shift conjecture]]
* [[filtrations on algebraic K-theory]]
[[!include noncommutative motives - table]]
## References
### Introductions
Surveys with accounts of the historical development include
* {#Arlettaz04} [[Dominique Arlettaz]], _Algebraic K-theory of rings from a topological viewpoint_ ([pdf](http://www.math.uiuc.edu/K-theory/0420/Arlettaz-survey.pdf))
* {#Grayson13} [[Daniel Grayson]], _Quillen's work in algebraic K-theory_, J. K-Theory 11 (2013), 527–547 [pdf](http://www.math.uiuc.edu/~dan/Papers/qs-published-final.pdf)
An introductory textbook account is in
* {#Weibel} [[Charles Weibel]], _The K-Book: An introduction to algebraic K-theory_ ([web](http://www.math.rutgers.edu/~weibel/Kbook.html))
Further review includes
* {#Isely05} Olivier Isely, _Algebraic $K$-theory_, 2005-06 ([[IselyKTheory.pdf:file]])
* {#Gerhardt14} [[Teena Gerhardt]], _Computations in algebraic K-theory_, talk at [CUNY Workshop on differential cohomologies 2014](http://qcpages.qc.cuny.edu/~swilson/cunyworkshop14.html) ([video recording](http://videostreaming.gc.cuny.edu/videos/video/1800/in/channel/55/))
Review of the relation to [[Dennis trace]], [[topological cyclic homology]] and [[topological Hochschild homology]] is in
* {#DundasGoodwillieMcCarthy13} [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], *The local structure of algebraic K-theory*, Springer (2013) [[doi:10.1007/978-1-4471-4393-2](https://doi.org/10.1007/978-1-4471-4393-2)]
### Classical
Original articles include
* {#Quillen73} [[Daniel Quillen]], _Higher algebraic K-theory_, in Higher K-theories, pp. 85–147, Proc. Seattle 1972, Lec. Notes Math. 341, Springer 1973.
([pdf](http://math.mit.edu/~hrm/kansem/quillen-higher-algebraic-k-theory.pdf))
also: [[Daniel Grayson]], _Higher algebraic K-theory II, [after Daniel Quillen]_ ([pdf](http://www.math.illinois.edu/~dan/Papers/HigherAlgKThyII.pdf))
* {#BrownGersten73} [[Kenneth Brown]], Stephen M. Gersten, _Algebraic K-theory as generalized sheaf cohomology_, Higher K-Theories, Lecture Notes in Mathematics Volume 341, 1973, pp 266-292.
* {#Waldhausen85} [[F. Waldhausen]], _Algebraic K-theory of spaces_, Alg. and Geo. Top., Springer Lect. Notes Math. 1126 (1985), 318-419, [pdf](http://www.maths.ed.ac.uk/~aar/surgery/rutgers/wald.pdf).
* {#Thomason85} [[R. W. Thomason]], _Algebraic K-theory and étale cohomology_, Ann. Sci. Ecole Norm. Sup. 18 (4), 1985, pp. 437–552.
* {#Nisnevich89} [[Yevsey Nisnevich]], _The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory_, Algebraic K-theory: connections with geometry and topology, 1989, pp 241-341.
* {#ThomasonTrobaugh90} [[R. W. Thomason]], Thomas Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_, _The Grothendieck Festschrift_, 1990, 247-435.
* {#Rosenschon06} Andreas Rosenschon, P.A. Ostvær, _Descent for K-theories_, Journal of Pure and Applied Algebra 206, 2006, pp 141–152.
For [[complex varieties]]:
* {#PedriniWeibel01} Claudio Pedrini, [[Charles Weibel]], _The higher K-theory of complex varieties_, K-theory 21 (2001), 367-385 ([web](http://www.math.uiuc.edu/K-theory/0403/))
* Michael Paluch, _Algebraic K-theory and topological spaces_ ([pdf](http://www.math.uiuc.edu/K-theory/0471/alg-top.pdf))
For discussion of stable phenomena in algebraic K-theory, see section 4 of
* {#Cohen} [[Ralph Cohen]], _Stability phenomena in the topology of moduli spaces_ ([pdf](http://arxiv.org/PS_cache/arxiv/pdf/0908/0908.1938v2.pdf))
Discussion of the [[comparison map between algebraic and topological K-theory]] includes
* [[Jonathan Rosenberg]], _Comparison between algebraic and topological K-theory for Banach algebras and $C^\ast$-algebras_ ([pdf](http://www2.math.umd.edu/~jmr/algtopK.pdf))
For [[smooth manifolds]]:
* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_, Journal of Homotopy and Related Structures October 2014 ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))
* {#Bunke14} [[Ulrich Bunke]], _A regulator for smooth manifolds and an index theorem_ ([arXiv:1407.1379](http://arxiv.org/abs/1407.1379))
### Algebraic K-theory of quotient stacks
{#ReferencesAlgebraicKTheoryForQuotientStacks}
Discussion of algebraic K-theory for [[algebraic stacks]] (generalizing algebraic [[equivariant K-theory]]) is in
* [[Robert Thomason]], _Algebraic K-theory of group scheme actions_, Algebraic Topology and Algebraic K-theory, Ann. Math. Stud., Princeton, 113, (1987), 539-563.
* [[Amalendu Krishna]], Charanya Ravi, _On the K-theory of schemes with group scheme actions_ ([arXiv:1509.05147](http://arxiv.org/abs/1509.05147))
See also at _[universal Chern-Simons 3-bundle -- For reductive groups](universal+Chern-Simons+line+3-bundle#ForReductiveAlgebraicGroups)_.
### Algebraic K-theory of ring spectra
The [[algebraic K-theory]] of [[ring spectra]]:
* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], chapter VI of _[[Rings, modules and algebras in stable homotopy theory]]_, AMS Mathematical Surveys and Monographs Volume 47 (1997) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))
* {#BlumbergGepnerTabuada10} [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], Section 9.5 of: _A universal characterization of higher algebraic K-theory_, Geom. Topol. 17 (2013) 733-838 ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282), [doi:10.2140/gt.2013.17.733](http://dx.doi.org/10.2140/gt.2013.17.733))
* [[Jacob Lurie]], _Algebraic K-Theory of Ring Spectra_, Lecture 19 of _[Algebraic K-Theory and Manifold Topology](https://www.math.ias.edu/~lurie/281.html)_, 2014 ([pdf](http://people.math.harvard.edu/~lurie/281notes/Lecture19-Rings.pdf))
The algebraic K-theory of specifically of [[suspension spectra]] of [[loop spaces]] (Waldhausen's _[[A-theory]]_) is originally due to
* [[Friedhelm Waldhausen]], _Algebraic K-theory of spaces_, In: A. Ranicki N., Levitt, F. Quinn (eds.), Algebraic and Geometric Topology, Lecture Notes in Mathematics, vol 1126. Springer, Berlin, Heidelberg (1985) ([doi:10.1007/BFb0074449](https://doi.org/10.1007/BFb0074449))
See also:
([Thomason-Trobaugh 90](#ThomasonTrobaugh90))
### Via stable $(\infty,1)$-categories
The [[stable (∞,1)-category]] theory picture is discussed in
* {#BlumbergGepnerTabuada10} [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _A universal characterization of higher algebraic K-theory_, Geom. Topol. 17 (2013) 733-838 ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282), [doi:10.2140/gt.2013.17.733](http://dx.doi.org/10.2140/gt.2013.17.733))
(in terms of [[noncommutative motives]]) and in
* [[Clark Barwick]], _On the algebraic K-theory of higher categories, I. The universal property of Waldhausen K-theory_ ([arXiv:1204.3607](http://de.arxiv.org/abs/1204.3607))
### Via symmetric monoidal $(\infty,1)$-categories
The perspective of [[algebraic K-theory of a symmetric monoidal (∞,1)-category]] is developed in
* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], section 2.1 of _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451))
* {#Nikolaus13} [[Thomas Nikolaus]] _Algebraic K-Theory of $\infty$-Operads_ ([arXiv:1303.2198](http://arxiv.org/abs/1303.2198))
* {#BunkeTamme13} [[Ulrich Bunke]], [[Georg Tamme]], _Multiplicative differential algebraic K-theory and applications_ ([arXiv:1311.1421](http://arxiv.org/abs/1311.1421))
* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], def. 6.1 in _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))
* {#GepnerGrothNikolaus13} [[David Gepner]], [[Moritz Groth]], [[Thomas Nikolaus]], _Universality of multiplicative infinite loop space machines_, [arXiv:1305.4550](http://arxiv.org/abs/1305.4550).
### K-theory stacks
The system of [[infinite loop spaces]] of the algebraic K-theory spectrum regarded as an [[∞-stack]] on the [[Nisnevich site]] and the [[principal ∞-bundles]] over it is considered in
* {#Saito14} [[Sho Saito]], _Higher Tate central extensions via K-theory and infinity-topos theory_ ([arXiv:1405.0923](http://arxiv.org/abs/1405.0923))
implementing a suggestion stated in
* {#Drinfeld03} [[Vladimir Drinfeld]], _Infinite-dimensional vector bundles in algebraic geometry (an introduction)_ ([arXiv:math/0309155](http://arxiv.org/abs/math/0309155))
### Equivariant versions
Refinement to [[global equivariant stable homotopy theory]]:
* [[Stefan Schwede]], _Global algebraic K-theory_ ([arXiv:1912.08872](https://arxiv.org/abs/1912.08872))
### Examples
* {#Ananyevskiy} Alexey Ananyevskiy, _On the algebraic $K$-theory of some homogeneous varieties_ ([pdf](http://www.math.uni-bielefeld.de/lag/man/431.pdf))
[[!redirects algebraic K-theory of a stable (∞,1)-category]]
[[!redirects algebraic K-theory of a stable (infinity,1)-category]] |
algebraic K-theory of smooth manifolds | https://ncatlab.org/nlab/source/algebraic+K-theory+of+smooth+manifolds |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
The construction that sends a [[smooth manifold]] $X$ to the [[algebraic K-theory]] [[spectrum]] $K(C^\infty(X,\mathbb{C}))$ of its [[ring]] of [[smooth functions]] (with values in the [[complex numbers]]) presents (after [[infinity-stackification]]) a [[sheaf of spectra]] on the [[site]] of [[smooth manifolds]], hence a [[smooth spectrum]]
$$
\mathbf{K} \in Stab(Smooth\infty Grpd) \simeq T_\ast Smooth \infty Grpd
\,,
$$
i.e. an object of the [[tangent cohesive (∞,1)-topos]] of [[Smooth∞Grpd]]. (See also [this definition](differential+cohomology+diagram#KTheoryOfStackOfVectorBundles) at _[[differential cohomology hexagon]]_.)
## Properties
### Shape and relation to topological K-theory
The [[shape modality|shape]] of this $\mathbf{K}$ is the [[topological K-theory]] spectrum $ku$ ([Bunke-Nikolaus-Voelkl 13, lemma 6.3](#BNV13), [Bunke 14, (48) with def. 2.21](#Bunke14)):
$$
ʃ \mathbf{K} \simeq ku
\,.
$$
Hence $\mathbf{K}$ is a [[differential cohomology]] refinement of $ku$, a form of [[differential K-theory]].
### Regulator and relation to differential K-theory
{#Regulator}
There is also the more standard [[differential K-theory]] refinement $\mathbf{ku}_{conn}$ of $ku$ ([[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 05]], [Bunke-Nikolaus-Voelkl 13](differential+K-theory#BunkeNikolausVoelkl13)) which is obtained by pulling back suitable sheaves of ($\mathbb{C}$-valued) [[differential forms]] $\mathbf{DD}^-$ along the usual [[Chern character map]] $ch \colon ku \longrightarrow DD^{per}$. This Chern character lifts through the [[shape modality]] to a [[regulator]] map ([Bunke 14, (50)](#Bunke14))
$$
\array{
\mathbf{K} &\stackrel{\mathbf{reg}}{\longrightarrow}& \mathbf{DD}^-
\\
\downarrow^{\mathrlap{\eta^{ʃ}}} && \downarrow^{\mathrlap{\eta^{ʃ}}}
\\
ku &\stackrel{ch}{\longrightarrow}& DD^{per}
}
$$
Moreover, this induces a differential [[regulator]] ([BNV 13, p.40 and example 6.9](#BNV13), [Bunke 14, def. 2.29](#Bunke14)):
$$
\mathbf{reg}_{conn} \;\colon\; \mathbf{K} \longrightarrow \mathbf{ku}_{conn}
\,.
$$
See also [this proposition](differential+cohomology+diagram#SmoothRegulator) at _[[differential cohomology diagram]]_.
### Moduli stacks
In ([Bunke 14](#Bunke14)) all this is generalized to the [[mapping spaces]] $[X,\mathbf{K}]$ out of a [[smooth manifold]] $X$, hence to the [[moduli stacks]] (here: moduli spectra) of algebraic K-theory cocycles on $X$.
The regulator from [above](#Regulator) induces a map
$$
[X,\mathbf{reg}_{conn}]
\;\colon\;
[X,\mathbf{K}]
\longrightarrow
[X,\mathbf{ku}_{conn}]
\,.
$$
(...)
## References
* {#BNV13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], section 6 of _Differential cohomology theories as sheaves of spectra_, Journal of Homotopy and Related Structures October 2014 ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))
* {#Bunke14} [[Ulrich Bunke]], _A regulator for smooth manifolds and an index theorem_ ([arXiv:1407.1379](http://arxiv.org/abs/1407.1379))
* [[Ulrich Bunke]], _Smooth aspects of algebraic K-theory_ ([pdf](http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/smooth-asp.pdf))
|
algebraic K-theory spectrum | https://ncatlab.org/nlab/source/algebraic+K-theory+spectrum | The term _algebraic K-theory spectrum_ refers to one of the following two notions:
* an object $K(C) \in \Spt$ in the [[stable (infinity,1)-category of spectra]], associated to a [[stable (infinity,1)-category]] or [[symmetric monoidal (infinity,1)-category]] $C$, whose [[homotopy groups]] give the [[algebraic K-theory]] of $C$;
* a [[motivic spectrum]] $KGL \in SH(S)$ in the [[stable motivic homotopy category]], representing an $\mathbf{A}^1$-[[homotopy invariant]] version of [[algebraic K-theory]], where $S$ is a scheme.
This entry is about the latter notion; for the former one, see [[algebraic K-theory]].
## Idea
Suppose that $S$ is a [[regular scheme]]. Then there exists a [[motivic spectrum]] $KGL_S$ with the property that, for every $X\in Sm/S$,
$$ KGL^{p,q}(X) = K_{2q-p}(X), $$
where $K_*(X)$ are the [[algebraic K-theory]] groups of $X$ defined by [[Daniel Quillen|Quillen]]. In particular, $KGL$-cohomology is $(2,1)$-periodic: this is _Bott periodicity_ for algebraic K-theory.
The multiplicative structure of algebraic K-theory makes $KGL$ into a ring spectrum (up to homotopy), which comes from a unique structure of $E_\infty$-algebra (see [Naumann-Spitzweck-Ostvaer](#NSO11)).
Over non-regular schemes, the motivic spectrum $KGL$ is also defined and it represents Weibel's homotopy invariant version of [[algebraic K-theory]] (see [Cisinski13](#Cis13)).
## Related concepts
* [[algebraic K-theory]]
* [[motivic homotopy theory]]
* [[motivic cohomology]]
* [[algebraic cobordism]]
## References
* [[Fabien Morel]], [[Vladimir Voevodsky]], _$\mathbb{A}^1$-homotopy theory of schemes_ K-theory, 0305 ([web](http://www.math.uiuc.edu/K-theory/0305/) [pdf](http://www.math.uiuc.edu/K-theory/0305/nowmovo.pdf))
{#MorelVoevodsky}
* {#Voevodsky98} [[Vladimir Voevodsky]], _$\mathbf{A}^1$-Homotopy Theory_, Doc. Math., Extra Vol. ICM 1998(I), 417-442, [web](http://www.mathematik.uni-bielefeld.de/documenta/xvol-icm/00/Voevodsky.MAN.html).
* {#Cis13} [[Denis-Charles Cisinski]], _Descente par éclatements en K-théorie invariante par homotopie_, Ann. of Math. (2) 177 (2013), no. 2, pp. 425–448 ([pdf](http://hal.archives-ouvertes.fr/docs/00/52/56/40/PDF/descente3.pdf))
* {#NSO11} [[Niko Naumann]], [[Markus Spitzweck]], [[Paul Arne Østvær]], _Existence and uniqueness of E-infinity structures on motivic K-theory spectra_, ([arXiv](http://arxiv.org/abs/1010.3944))
* [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], _On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory_, Inventiones mathematicae, 2009, 175(2), 435-451, [DOI](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.244.7301), [arXiv](http://arxiv.org/abs/0709.4124).
* {#GepnerSnaith08} [[David Gepner]], [[Victor Snaith]], _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Documenta Math. 2008 ([arXiv:0712.2817](http://arxiv.org/abs/0712.2817)).
[[!redirects algebraic K-theory spectra]]
[[!redirects motivic algebraic K-theory spectrum]]
[[!redirects KGL]] |
algebraic K-theory, a historical perspective | https://ncatlab.org/nlab/source/algebraic+K-theory%2C+a+historical+perspective | [[!redirects Algebraic K-theory, a historical perspective]]
## A little history: the beginnings
[[algebraic K-theory|Algebraic K-theory]] grew out of two apparently unrelated areas of algebraic geometry and algebraic
topology. The second of these, historically, was the development by Grothendieck of (geometric
and topological) K-theory based on projective modules over a ring, or finite dimensional vector
bundles on a space, that is the [[Grothendieck group]] of the given ring or of the ring of functions on the space. (The connection between these is that the space of global sections of a finite
dimensional vector bundle on a nice enough space, $X$ is a finitely generated projective module over
the ring of continuous real or complex functions on $X$ . This latter aspect is where the link with [[topological K-theory]] comes in.)
The second input is from of [[simple homotopy theory]]. [[J. H. C. Whitehead]], following on from earlier ideas of [[Reidemeister]], looked at possible extensions of combinatorial
group theory, with its study of presentations of groups, to give a [[combinatorial homotopy theory]]. This would take the form of an 'algebraic homotopy theory' giving good algebraic models
for homotopy types, and would hopefully ease the determination of homotopy equivalences for instance of polyhedra. The 'combinatorial' part was exemplified by his two papers on 'Combinatorial
Homotopy Theory', but raised an interesting question.
_Could one give a 'combinatorial way of generating all homotopy equivalences, (up to homotopy), starting with some 'elementary expansions' and 'contractions'?_
He showed the answer was negative, and there was an invariant ([[Whitehead torsion]]) whose vanishing was a necessary and sufficient condition for a homotopy equivalence to be so constructible. That invariant was an element of a group constructed from the stable general linear group over the [[group ring]] of the [[fundamental group]] of the domain space. This was a quotient of what became known as $K_1(\mathbb{Z}\pi_1X)$, that is the abelianisation of $GL(\mathbb{Z}\pi_1X)$.
This ties in, even at this basic level with the [[nPOV]] and the processes around categorification. For a ring, $R$, the [[Grothendieck group]], $K_0(R)$, looks at the [[core]] of the category of finitely generated projective modules, and takes its set of connected components, which is just the set of isomorphism classes. This becomes an abelian monoid under direct sum and then a group after group completion. For $K_1(R)$, you are taking the core again, but looking at the category of morphisms in it. If we take $K_0$ of that category we get $K_1$ of the ring. So $K_1$ looks at the loops whilst $K_0$ at the connected components.
## References
One of the best accounts of the history of K-theory is by [[Chuck Weibel]]:
* {#Weibel99} [[Charles A. Weibel]], [The development of algebraic K-theory before 1980](http://www.math.rutgers.edu/~weibel/papers-dir/khistory.pdf) published in _Algebra, K-theory, groups, and education:
on the occasion of Hyman Bass's 65th birthday_, a volume of _Contemporary
Mathematics, American Mathematical Soc, 1999.
Other historical accounts include
* {#Arlettaz04} [[Dominique Arlettaz]], _Algebraic K-theory of rings from a topological viewpoint_ ([pdf](http://www.math.uiuc.edu/K-theory/0420/Arlettaz-survey.pdf))
* {#Grayson13} [[Daniel Grayson]], _Quillen's work in algebraic K-theory_, J. K-Theory 11 (2013), 527–547 [pdf](http://www.math.uiuc.edu/~dan/Papers/qs-published-final.pdf)
|
algebraic Kan complex | https://ncatlab.org/nlab/source/algebraic+Kan+complex | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _algebraic Kan complex_ is an [[algebraic definition of higher categories|algebraic definition of]] [[∞-groupoid]]s.
It builds on the classical [[geometric definition of higher categories|geometric definition]] of $\infty$-groupoids in terms of [[Kan complex]]es. A Kan complex is like an algebraic $\infty$-groupoid in which we have forgotten what precisely the [[composition]] operation and what the [[inverse]]s are, and only know that they do exist. This becomes an _algebraic model_ for $\infty$-groupoids by adding the specific choices of composites back in.
The nontrivial aspect of the definition of algebraic Kan complexes is that they do still [[presentable (∞,1)-category|present]] the full [[(∞,1)-category]] [[∞Grpd]]. Notably the [[homotopy hypothesis]] is true for algebraic Kan complexes.
## Definition
An **algebraic Kan complex** is a [[Kan complex]] equipped with a _choice_ of [[horn]] fillers for all horns.
A morphism of algebraic Kan complexes is a morphism of the underlying Kan complexes that sends chosen fillers to chosen fillers.
This defines the category $Alg Kan$ of algebraic Kan complexes.
For more see the section [Algebraic fibrant models for higher categories](http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects#AlgebaicHigherCategories) at [[model structure on algebraic fibrant objects]].
A slight variant of this definition is that of a [[simplicial T-complex]].
## Properties
### Monadicity
The category $Alg Kan$ is the category of algebras over a [[monad]]
$$
sSet \stackrel{\leftarrow}{\to} Alg sSet
\,.
$$
This means that algebraic Kan complexes are formally an _algebraic model_ for higher categories.
See [[model structure on algebraic fibrant objects]] for details.
### Homotopy hypothesis-theorem
The [[homotopy hypothesis]] is true for algebraic Kan complexes:
there is a [[model category]] structure on $Alg Kan$ -- the [[model structure on algebraic fibrant objects]] -- and a [[Quillen equivalence]] to the standard [[model structure on simplicial sets]].
Moreover, there is a direct [[Quillen equivalence]]
$$
\Pi_\infty : Top \stackrel{\leftarrow}{\to} AlgKan : |-|_r
\,,
$$
to the standard [[model structure on topological spaces]], where the [[left adjoint]] $|-|_r$ is a quotient of the [[geometric realization]] of the underlying Kan complexes and $\Pi_\infty$ is a version of the [[fundamental ∞-groupoid]]-functor with values in algebraic Kan complexes.
See <a href="http://ncatlab.org/nlab/show/homotopy+hypothesis#ForAlgebraicKanComplexes">homotopy hypothesis -- for algebraic Kan complexes</a> for details.
### Algebraicization
If we assume the [[axiom of choice]], then any [[Kan complex]] can be made into an algebraic Kan complex by making a simultaneous choice of a filler for every horn.
In the absence of AC, one might argue that algebraic Kan complexes are a better model of $\infty$-groupoids than non-algebraic ones. For instance, an algebraic Kan complex always has the right lifting property with respect to all [[anodyne morphisms]], whereas for a non-algebraic Kan complex this fact requires choice.
## References
* {#Nikolaus} [[Thomas Nikolaus]], _Algebraic models for higher categories_, Indagationes Mathematicae
Volume 21, Issues 1–2, July 2011, Pages 52-75
([arXiv/1003.1342](http://arxiv.org/abs/1003.1342), [doi:10.1016/j.indag.2010.12.004](https://doi.org/10.1016/j.indag.2010.12.004))
[[!redirects algebraic Kan complex]]
[[!redirects algebraic Kan complexes]]
|
algebraic lattice | https://ncatlab.org/nlab/source/algebraic+lattice |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition {#Def}
+-- {: .num_defn}
###### Definition
An **algebraic lattice** is a [[lattice]] which is
* a [[complete lattice]];
* such that every element is a [[join]] of [[compact elements]].
=--
An __algebraic lattice__ is a [[complete lattice]] (equivalently, a [[suplattice]], or in different words a [[poset]] with the [[extra property|property]] of having arbitrary [[colimits]] but with the [[structure]] of [[directed colimits]]/[[directed joins]]) in which every element is the [[supremum]] of the [[compact element]]s below it (an element $e$ is compact if, for every subset $S$ of the lattice, $e$ is less than or equal to the supremum of $S$ just in case $e$ is less than or equal to the supremum of some finite subset of $S$).
Here is an alternative formulation:
+-- {: .num_defn}
###### Definition
An algebraic lattice is a [[poset]] which is [[locally finitely presentable category|locally finitely presentable]] as a category.
=--
This formulation suggests a useful way of viewing algebraic lattices in terms
of [[Gabriel-Ulmer duality]] (but with regard to enrichment in [[truth values]], instead of in $Set$).
As this last formulation suggests, algebraic lattices typically arise as [[subobject lattices]] for objects in locally finitely presentable categories. As an example, for any (finitary) [[Lawvere theory]] $T$, the subobject lattice of an object in $T$-$Alg$ is an algebraic lattice (this class of examples explains the origin of the term "algebraic lattice", which is due to Garrett Birkhoff). In fact, all algebraic lattices arise this way (see Theorem \ref{GS} below).
It is trivial that every finite lattice is algebraic.
## Properties
### The category of algebraic lattices
The [[morphisms]] most commonly considered between algebraic lattices are the [[finitary functors]] between them, which is to say, the [[Scott topology|Scott-continuous]] functions between them; i.e., those functions which preserve directed joins (hence the parenthetical remarks [above](#Def)).
The resulting category __AlgLat__ is [[cartesian closed]] and is dually equivalent to the category whose objects are [[meet semilattices]] (construed as categories with [[finite limits]] [[enriched category|enriched]] over [[truth values]]) and whose morphisms are meet-preserving [[profunctors]] between them (using the convention that a $V$-enriched profunctor from $C$ to $D$ is a functor $D^{op} \times C \rightarrow V$; of course, with an opposite convention, one could similarly state a covariant equivalence).
There is a _full_ [[full embedding|embedding]]
$$i \colon AlgLat \to Top_0$$
to the category of $T_0$-[[separation axioms|spaces]], taking an algebraic lattice $L$ to the space whose points are elements of $L$, and whose [[open sets]] $U$ are defined by the property that their [[characteristic maps]]
$$\chi_U: L \to \mathbf{2}$$
($\chi_U(a) = 1$ if $a \in U$, else $\chi_U(a) = 0$) are poset maps that preserve [[directed colimits]]. The [[specialization order]] of $i(L)$ is $L$ again.
Every $T_0$-space $X$ occurs as a [[subspace]] of some space $i(L)$ associated with an algebraic lattice. Explicitly, let $L(X)$ be the [[power set]] of the underlying set of the [[topology]], $P{|\mathcal{O}(X)|}$, and define
$$X \to (i\circ L)(X)$$
to take $x$ to $N(x) \coloneqq \{U \in \mathcal{O}(X): x \in U\}$. This gives a topological embedding of $X$ in $i(L(X))$.
+-- {: .un_remark}
###### Remark
On similar grounds, if $U \colon AlgLat \to Set$ is the forgetful functor, then the [2-image](http://ncatlab.org/nlab/show/stuff%2C+structure%2C+property#a_factorisation_system_14) of the projection functor $\pi \colon Set\downarrow U \to Set$ is the category of topological spaces $Top$. In more nuts-and-bolts terms, an object $(S, L, f \colon S \to U(L))$ gives a space with underlying set $S$ and open sets those of the form $f^{-1}(O)$, where $O$ ranges over the Scott topology on $L$. Notice that if $(f \colon S \to S', g \colon L \to L')$ is a morphism in $Set \downarrow U$, then $f$ is continuous with respect to these topologies. Therefore the projection $\pi \colon Set \downarrow U \to Set$ factors through the faithful forgetful functor $Top \to Set$. Thus, working in the factorization system (eso+full, faithful) on $Cat$, we have a faithful functor $2$-$im(\pi) \to Top$ filling in as the diagonal
$$\array{
Set \downarrow U & \to & Top \\
\downarrow & \nearrow & \downarrow \\
2\text{-}im(\pi) & \to & Set.
}$$
But notice also that $Set \downarrow U \to Top$ is [eso and full](http://ncatlab.org/nlab/show/ternary+factorization+system#examples_9). It is eso because any topology $\mathcal{O}(S)$ on $S$ can be reconstituted from the triple $(S, P{|\mathcal{O}(S)|}, x \mapsto N(x) \colon S \to P{|\mathcal{O}(S)|})$. We claim it is full as well. For, every continuous map $X \to X'$ between topological spaces induces a continuous map between their $T_0$ reflections $X_0 \to X_{0}'$, and since algebraic lattices like $P{|\mathcal{O}(X)|}$ (being continuous lattices) are [[injective objects]] in the category of $T_0$ spaces, we are able to complete to a diagram
$$\array{
X & \to & X_0 & \to & P{|\mathcal{O}(X)|} \\
\downarrow & & \downarrow & & \downarrow \\
X' & \to & X_{0}' & \to & P{|\mathcal{O}(X')|}
}$$
where the rightmost vertical arrow is Scott-continuous (and the horizontal composites are of the form $x \mapsto N(x)$). Finally, since $Set \downarrow U \to Top$ is eso and full, it follows that $2$-$im(\pi) \to Top$ is eso, full, and faithful, and therefore an equivalence of categories.
This connection is explored in more depth with the category of [[equilogical spaces]], which can be seen either as a category of (set-theoretic) [[equivalence relation|partial equivalence relations]] over $AlgLat$, or equivalently of (set-theoretic) total [[equivalence relations]] on $T_0$ topological spaces.
=--
### Relation to locally finitely presentable categories
{#RelationToLocallyFinitelyPresentableCategories}
One of our definitions of algebraic lattice is: a poset $L$ which is locally finitely presentable when viewed as a category. The completeness of $L$ means that right adjoints $L \to Set$ are representable, given by $L(p, -) \colon L \to Set$, and we are particularly interested in those representable functors that preserve [[filtered colimits]]. These correspond precisely to finitely presentable objects $p$, which in lattice theory are usually called compact elements. These compact elements are closed under finite joins.
By [[Gabriel-Ulmer duality]], $L$ is determined from the join-semilattice of compact elements $K$ by $L \cong Lex(K^{op}, Set)$. Since the elements of $K^{op}$ are subterminal, we can also write $L \cong Lex(K^{op}, 2)$ where $2 = Sub(1)$.
+-- {: .num_theorem}
###### Theorem
**(Porst)**
If $C$ is a [[locally finitely presentable category]] and $X$ is an object of $C$, then
* The lattice of subobjects $Sub(X)$,
* the lattice of quotient objects (equivalence classes of epis sourced at $X$) $Quot(X)$,
* the lattice of congruences (internal equivalence relations) on $X$
are all algebraic lattices.
=--
This is due to [Porst](#Porst). Of course if $C$ is the category of algebras of an Lawvere theory, then the lattice of quotient objects of an algebra is isomorphic to its congruence lattice, as such $C$ is an [[exact category]].
### Congruence lattices
The following result is due to Grätzer and Schmidt:
+-- {: .num_theorem #GS}
###### Theorem
Every algebraic lattice is isomorphic to the congruence lattice of some [[model]] $X$ of some finitary algebraic theory.
=--
In particular, since every finite lattice is algebraic, every finite lattice arises this way. Remarkably, it is not known at this time whether every finite lattice arises as the congruence lattice of a *finite* algebra $X$. It has been conjectured that this is in fact **false**: see this [MO discussion](http://mathoverflow.net/a/196074/2926).
Another problem which had long remained open is the congruence lattice problem: is every *distributive* algebraic lattice the congruence lattice (or lattice of quotient objects) of some lattice $L$? The answer is negative, as shown by Wehrung in 2007: see this [Wikipedia article](http://en.m.wikipedia.org/wiki/Congruence_lattice_problem).
### Completely distributive lattices
+-- {: .num_prop}
###### Proposition
The category of [[Alexandroff locales]] is equivalent to that of [[completely distributive lattice|completely distributive]] algebraic lattices.
=--
This appears as ([Caramello, remark 4.3](#Caramello)).
The [[completely distributive lattice|completely distributive]] algebraic lattices form a [[reflective subcategory]] of that of all distributive lattices. The reflector is called _[[canonical extension]]_.
## Related concepts
See also [[compact element]], [[compact element in a locale]].
[[!include locally presentable categories - table]]
## References
* [[Andrej Bauer]], [[Lars Birkedal]], [[Dana Scott]], _Equilogical Spaces_, Theoretical Computer Science, 315(1):35-59, 2004. ([web](http://math.andrej.com/2002/07/05/equilogical-spaces/))
* Olivia Caramello, _A topos-theoretic approach to Stone-type dualities_ ([arXiv:1103.3493](http://arxiv.org/abs/1103.3493))
{#Caramello}
The relation to [[locally finitely presentable categories]] is discussed in
* [[Hans Porst]], _Algebraic lattices and locally finitely presentable categories_, Algebra Univers. **65**, 285–298 (2011). <https://doi.org/10.1007/s00012-011-0129-0> ([pdf](http://www.math.uni-bremen.de/~porst/dvis/PORST_AlgebraicLattices_revfinAU.pdf))
{#Porst}
That every algebraic lattice is a congruence lattice is proved in
* G. Grätzer and E. T. Schmidt, _Characterizations of congruence lattices of abstract algebras_, Acta Sci. Math. (Szeged) 24 (1963), 34–59.
[[!redirects algebraic lattice]]
[[!redirects algebraic lattices]] |
algebraic Lefschetz formula | https://ncatlab.org/nlab/source/algebraic+Lefschetz+formula | Let $(C,d)$ be a nonnegative [[cochain complex]] of [[vector space]]s over a [[field]] of (total) finite dimension $dim C = \sum_{p=0}^\infty dim C^p \lt \infty$ and $f = (f^p)_{p\geq 0} :(C,d)\to (C,d)$ an endomorphism of cochain complexes.
The **algebraic Lefschetz formula** is the statement
$$
\sum_{p\geq 0} (-1)^p tr (f^p :C^p\to C^p) = \sum_{p\geq 0} (-1)^p tr (H^p(f):H^p(C)\to H^p(C)).
$$
Its special case for $f = id$ is the **Euler-Poincaré formula**
$$
\sum_{p\geq 0} (-1)^p dim C^p = \sum_{p\geq 0} (-1)^p dim H^p(C).
$$ |
algebraic limit field | https://ncatlab.org/nlab/source/algebraic+limit+field | [[!redirects algebraic limit fields]]
[[!redirects algebraic limit theorem]]
[[!redirects algebraic limit theorems]]
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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea ##
A [[field]] with a notion of a [[limit of a sequence|limit]] of a [[function]] that satisfy the [[algebraic limit theorems]].
## Definition ##
Let $F$ be a [[Heyting field]] and a [[Hausdorff function limit space]], where $x^{-1}$ is another notation for the [[reciprocal function]] $\frac{1}{x}$. $F$ is a __algebraic limit field__ if the __algebraic limit theorems__ are satisfied, i.e. if the limit preserves the field operations:
* for all elements $c \in S$,
$$\lim_{x \to c} 0(x) = 0$$
* for all elements $c \in S$ and functions $f:S \to C$ and $g:S \to C$ such that
$$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$
$$\lim_{x \to c} f(x) + g(x) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$$
* for all elements $c \in S$ and functions $f:S \to C$ such that
$$\lim_{x \to c} f(x) = c$$
$$\lim_{x \to c} -f(x) = -\lim_{x \to c} f(x)$$
* for all elements $c \in S$ and functions $f:S \to C$ and $g:S \to C$ such that
$$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$
$$\lim_{x \to c} f(x) - g(x) = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$$
* for all elements $c \in S$, integers $a \in \mathbb{Z}$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
$$\lim_{x \to c} a f(x) = a \lim_{x \to c} f(x)$$
* for all elements $c \in S$ and $a \in S$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
$$\lim_{x \to c} a f(x) = a \lim_{x \to c} f(x)$$
* for all elements $c \in S$,
$$\lim_{x \to c} 1(x) = 1$$
* for all elements $c \in S$ and functions $f:S \to C$ and $g:S \to C$ such that
$$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$
$$\lim_{x \to c} f(x) \cdot g(x) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$$
* for all elements $c \in S$, natural numbers $n \in \mathbb{N}$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
$$\lim_{x \to c} {f(x)}^n = {\left(\lim_{x \to c} f(x)\right)}^n$$
* for all elements $c \in S$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
if
$$\lim_{x \to c} f(x) \# 0$$
then
$$\lim_{x \to c} {f(x)}^{-1} = {\left(\lim_{x \to c} f(x)\right)}^{-1}$$
* for all elements $c \in S$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
if
$$\lim_{x \to c} f(x) \# 0$$
then
$$\lim_{x \to c} f(x) \cdot {f(x)}^{-1} = 1$$
## See also ##
* [[function limit space]]
* [[difference quotient]]
* [[algebraic limit vector space]]
* [[Newton-Leibniz operator]]
* [[differentiable space]] |
algebraic limit vector space | https://ncatlab.org/nlab/source/algebraic+limit+vector+space | [[!redirects algebraic limit vector spaces]]
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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition ##
Given an [[algebraic limit field]] $F$, an __algebraic limit vector space__ is a $F$-[[vector space]].
## See also ##
* [[algebraic limit field]]
* [[difference quotient]]
* [[Newton-Leibniz operator]] |
algebraic line bundle | https://ncatlab.org/nlab/source/algebraic+line+bundle |
#Contents#
* table of contents
{:toc}
## Idea
The generalization of the concept of _[[holomorphic line bundle]]_ from [[complex analytic geometry]] to more general [[algebraic geometry]].
Algebraic line bundles on some [[variety]]/[[scheme]] form its [[Picard group]]/[[Picard scheme]].
## Related concepts
* [[divisor (algebraic geometry)]]
* [[algebraic line n-bundle]]
[[!redirects algebraic line bundles]]
|
algebraic line n-bundle | https://ncatlab.org/nlab/source/algebraic+line+n-bundle |
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### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of [[line n-bundle]] in [[algebraic geometry]], classified by maps into the $n$-fold [[delooping]] $\mathbf{B}^n \mathbb{G}_m$ of the [[multiplicative group]].
* for $n = 0$, classified by the [[group of units]];
* for $n= 1$ these are [[algebraic line bundles]], classified by [[Picard group]], modulated by [[Picard stack]];
* for $n =2$ these are [[algebraic line 2-bundles]], classified by [[Brauer group]], modulated by [[Brauer stack]]
## Properties
{#Properties}
According to ([Grothendieck 64, prop. 1.4](#Grothendieck64)) for $X$ a [[Noetherian scheme]] whose [[local rings]] have strict [[Henselian ring|Henselisations]] that are factorial (...explain...) then the [[cohomology groups]]
$$
H^n(X,\mathbb{G}_m) = \pi_0 \mathbf{H}(X,\mathbf{B}^n \mathbb{G}_m)
$$
are all [[torsion groups]] for $n \geq 2$. (For $n = 2$ this is the [[Brauer group]].) See also [this MO discussion](http://mathoverflow.net/q/171638/381).
See also at _[[Friedlander-Milnor isomorphism conjecture]]_.
## Related concepts
* [[holomorphic line n-bundle]]
## References
* {#Grothendieck58} [[Alexander Grothendieck]], _Torsion homologique et sections rationnelles_, Séminaire Claude Chevalley, 3 (1958), Exp. No. 5, 29 p. ([Numdam](http://www.numdam.org/item?id=SCC_1958__3__A5_0))
* {#Grothendieck64} [[Alexander Grothendieck]], _Le groupe de Brauer : II. Théories cohomologiques_. Séminaire Bourbaki, 9 (1964-1966), Exp. No. 297, 21 p. ([Numdam](http://www.numdam.org/item?id=SB_1964-1966__9__287_0))
[[!redirects algebraic line n-bundles]]
[[!redirects algebraic line 2-bundle]]
[[!redirects algebraic line 2-bundles]]
|
algebraic microlocalization | https://ncatlab.org/nlab/source/algebraic+microlocalization | #Contents#
* automatic table of contents goes here
{:toc}
# Idea
As an analogue of the [[microlocalization]] in operator theory, T. Springer has introduced an algebraic microlocalization in the theory of filtered noncommutative rings.
[[microlocal analysis|Microlocal analysis]] using [[hyperfunctions]] instead of [[Schwartz distributions]] is also called algebraic microlocal analysis.
#References
An alternative way to algebraic microlocalization is given in
* Maria J. Asensio, [[Michel Van den Bergh]], [[Freddy Van Oystaeyen]], _A new algebraic approach to microlocalization of filtered rings_, Trans. Amer. Math. Soc. __316__, 2 (Dec. 1989) 537--553 [jstor](http://www.jstor.org/pss/2001360)
This is used in comparison to Kapranov's noncommutative geometry based on commutator expansion in
* [[Lieven Le Bruyn]], _Formal structures and representation spaces_, J. Algebra __247__, 616--635 (2002) [doi](https://doi.org/10.1006/jabr.2001.9019)
An introduction to the [[microlocal analysis]] of [[hyperfunctions]] is this:
* Goro Kato, Daniele C. Struppa: _Fundamentals of algebraic microlocal analysis_ ([ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0924.35001&format=complete))
[[!redirects algebraic microlocal analysis]] |
algebraic model category | https://ncatlab.org/nlab/source/algebraic+model+category |
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#### Model category theory
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[[!include model category theory - contents]]
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=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The structure of an _algebraic model category_ is a refinement of that of a [[model category]].
Where a bare [[model category]] structure is a [[category with weak equivalences]] refined by two [[weak factorization systems]] ((cofibrations, acyclic fibrations) and (acyclic cofibrations, fibrations)) in an algebraic model structure these are refined further to [[algebraic weak factorization systems]] plus a bit more.
This extra structure supplies more control over constructions in the model category. For instance its choice induces a [[weak factorization system]] also in every [[diagram]] [[category]] of the given model category.
## Definition
An _algebraic model structure_ on a [[homotopical category]] $(M,W)$ consists of a pair of algebraic weak factorization systems $(C_t, F)$, $(C,F_t)$ together with a _morphism of algebraic weak factorization systems_
$$(C_t,F) \to (C,F_t)$$
such that the underlying weak factorization systems form a model structure on $M$ with weak equivalences $W$.
A _morphism of algebraic weak factorization systems_ consists of a natural transformation
$$\array{ & \text{dom} f & \\ {}^{C_{t}f}\swarrow & & \searrow {}^{{C}{f}} \\ Rf & \stackrel{\xi_f}{\to} & Qf \\ {}_{{F}{f}}\searrow & & \swarrow {}_{F_{t}f} \\ & \text{cod} f & }$$
comparing the two functorial factorizations of a map $f$ that defines a [[monad|colax comonad morphism]] $C_t \to C$ and a lax monad morphism $F_t \to F$.
## Properties
Every [[cofibrantly generated model category]] structure can be lifted to that of an algebraic model category. It is not clear whether or not this is true for any [[accessible model category]].
Any algebraic model category has a fibrant replacement monad $R$ and a cofibrant replacement comonad $Q$. There is also a canonical [[distributive law]] $RQ \to QR$ comparing the two canonical bifibrant replacement functors.
## Related pages
* [[algebraic weak factorization system]]
* [[accessible model category]]
[[!include algebraic model structures - table]]
## References
The notion was introduced in:
* [[Emily Riehl]], _Algebraic model structures_, New York J. Math. 17 (2011) 173-231 ([journa](http://nyjm.albany.edu/j/2011/17-10.html), [arXiv](http://arxiv.org/abs/0910.2733))
The algebraic analog of [[monoidal model categories]] is discussed in
* [[Emily Riehl]], _Monoidal algebraic model structures_ ([arXiv:1109.2883](http://arxiv.org/abs/1109.2883))
Review:
* [[Emily Riehl]], §12.9 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]
See also:
* Patrick M. Schultz, *Algebraic Weak Factorization Systems in Double Categories*, PhD thesis, University of Orego (2014) [[hdl:1794/18429](http://hdl.handle.net/1794/18429), [pdf](https://scholarsbank.uoregon.edu/xmlui/bitstream/handle/1794/18429/Schultz_oregon_0171A_11048.pdf?sequence=1&isAllowed=y)]
* Gabriel Bainbridge, *Some Constructions of Algebraic Model Categories*, PhD thesis, Ohio State University (2021) [[pdf](https://etd.ohiolink.edu/apexprod/rws_etd/send_file/send?accession=osu1620719585729611&disposition=inline), [[Bainbridge-AlgebraicModelCategories.pdf:file]]]
[[!redirects algebraic model categories]]
[[!redirects algebraic model structure]]
[[!redirects algebraic model structures]]
|
algebraic model for modal logics | https://ncatlab.org/nlab/source/algebraic+model+for+modal+logics |
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### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
# Algebras for modal logics
* table of contents
{:toc}
## Idea
Classical [[propositional calculus]] has an algebraic [[model]], namely a [[Boolean algebra]]. With a bit of imagination, one can give it a combinatorial model in the line of [[Kripke semantics]]. As this ordinary propositional logic has no [[modal operators]], then the corresponding [[frame (modal logic)|frames]] have no relations, so are just sets. If $W$ is such a set, (of worlds), a valuation $V: Prop \to 2^W$ just assigns to each $p \in Prop$ and $w\in W$ a truth value, $\top$ or $\bot$, (true or false). We know however that there is a Boolean algebra structure around in that power set $2^W$ and the [[semantics]] extends the assignment given by $V$ to a map of algebras, from the term algebra based on the basic propositional language to the Boolean algebra of subsets of $W$. That is just to say that it builds up that extension of $V$ bit-by-bit on the terms. This gives an algebraic interpretation or representation of the terms of the logic in terms of the algebra of subsets of $W$, in other words an algebraic semantics.
[[modal logic|Modal logics]] also have an algebraic semantics based on a Boolean algebra, but with additional operators that model the [[modal operators]]. This is as well as the [[geometric models for modal logics|geometric semantics]] using [[frames (modal logic)|frames]].
## Boolean algebras with operators (BAOs)
### Boolean algebras
We will, here, consider a [[Boolean algebra]], $\mathbb{B }$, as an algebra, and in the notation,
$$\mathbb{B} = (B, +, \cdot, \overline{},0,1)$$
so, for example, for a set $S$, the [[power set]] Boolean algebra will be
$$\mathbb{P}(S) = (\mathcal{P}(S), \cup, \cap,-,\emptyset, S),$$
where $-A$ is shorthand for the complement, $S- A$, of $A$.
### Operators
The __operators__ that we need to add into the Boolean algebras do not always preserve all the structure:
+-- {: .num_defn}
###### Definition
A [[function]], $m : B\to B$ is called an __operator__ on the Boolean algebra, $\mathbb{B}$, if it is _additive_
$$m(x+y) = m x + m y.$$
The operator, $m$, is called __normal__ if $m(0)=0$.
=--
Any operator, $m$, in this sense has a dual $l : B\to B$ given by
$$l(x) = (m(x^-))^-.$$
As $m$ is additive, $l$ is __multiplicative__
$$l(x\cdot y) = l(x)\cdot l(y),$$
and has $l(1) = 1$ if $m$ is normal.
+-- {: .num_remark}
###### Remark
One of the myriad notations used for the generic modal operators $\lozenge$ and $\Box$, are $M$ and $L$, whence $M$ is 'possibility, and $L$ is 'necessity", and these gave the names to the operators above.
=--
+-- {: .num_defn}
###### Definition
A **Boolean algebra with operators,** or BAO, of type $n$ consists of a Boolean algebra $\mathbb{B}$, and a set, $m_i$, $i = 1,\ldots, n$ of operators on $B$.
=--
BAOs are sometimes called **modal algebras**, especially in the case that $n = 1$. The term **polymodal algebra** is then used for the general case.
There is no need in the definition of BAOs to restrict to finitely many operators nor to have all the operators being unary. The general theory is discussed in the Survey by Goldblatt (see the references).
## Examples
+-- {: .num_example}
###### Example
**BAOs from frames.**
Let $\mathfrak{F} = (W ,R)$ be a [[frame (modal logic)|frame]]. We define on the power set Boolean algebra, $\mathbb{P}(W)$, the operator $m$ by, if $T\subseteq W$,
$$m(T) = \{w \in W : \exists t\in T, R w t \}$$
It perhaps pays to interpret this in the case where $R$ is a [[preorder]] and when it is an equivalence relation. In the first case, this will be the set of states less than or equal to something in $T$, in the second it is the union of all equivalence classes that contain an element of $T$.
+-- {: .num_lemma}
###### Lemma
The function $m$ is a normal operator.
=--
The proof is a simple manipulation of the definitions.
The dual operator $l$ is given by $l(T) = \{w\in W\mid \forall t\in T \neg R w t\}$. (Again look at this for the preorder and equivalence frame cases.)
It is easy to extend this example to $\mathfrak{F} = (W ,R_1,\ldots, R_n)$ with the result being a BAO of type $n$.
=--
+-- {: .num_example}
###### Example
**The Lindenbaum-Tarski algebra of a modal logic.**
Suppose $\Lambda \subseteq \mathcal{L}_\omega(n)$ is a [[normal modal logic]], then its [[Lindenbaum-Tarski algebra]] has a natural BAO structure, for which see the above page.
=--
##Varieties of modal and polymodal algebras
The following is a list of some of the main equationally defined classes of (poly)modal algebras. (For convenience each has been given a separate entry.)
* [[temporal algebras]];
* [[closure algebras]];
* [[monadic algebras]].
## Related concepts
* [[geometric model for modal logic]]
* [[modal type theory]]
## References
General books on modal logics that include information on algebraic models include:
* [[Patrick Blackburn]], M. de Rijke and Y. [[Venema]], _Modal Logic_, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.
* [[Marcus Kracht]], _Tools and Techniques in Modal Logic,_ Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999.
There is an excellent short survey article (versions of which are available on the web):
* [[Robert Goldblatt]], _Algebraic Polymodal Logic: A Survey_, the Logic Journal of the
IGPL, 8, (2000) pages 393–450, Special Issue on Algebraic Logic, edited by Istvan Nemeti and
Ildiko Sain.
Discussion of modal logic in terms of [[coalgebra]] and [[terminal coalgebra of an endofunctor]] is in
* Corina Cirstea, Alexander Kurz, Dirk Pattinson, Lutz Schröder and Yde Venema, _Modal logics are coalgebraic_ ([pdf](http://eprints.soton.ac.uk/267144/1/ModalCoalgRev.pdf))
[[!redirects modal algebra]]
[[!redirects modal algebras]]
[[!redirects polymodal algebra]]
[[!redirects polymodal algebras]]
[[!redirects algebraic model for modal logic]]
[[!redirects algebraic model for modal logics]]
[[!redirects algebraic models for modal logic]]
[[!redirects algebraic models for modal logics]]
|
algebraic model structures - table | https://ncatlab.org/nlab/source/algebraic+model+structures+-+table | **Algebraic model structures:** [[Quillen model structures]], mainly on [[locally presentable categories]], and their constituent [[categories with weak equivalences]] and [[weak factorization systems]], that can be equipped with further algebraic structure and "freely generated" by small data.
| structure | small-set-generated | small-category-generated | algebraicized |
| -- | -- | -- | -- |
| [[weak factorization system]] | [[combinatorial wfs]] | [[accessible wfs]] | [[algebraic wfs]] |
| [[model category]] | [[combinatorial model category]] | [[accessible model category]] | [[algebraic model category]] |
| [[transfinite construction of free algebras|construction method]] | [[small object argument]] | same as $\to$ | [[algebraic small object argument]] |
|
algebraic number | https://ncatlab.org/nlab/source/algebraic+number |
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### Context
#### Arithmetic
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[[!include arithmetic geometry - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
An **algebraic number** is a [[root]] of a non-zero [[polynomial]] with [[integer]] [[coefficients]] (or, equivalently, with [[rational number|rational]] coefficients). Equivalently, an element $\alpha$ of a [[field extension]] $K$ of the [[rational numbers]] $\mathbb{Q}$ is *algebraic* if the [[subfield]] $\mathbb{Q}(\alpha)$ is a finite degree [[field extension|extension]], i.e., is [[finite dimensional vector space|finite-dimensional]] as a vector space over $\mathbb{Q}$.
Since the [[rational numbers]] are a [[subfield]] of the [[complex numbers]], and since the [[complex numbers]] are an [[algebraically closed field]], algebraic numbers are naturally regarded as a sub-field of [[complex numbers]]
$$
\mathbb{Q} \hookrightarrow \mathbb{C}
\,.
$$
$,$
<center>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Algebraicszoom.png/1200px-Algebraicszoom.png" width="490"/>
</center>
> **Visualisation of the (countable) field of algebraic numbers in the complex plane.** Colours indicate degree of the polynomial the number is a root of (red = linear, i.e. the rationals, green = quadratic, blue = cubic, yellow = quartic...). Points becomes smaller as the integer polynomial coefficients become larger. View shows integers 0,1 and 2 at bottom right, $+i$ near top.
> due to Stephen J. Brooks [here](https://en.wikipedia.org/wiki/File:Algebraicszoom.png)
But the collection of all algebraic numbers forms itself already an [[algebraically closed field|algebraically closed]] [[field]], typically denoted $\overline{\mathbb{Q}}$, as this is the [[algebraic closure]] of the field $\mathbb{Q}$ of [[rational numbers]]. This also follows easily from the equivalent definition of algebraic numbers in terms of finite degree extensions. The [[absolute Galois group]] $Gal(\overline{\mathbb{Q}}, \mathbb{Q})$ is peculiar, see [there](absolute+Galois+group#OfTheRationalNumbers).
Given a [[field]] $k$, an algebraic __[[number field]]__ $K$ over $k$ is a finite-degree extension of $k$. By default, the term "algebraic number field" means an algebraic number field over the rational numbers. If $\alpha$ is an algebraic number over $\mathbb{Q}$ then $\mathbb{Q}[\alpha]$ is a number field, however the field of all algebraic numbers is *not* a number field.
An [[algebraic integer]] is a root of a [[monic polynomial]] with integer coefficients. Equivalently, an element $\alpha$ of a field extension $K$ of $\mathbb{Q}$ is an algebraic integer if the [[ring]] $\mathbb{Z}[\alpha]$ is of finite [[rank]] as a $\mathbb{Z}$-module. It follows easily from this characterization that the collection of all algebraic integers forms a commutative ring.
## Related concepts
* [[algebraic number theory]]
* **algebraic number**, [[algebraic integer]]
* [[number field]]
* [[transcendental number]]
## References
See also
* Wikipedia, _[Algebraic numbers](https://en.wikipedia.org/wiki/Algebraic_number)_
[[!redirects algebraic number]]
[[!redirects algebraic numbers]] |
algebraic number theory | https://ncatlab.org/nlab/source/algebraic+number+theory |
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#### Algebra
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[[!include higher algebra - contents]]
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#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Algebraic number theory studies [[algebraic numbers]], [[number fields]] and related [[algebra|algebraic]] [[structures]].
An [[algebraic number]] is a [[root]] of a [[polynomial]] [[equation]] with [[integer]] [[coefficients]] (or, equivalently with [[rational number|rational]] coeffients). An [[algebraic integer]] is a root of a [[monic polynomial]] with integer coefficients. Given a [[field]] $k$ a (algebraic) __[[number field]]__ $K = k[P]$ over $k$ is the minimal [[field]] containing all the roots of a given polynomial $P$ with coefficients in $k$. Usually one considers algebraic number fields over rational numbers.
The main direction in algebraic number theory is the [[class field theory]] which roughly studies finite abelian extensions of number fields. The one dimensional class field theory stems from the ideas of Kronecker and Weber, and results of Hilbert soon after them. Main results of the theory belong to the first half of the 20th century (Hilbert, Artin, Tate, Hasse...) and are quite different for the [[local field]] from the [[global field]] case. Generalizations for higher dimensional fields came later under now active higher class field theory, which is usually formulated in terms of algebraic K-theory and is closely related to deep questions of algebraic geometry (Tate, Kato, Saito etc.).
## Related entries
The circle of $n$Lab entries belonging or related closely to algebraic number theory is in its infancy, and the partial list of entries some of which are started and most of which are to be created should include (the entries grouped by similarity)
* [[field]], [[characteristic]], [[division ring]], [[ideal]]
* [[integer]], [[rational number]], [[irrational number]], [[period]]
* [[p-adic number]], [[profinite group]], [[idele]], [[adele]]
* [[separable extension]], [[normal extension of fields]], [[Galois group]], [[Galois extension]], [[abelian extension of fields]], [[cyclotomic field]]
* [[Brower group]], [[Galois cohomology]]
* [[Dedekind ring]], [[principal ideal ring]], [[unique factorization domain]], [[integral closure]]
* [[algebraic number]], [[algebraic closure]], [[number field]]
* [[discriminant]], [[resultant]], [[Euclid algorithm]]
* [[Diophantine equation]], [[Matiyasevich theorem]]
* [[function field]]
* [[discrete valuation]], [[valuation ring]], [[valuation ideal]], [[archimedean valuation]]
* [[local field]], [[global field]], [[complete field]]
* [[conductor]], [[ideal class group]], [[Picard group]], [[Milnor K-group]]
* [[class field theory]], [[global class field theory]], [[local class field theory]], [[higher class field theory]], [[Hilbert class field]]
* [[reciprocity law]], [[Artin reciprocity law]], [[Weil reciprocity law]]
* [[arithmetic scheme]], [[Arakelov geometry]], [[field with one element]]
* [[L-function]], [[motive]], [[Riemann conjecture]], [[algebraic K-theory]], [[Grothendieck Galois theory]]
## Literature
* {#FroehlichCassels67} [[Albrecht Fröhlich]], [[J. W. S. Cassels]] (eds.), _Algebraic number theory_, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, [pdf](https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf), [errata pdf](https://www.ma.imperial.ac.uk/~buzzard/errata.pdf) by [[Kevin Buzzard]])
* {#Cassels86} [[J. W. S. Cassels]], *Local Fields*, Cambridge University Press, 1986 (ISBN:9781139171885, [doi:10.1017/CBO9781139171885](https://doi.org/10.1017/CBO9781139171885))
* {#Neukirch92} [[Jürgen Neukirch]], _Algebraische Zahlentheorie_ (1992), English translation _Algebraic Number Theory_, Grundlehren der Mathematischen Wissenschaften 322, 1999 ([pdf](http://www.plouffe.fr/simon/math/Algebraic%20Number%20Theory.pdf))
* [[Albrecht Fröhlich]], Martin J. Taylor, _Algebraic number theory_, Cambridge Studies in Advanced Mathematics 27, 1993
* [[James Milne]], *Algebraic number theory*, 2020 ([pdf](https://www.jmilne.org/math/CourseNotes/ANT.pdf))
The following survey of Connes-Marcolli work has an accessible quick introduction to algebraic number theory
* P. Almeida, _Noncommutative geometry and arithmetics_, Russian Journal of Mathematical Physics 16, No. 3, 2009, pp. 350–362, [doi](http://dx.doi.org/10.1134/S1061920809030030), see also nLab:arithmetic and noncommutative geometry
See also
* Alexander Schmidt, _Higher dimensional class field theory from a topological point of view_, [page](http://www.mathi.uni-heidelberg.de/~schmidt/papers/schmidt21-en.html)
|
algebraic quasi-category | https://ncatlab.org/nlab/source/algebraic+quasi-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
#### Quasi-category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
An **algebraic quasi-category** is a [[quasi-category]] equipped with a _choice_ of (inner) [[horn]] fillers.
Algebraic quasi-categories give a [[algebraic definition of higher categories|algebraic definition of]] [[(∞,1)-categories]].
For more see the section [Algebraic fibrant models for higher categories](http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects#AlgebaicHigherCategories) at [[model structure on algebraic fibrant objects]].
[[!redirects algebraic quasi-category]]
[[!redirects algebraic quasi-categories]]
[[!redirects algebraic quasicategory]]
[[!redirects algebraic quasicategories]]
|
algebraic set theory | https://ncatlab.org/nlab/source/algebraic+set+theory | [[!redirects Algebraic set theory]]
#Contents#
* table of contents
{:toc}
## Idea
The new insight taken as a starting point in _algebraic set theory_ (AST) is that [[models]] of [[set theory]] are in fact [[algebra over an algebraic theory|algebras]] for a suitably presented [[algebraic theory]], and that many familiar [[set theory|set-theoretic]] conditions (such as well-foundedness) are thereby related to familiar algebraic ones (such as [[free functor|freeness]]). ([Awodey](#Awodey))
##Description of the project##
Also sometimes called _categorical_ (meaning category-theoretic) _set theory_, _algebraic set theory_ started to be developed in 1988 by [[André Joyal]] and [[Ieke Moerdijk]] and was first presented in detail as a book in 1995 by them. AST is a robust framework based on [[category theory]] to study and organize [[set theory|set theories]] and to construct [[model of a set theory|models of set theories]]. The aim of AST is to provide a uniform categorical semantics or description of set theories of different kinds (classical or [[constructive mathematics|constructive]]; bounded, [[predicative mathematics|predicative]] or impredicative; [[well-founded set|well founded]] or ill founded, ...), the various constructions of the [[von Neumann hierarchy|cumulative hierarchy]] of [[pure set]]s, [[forcing]] models, [[sheaf]] models and [[realizability|realisability]] models.
Instead of focusing on categories of sets AST focuses on categories of [[class]]es. The basic tool of AST is the notion of a [[category with class structure]] (a category of classes equipped with a class of small maps (the intuition being that their fibres are small in some sense), powerclasses and a [[universe|universal object]]) which provides an axiomatic framework in which models of set theory can be constructed. The notion of a class category permits both the definition of [[Zermelo-Fraenkel algebra|ZF-algebras]] and related structures expressing the idea that the hierarchy of sets is an algebraic structure on the one hand and the interpretation of the first order logic of elementary set theory on the other. The subcategory of sets in a class category is an [[topos|elementary topos]] and every elementary topos occurs as the topos of sets in some class category.
The class category itself always embeds into the [[ideal completion]] of a topos. The interpretation of the logic is that in every class category the universe is a model of basic intuitionistic set theory $\mathbf{BIST}$ that is logically [[complete model|complete]] with respect to class category models. Therefore class categories generalize both topos theory and intuitionistic set theory. AST founds and formalizes set theory on the ZF-algebra with operations union and successor (singleton) instead of on the membership relation. The [[ZFC|ZF axioms]] are nothing but a description of the free ZF-algebra just as the [[natural numbers object|Peano axioms]] are a description of the free [[monoid]] on one generator. In this perspective the models of set theory are algebras for a suitably presented [[algebraic theory]]. Using an auxiliary notion of small map it is possible to extend the axioms of a topos and provide a general theory for uniformly constructing models of set theory out of toposes.
##Naming##
There are two reasons for referring to this research as "algebraic set theory": The first reason is that the models of set theory that are produced by these methods are algebras for an abstractly presented "theory", in a precise, technical sense known to category theorists as a [[monad]]. The notion of an algebra for a monad subsumes and generalizes that of a model for a conventional algebraic theory, such as groups, rings, modules, etc. Indeed, the first significant work in this style on the applications of [[category theory]] to the study of [[set theory]] was the monograph ([Joyal-Moerdijk 1995](#JoyalMoerdijk)) _Algebraic set theory_.
The second reason is that we believe the locution "algebraic logic" should properly refer to categorical logic rather than just the logic of Boole and his modern proponents, since categorical logic subsumes such lattice theoretic methods and not the other way around. Hence the term "algebraic set theory" rather than "categorical set theory". This is in keeping with the use of "algebraic" to mean, essentially, "functorial" in modern algebraic topology, algebraic geometry, etc. (Awodey, [Why "algebraic set theory"?](http://www.phil.cmu.edu/projects/ast/whyast.html))
### Stack semantics
[[michaelshulman:stack semantics]] provides a structural and somewhat more uniform way of treating "classes" in [[topos]] theory.
## See also
* [[category of classes]]
* [[category with class structure]]
## References
Introductions and further pointers are at
* [Algebraic Set Theory Website](http://www.phil.cmu.edu/projects/ast/index.html)
Two introductions are:
* [[Steve Awodey]], _An Outline of Algebraic Set Theory_ ([archived pdf](https://web.archive.org/web/20070614063821/https://caae.phil.cmu.edu/projects/ast/Papers/awodey_outline.pdf))
* [[Benno van den Berg]], [[Ieke Moerdijk]], _A Unified Approach to Algebraic Set Theory_ [arXiv](http://arxiv.org/abs/0710.3066)
A standard textbook is
* [[André Joyal]], [[Ieke Moerdijk]], _Algebraic set theory_, Cambridge University Press (1995)
{#JoyalMoerdijk}
A reference for BIST is
* [[Steve Awodey]], [[Carsten Butz]], [[Alex Simpson]], [[Thomas Streicher]], ["Relating first-order set theories, toposes and categories of classes"](https://doi.org/10.1016/j.apal.2013.06.004), _Annals of Pure and Applied Logic_ 165 (2014)
[[!redirects algebraic set theories]]
[[!redirects small maps]]
[[!redirects small map]]
|
algebraic small object argument | https://ncatlab.org/nlab/source/algebraic+small+object+argument | # The algebraic small object argument
* table of contents
{: toc}
## Idea
The ordinary [[small object argument]] is a way of constructing (often [[combinatorial wfs|combinatorial]]) [[weak factorization systems]] from a set of "generators". While powerful and useful, it has several defects, such as:
* Its result is not uniquely determined. In particular, it does not converge: we just go on until we've gone on long enough, but going on longer would produce a different result.
* Relatedly, it has no universal property. This makes it hard to deal with category-theoretically.
* It does not suffice to construct every weak factorization system, not even every [[accessible wfs|accessible]] one.
The algebraic small object argument is a refinement of the small object argument, due to [Garner](#Garner), that remedies these defects.
## Definition
...
## Properties
* The result of the algebraic small object argument is an [[algebraic weak factorization system]], which is "freely generated" by the input data in an appropriate sense.
* If the input data is a set of arrows (rather than a category or double category), then the algebraic right class consists of the [[algebraically injective objects]] for the generating class of arrows.
* If the underlying category is [[locally presentable category|locally presentable]], then this awfs is in particular an [[accessible weak factorization system]]. Conversely, any accessible wfs can be generated by the algebraic small object argument; see [Rosicky](#Rosicky).
* Not every accessible *algebraic* wfs can be generated by the algebraic small object argument as above: every accessible wfs admits *some* algebraic realization that's generated by the algebraic SOA above, but it could admit other algebraic realizations that are not. However, there is a further refinement of the algebraic SOA, due to [Bourke and Garner](#BourkeGarnerI), that takes as input a *double* category, and does suffice to generate all accessible algebraic wfs.
## Related concepts
* [[small object argument]]
* [[algebraically injective object]]
* [[accessible weak factorization system]]
* [[algebraic weak factorization system]]
[[!include algebraic model structures - table]]
## References
* {#Garner} [[Richard Garner]], _Understanding the small object argument_, [arXiv](http://arxiv.org/abs/0712.0724).
* [[Emily Riehl]], _Algebraic model structures_, ([arXiv:0910.2733](http://arxiv.org/abs/0910.2733)).
* [[Thomas Athorne]], _The coalgebraic structure of cell complexes_, [TAC](http://www.tac.mta.ca/tac/volumes/26/11/26-11abs.html)
* [[Tobias Barthel]] and [[Emily Riehl]], *On the construction of functorial factorizations for model categories*, Algebr. Geom. Topol. Volume 13, Number 2 (2013), 1089-1124, [projecteuclid](https://projecteuclid.org/euclid.agt/1513715550), [arxiv](https://arxiv.org/abs/1204.5427)
* {#BourkeGarnerI} [[John Bourke]] and [[Richard Garner]], _Algebraic weak factorisation systems I: accessible AWFS_, [arXiv](http://arxiv.org/abs/1412.6559).
* [[John Bourke]] and [[Richard Garner]], _Algebraic weak factorisation systems II: categories of weak maps_, [arXiv](http://arxiv.org/abs/1412.6560).
* {#Rosicky} J. Rosicky, _Accessible model categories_, [arxiv](https://arxiv.org/abs/1503.05010)
[[!redirects Garner's small object argument]]
[[!redirects algebraic SOA]]
|
algebraic space | https://ncatlab.org/nlab/source/algebraic+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _algebraic space_ is an [[object]] in the [[sheaf topos]] over the [[fppf-site]], that has [[representable morphism of stacks|representable]] [[diagonal]] and an [[étale cover]] ([[atlas]]) by a [[scheme]].
In [[algebraic geometry]] one can glue [[affine schemes]] in various topologies; this way one obtains various kinds of locally affine [[ringed space]]s. For example, [[scheme]]s locally affine in [[Zariski topology]]. [[etale topology|Étale topology]] is finer than Zariski, hence the category of locally affine (ringed) spaces in étale topology is larger than the category of schemes. __Algebraic spaces__ make a category which includes the category of all schemes and is close to the category of locally affine spaces in [[étale topology]], namely it consists of those [[ringed space]]s which may be obtained as a [[quotient object|quotient]] of a scheme $S$ by an [[equivalence relation]] $R\subset S\times S$ which is a closed subscheme, and whose projections $p_1,p_2: R\to S$ are étale morphisms of schemes.
## Definition
Write $C_{fppf}$ for the [[fppf-site]] (over some [[scheme]], as desired).
+-- {: .num_defn}
###### Definition
An **algebraic space** is
* an object $X \in Sh(C_{fppf})$ in the [[sheaf topos]];
* whose [[diagonal]] morphism $X \to X \times X$ is [[representable morphism of stacks|representable]];
* and for which there exists $U \in C$ and a morphism $U \to X$ which is
* surjective;
* [[étale morphism|étale]].
=--
In this form this appears as [de Jong, def. 35.6.1](#deJong).
## Properties
### Characterization as presheaf on affine schemes
* [[Artin representability theorem]]
## Related concepts
* [[algebraic stack]]
## References
Algebraic spaces are the topic of part 4 (tag 0ELT) in
* [[Aise Johan de Jong]], _[[The Stacks Project]]_
{#deJong}
Monographs:
* [[Donald Knutson]], *Algebraic spaces*, Lecture Notes in Mathematics, **203**, Springer (1971) [[doi:10.1007/BFb0059750](https://link.springer.com/book/10.1007/BFb0059750)]
* [[Martin Olsson]], *Algebraic Spaces and Stacks*, Colloquium Publications **62** (2016) [[doi:10.1090/coll/062](https://doi.org/10.1090/coll/062), [ISBN:978-1-4704-2798-6](https://bookstore.ams.org/coll-62)]
Lecture notes:
* [[James Milne]], section 7 of _[[Lectures on Étale Cohomology]]_
* G. B. Winters, _An elementary lecture on algebraic spaces_, in: P. Salmon (eds) Categories and Commutative Algebra. C.I.M.E. Summer Schools __58__ (Varena 1971), C.I.M.E., Ed. Cremonese, Roma 1973; reprint Springer 2010, [doi 2010](https://doi.org/10.1007/978-3-642-10979-9_9)
Definition in [[E-∞ geometry]] is in
* [[Jacob Lurie]], section 1.3 of _[[Quasi-Coherent Sheaves and
Tannaka Duality Theorems]]_
Some related MO questions:
* [why-is-this-not-an-algebraic-space](http://mathoverflow.net/questions/9043/why-is-this-not-an-algebraic-space),
* [Can an algebraic space fail to have a universal map to a scheme?](http://mathoverflow.net/questions/4587/can-an-algebraic-space-fail-to-have-a-unviersal-map-to-a-scheme)
* [What are the Benefits of Using Algebraic Spaces over Schemes?](http://mathoverflow.net/q/3194/447)
* [Commutative rings to algebraic spaces in one jump?](http://mathoverflow.net/q/11226/447)
[[!redirects algebraic spaces]] |
algebraic spaces > history | https://ncatlab.org/nlab/source/algebraic+spaces+%3E+history | < [[algebraic spaces]]
[[!redirects algebraic spaces -- history]] |
algebraic stack | https://ncatlab.org/nlab/source/algebraic+stack |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
An **algebraic stack** is essentially a [[geometric stack]] on the [[étale site]].
Depending on details, this is a [[Deligne-Mumford stack]] or a more general [[Artin stack]] in the traditional setup of [[algebraic space]]s.
## Definition
Let $C_{fppf}$ be the [[fppf-site]] and $\mathcal{E} = Sh_{(2,1)}(C_{fppf})$
the [[(2,1)-topos]] of [[stack]]s over it.
+-- {: .un_defn}
###### Definition
An **algebraic stack** is
* an object $\mathcal{X}\in Sh_{(2,1)}(C_{fppf})$;
* such that
1. the [[diagonal]] $\mathcal{X} \to \mathcal{X} \times \mathcal{X}$
is [[representable morphism of stacks|representable]] by [[algebraic space]]s;
1. there exists a [[scheme]] $U \in Sh(C_{fppf}) \hookrightarrow Sh_{(2,1)}(C_{fppf})$ and a morphism $U \to \mathcal{X}$ which is a surjective and
[[smooth morphism of schemes|smooth morphism]].
=--
This appears in this form as ([deJong, def. 47.12.1](#deJong)).
+-- {: .un_defn}
###### Definition
A **smooth algebraic groupoid** is an [[internal groupoid]] in [[algebraic space]]s such that source and target maps are [[smooth morphism of schemes|smooth morphisms]].
=--
This appears as ([deJong, def. 47.16.2](#deJong)).
Notice that every [[internal groupoid]] in [[algebraic spaces]] represents a [[(2,1)-presheaf]] on the [[fppf-site]]. We shall not distinguish between the groupoid and the [[stackification]] of this presheaf, called the **quotient stack** of the groupoid.
+-- {: .un_theorem}
###### Theorem
Every algebraic stack is equivalent to a smooth algebraic groupoid and every smooth algebraic groupoid is an algebraic stack.
=--
This appears as ([deJong, lemma 47.16.2, theorem 47.17.3](#deJong)).
## Properties
* [[Tannaka duality for geometric stacks]].
## Examples
[[orbifold|Orbifolds]] are an example of an [[Artin stack]]. For orbifolds the stabilizer groups are [[finite group]]s, while for Artin stacks in general they are [[algebraic group]]s.
## Generalizations
### Noncommutative spaces
A noncommutative generalization for [[Q-category|Q-categories]] instead of [[Grothendieck topology|Grothendieck topologies]], hence applicable in noncommutative geometry of Deligne--Mumford and Artin stacks can be found in ([KontsevichRosenberg](#KontsevichRosenberg)).
## Examples
* [[projective stack]]
## Related concepts
* [[stack]], [[moduli stack]]
* [[lisse-étale site]]
* [[geometric stack]]
* **algebraic stack**
* [[topological stack]]
* [[differentiable stack]]
* [[complex analytic stack]]
* [[geometric ∞-stack]]
## References
The original articles:
* {#DeligneMumford69} [[Pierre Deligne]], [[David Mumford]], *The irreducibility of the space of curves of given genus*, Publications Mathématiques de l'IHÉS (Paris) **36** (1969) 75-109 [[doi:10.1007/BF02684599](https://doi.org/10.1007/BF02684599), [numdam:PMIHES_1969__36__75_0](http://www.numdam.org/item?id=PMIHES_1969__36__75_0)]
> (cf. *[[Deligne-Mumford stack]]*)
* [[Michael Artin]], *Versal deformations and algebraic stacks*, Invent Math **27** (1974) 165–189 [[doi:10.1007/BF01390174](https://doi.org/10.1007/BF01390174), [eudml:142310](https://eudml.org/doc/142310), [pdf](http://math.uchicago.edu/~drinfeld/Artin_on_stacks.pdf)]
> (cf. *[[Artin stack]]*)
Early review:
* [[Angelo Vistoli]], Appendix of: _Intersection theory on algebraic stacks and on their moduli spaces_, Inventiones mathematicae **97** (1989) 613–670 [[doi:10.1007/BF01388892](https://doi.org/10.1007/BF01388892)]
Monographs and review:
* {#LaumontMoret-Bailly} [[Gérard Laumon]], [[Laurent Moret-Bailly]], _Champs algébriques_, Ergebn. der Mathematik und ihrer Grenzgebiete **39**, Springer (2000) [[doi:10.1007/978-3-540-24899-6](https://doi.org/10.1007/978-3-540-24899-6)]
* [[Angelo Vistoli]], _Grothendieck topologies, fibered categories and descent theory_ [[math.AG/0412512](http://arxiv.org/abs/math/0412512), [MR2223406](http://www.ams.org/mathscinet-getitem?mr=2223406)] in: Fantechi et al. (eds.), _Fundamental algebraic geometry. Grothendieck's [[FGA explained]]_, Mathematical Surveys and Monographs __123__, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)]
> (focusing on the incarnation of stacks, under the [[Grothendieck construction]], as [[Grothendieck fibrations]])
* [[Bertrand Toen]], _[[Master course on algebraic stacks]]_ (2005)
* [[Frank Neumann]], *Algebraic Stacks and Moduli of Vector Bundles*, impa (2011) [[pdf](https://impa.br/wp-content/uploads/2017/04/PM_36.pdf), [pdf](https://www.cimat.mx/~luis/seminarios/Pilas-algebraicas/neumann-Stacks.pdf)]
* [[Michael Groechenig]], *Algebraic Stacks*, Lecture notes (2014) [[web](http://individual.utoronto.ca/groechenig/stacks.html), [pdf](http://individual.utoronto.ca/groechenig/stacks.pdf), [[Groechenig-AlgebraicStacks.pdf:file]]]
* [[Martin Olsson]], *Algebraic Spaces and Stacks*, Colloquium Publications **62** (2016) [[doi:10.1090/coll/062](https://doi.org/10.1090/coll/062), [ISBN:978-1-4704-2798-6](https://bookstore.ams.org/coll-62)]
* [[Daniel Halpern-Leistner]], *Moduli theory*, Lecture notes (2020) [[pdf](http://pi.math.cornell.edu/~danielhl/moduli_theory_notes.pdf), [[Halpern-Leistner_ModuliTheory.pdf:file]]]
See also:
* {#deJong} [[Aise Johan de Jong]], _[[The Stacks Project]]_, [[tag:026K](https://stacks.math.columbia.edu/tag/026K)]
* [[Fredrik Meyer]], *Notes on algebraic stacks* (2013) [[pdf](https://fredrikmeyer.net/uio-math/algstacks.pdf), [[Meyer_AlgebraicStacks.pdf:file]]]
Brief overview:
* Anatoly Preygel, _Algebraic stacks_, Seminar notes: Quantization of Hitchin's integrable system and Hecke eigensheaves, 2009, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept15-17%28stacks%29.pdf).
The noncommutative version is discussed in
* [[Maxim Kontsevich]], [[Alexander Rosenberg]], _Noncommutative stacks_ , preprint MPIM2004-37 ([dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=2305) [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2333))
{#KontsevichRosenberg}
[[!redirects algebraic stack]]
[[!redirects algebraic stacks]] |
algebraic structure | https://ncatlab.org/nlab/source/algebraic+structure | Algebraic structure may mean either
* the same as [[algebra]] in the sense of [[universal algebra]];
or
* the [[extra structure]] which such an algebra has with respect to the underlying object.
There are several notions of an __algebraic structure__ on an object of some category or higher category, which differ in generality. It may be to be an algebra over an [[algebraic theory]], [[algebra over an operad]] (or higher operad) or an [[algebra over a monad]], or over a [[PROP]], over a [[properad]] etc. See also [[variety of algebras]].
There is also an older notion of an algebraic structure/algebra as a model for a one-sorted [[theory]] where the only relation symbols in the language involved are $\epsilon$ and equality (with standard interpretation in models). This notion includes for example fields which are not an algebraic theory in the sense of monads (because there are no free objects in the case of fields, i.e. [[Field]], the category of fields is not [[monadic functor|monadic]] over the category of sets).
There is a [[forgetful functor]] from the category of algebras/algebraic structures of some type (in any of the above formalisms) to the original category. This functor __forgets structure__ in the sense of [[stuff, structure, property]]. We say that a functor in the base category __preserves some algebraic structure__ if it lifts to the corresponding category of algebras.
[[!redirects algebraic structures]] |
algebraic surface | https://ncatlab.org/nlab/source/algebraic+surface |
## Idea
An [[algebraic variety]] of [[dimension]] 2.
## Examples
* [[K3 surface]]
## Related concepts
* dimension 1-case: [[algebraic curve]]
* [[complex surface]]
[[!redirects algebraic surfaces]]
|
algebraic theories in functional analysis | https://ncatlab.org/nlab/source/algebraic+theories+in+functional+analysis |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--
At the moment, this is a "place holder" page. I ([[Andrew Stacey]]) want to learn about the appearance of [[algebraic theories]] in [[functional analysis]] and shall record what I learn here. A preliminary outline is to find out about the following statements:
1. The category of [[Banach spaces]] with linear [[short maps]] is not [[monadic category|monadic]] over Set. The "nearest" algebraic theory is that of [[totally convex spaces]].
2. The category of [[Banach algebras]] is also not algebraic.
3. The category of $C^*$-[[C-star-algebra|algebras]] is algebraic.
### Banach Spaces ###
We consider the category of [[Banach spaces]] with linear [[short maps]]. That is, this is the category $\operatorname{Ban}$ with:
* Objects: Banach spaces over $\mathbb{R}$
* Morphisms $E \to F$: Linear short maps. That is, bounded linear transformations $T \colon E \to F$ such that $\|T\| \le 1$
We define a functor $B \colon \operatorname{Ban} \to \operatorname{Set}$ sending a Banach space to its unit ball. Since linear short maps $E \to F$ take the unit ball of $E$ into the unit ball of $F$, this is well-defined.
There is a functor in the opposite direction which assigns to a set the "free" Banach space on that set. That is, it assigns to a set $X$ the Banach space $\ell^1(X)$ of all absolutely summable sequences indexed by elements of $X$. It is a standard result that such a sequence must have countable support, no matter how large $X$ is.
+-- {: .num_lemma #bspadj}
###### Lemma
$\ell^1$ is left adjoint to $B$.
=--
+-- {: .proof}
###### Proof
We need to define the adjunction natural transformations: $\eta_X \colon X \to B \ell^1(X)$ and $\epsilon_E \colon \ell^1(B E) \to E$. The first is the map which assigns to $x$ the sequence $(\delta_{x y})$ which is $1$ at $x$ and $0$ elsewhere. The second is the summation map which assigns to an absolutely summable sequence $(a_e)$ indexed by $e \in B E$ its sum, $\sum a_e e$.
=--
This adjunction defines a [[monad]] over $\operatorname{Set}$. Let us spell out the details. The functor $T \colon \operatorname{Set} \to \operatorname{Set}$ sends a set $X$ to the unit ball of $\ell^1(X)$. That is, an element of $T(X)$ is a weighted (formal) sum of elements of $X$, $\sum a_x$, such that $\sum |a_x| \le 1$. The unit for the monad sends an element $x \in X$ to the delta sequences in $T(X)$. The product, $\mu$, takes a "sum of sums" and evaluates them. That is, given a formal sum $\sum a_s$ where each $s$ is of the form $\sum s_x$, $\mu(\sum a_s) = \sum b_x$ where $b_x = \sum_s s_x$.
+-- {: .query}
AS: I think! I need to check exactly how the product works in this example but I'm just getting the basic sketch down first.
=--
The key question is whether or not $\operatorname{Ban}$ is (equivalent to) the category of algebras for this monad. That is, is $B \colon \operatorname{Ban} \to \operatorname{Set}$ _tripleable_? If not (as it will turn out), how close is it?
[[Beck's tripleability theorem]] gives three conditions for a functor to be tripleable. We already have one (the adjunction), let us show that the second also holds.
+-- {: .num_lemma #bsprefiso}
###### Lemma
$B \colon \operatorname{Ban} \to \operatorname{Set}$ reflects isomorphisms.
=--
+-- {: .proof}
###### Proof
Let $T \colon E \to F$ be a linear short map which induces an isomorphism on the unit balls of $E$ and $F$. It is evident that it is therefore a bijection from the underlying set of $E$ to that of $F$. Hence, by the open mapping theorem, it is a linear homeomorphism. It remains to show that $\|T(x)\| = \|x\|$ (so that its inverse is a short map as well). This is simple to show: if we had some $x \in E$ with $\|x\| = 1$ but $\|T(x)\| \lt 1$ (if it fails, it must fail that way as $T$ is short) then there would be some $\lambda \gt 1$ such that $\|T(\lambda x)\| \le 1$. As $B(T) \colon B E \to B F$ is surjective, there is some $y \in B E$ such that $T(y) = T(\lambda x)$. But as $\lambda \gt 1$, $\lambda x \notin B E$ so $\lambda x \ne y$, contradicting the injectivity of $T$. (Incidentally, this argument is valid [[constructive mathematics|constructively]]; it is a property of located [[real numbers]] that any number that is neither greater nor smaller than $1$ must equal $1$.)
=--
+-- {: .query}
AS: To be continued ...
=--
#### References: ####
The above is essentially [[Andrew Stacey|my]] "notes" on reading the following (and whatever necessary to understand the following):
Section 4.4 of _Toposes, Triples, and Theories_ by Barr and Wells ([TAC reprint](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html))
## References
* _On the equational theory of $C^*$-algebras_, Pelletier, J. Wick and Rosický, J., [MR1223636](http://www.ams.org/mathscinet-getitem?mr=1223636)
* Any more suggested by [this question on mathoverflow](http://mathoverflow.net/questions/9169/request-for-reference-banach-type-spaces-as-algebraic-theories) |
algebraic theory | https://ncatlab.org/nlab/source/algebraic+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
#### Higher algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
* table of contents
{: toc}
## Idea
An _algebraic [[theory]]_ is a concept in [[universal algebra]] that describes a specific type of algebraic gadget, such as [[group|groups]] or [[ring|rings]]. An individual group or ring is a _model_ of the appropriate theory. Roughly speaking, an algebraic theory consists of a specification of operations and laws that these operations must satisfy.
### Categorical formulation
Traditionally, algebraic theories were described in terms of [[logic|logical syntax]], as [[theory|first-order theories]] whose [[signatures]] have only function symbols, no relation symbols, and all of whose [[axioms]] are [[equational law]]s ([[universal quantifier|universally quantified]] [[equations]] between terms built out of these function symbols). Such descriptions may be viewed as _presentations_ of a theory, analogous to [[generators and relations]] as presentations of [[groups]]. In particular, different logical presentations can lead to equivalent mathematical objects.
In his thesis, [[Bill Lawvere]] undertook a more invariant description of (finitary) algebraic theories. Here _all_ the definable operations of an algebraic theory, or rather their equivalence classes modulo the equational axioms imposed by the theory, are packaged together to form the morphisms of a category with finite products, called a [[Lawvere theory]]. None of these operations are considered "primitive", so a Lawvere theory doesn't play favorites among operations.
This article is about generalized Lawvere theories. The article [[Lawvere theory]] treats the traditional notion of finitary, single-sorted Lawvere theories, with worked examples. The core of the present article is a working out of the precise connection between infinitary (multi-sorted) Lawvere theories and monads.
### Basic Intuitions
Intuitively, a Lawvere theory is the "generic category of products equipped with an object $x$ of given algebraic type $T$". For example, the Lawvere theory of groups is what you get by assuming a category with products and with a [[group object]] $x$ inside, and nothing more; $x$ can be considered "the generic group". Every object in the Lawvere theory is a finite power $x^n$ of the generic object $x$. The morphisms $x^n \to x$ are nothing but the $n$-ary operations it is possible to define on $x$.
In other words, if we abstract away from the usual set-theoretic semantics, and consider a model for the theory of groups to be _any_ category with finite products together with a specified group object inside, then the Lawvere theory of groups becomes a universal model of the theory, and carries all the information of the theory but independent of a particular presentation. In this way, theories and models of a theory are placed on an equal footing. A model of a Lawvere theory $T$ in a category with products $C$ is nothing but (i.e., is equivalent to) a product-preserving functor $T \to C$; where the generic object $x$ is sent to is the given model of $T$ in $C$. If $T$ is the Lawvere theory of groups, then a product-preserving functor $T \to Set$ is tantamount to an ordinary group.
The actual categorical construction of a Lawvere theory is described very easily and elegantly: it is the category opposite to the category of (finitely generated) free algebras of the theory. The free algebra on one generator becomes the generic object.
If theories and models are placed on an equal footing, then what feature sets "theories" _per se_ apart? In some very abstract sense, any category with products $C$ could be considered a theory, where the $C$-models in $D$ are product-preserving functors $C \to D$. Sometimes this is a useful point of view, but it is far removed from traditional syntactic considerations. To give a more "honest" answer, we remember that an ordinary (finitary, single-sorted) algebraic theory a la Lawvere is generated from a single object $x$, and that every other object should be (at least up to isomorphism) a finite power $x^n$. The exponent $n$ serves to keep track of arities of operations.
The generic "category of arities" $n$ is, in the finitary case, the category opposite to the category of finite sets (opposite because the $n$ appears contravariantly in powers $x^n$). This is also the Lawvere "theory of equality", or if you prefer the theory generated by an empty signature. The answer to the question "what sets theories apart" is that a Lawvere theory $T$ should come equipped with a product-preserving functor
$$x^{-}: FinSet^{op} \to T$$
that is essentially surjective (each object of $T$ is isomorphic to $x^n$ for some arity $n$). As we see below, this definition is a cornerstone to a very elegant theory of algebraic theories.
### Extensions
#### Infinitary operations
Lawvere's program can be extended to cover many theories with infinitary operations as well. In the best-behaved case, one has algebraic theories involving only operations of arity bounded by some [[cardinal number]] --- or, more precisely, belonging to some [[arity class]] --- and these can be understood category-theoretically with a suitable generalization of Lawvere theories. In this bounded case, the Lawvere theory can be described by a small category, and the category of models will be very well behaved, in particular it is a [[locally presentable category]]. In such cases there is a satisfying duality between syntax and semantics along the lines of [[Gabriel-Ulmer duality]].
Lawvere's program can to some degree be extended further: one can work with Lawvere theories which are locally small (not just small) categories. Here, the theory might not be bounded, but at least there is only a small set of operations of each arity. Examples of such large theories include
* The theory of algebras with arbitrary sums (one model of which is $[0,\infty]$),
* The theory of sup-lattices, in which there is one operation of each arity, and
* The theory of compact Hausdorff spaces, where the operations are parametrized by ultrafilters.
These examples go outside the bounded (small theory) case. Locally small theories in this sense are co-extensive with the notion of monad (on $Set$): there is a free-forgetful adjunction between $Set$ and the category of models, and algebras of the theory are equivalent to algebras of the monad.
In the worst case, there are algebraic theories where the number of definable operations explodes, so that there may be a proper class of operations of some fixed arity. In this case there are no free algebras, and Lawvere's reformulation no longer applies. An example is the theory of complete Boolean algebras. (Note: category theorists who define a category $U: A \to Set$ over sets to be [[algebraic category|algebraic]] if it is [[monadic functor|monadic]] would therefore not consider the variety of algebras in such cases to be "algebraic").
Further commentary on these aspects may be found in the dozen or so comments in [this thread](http://golem.ph.utexas.edu/category/2009/04/report_on_88th_peripatetic_sem.html#c023188), dated April 13 - May 7, 2009.
In summary, then, here is the connection between the logical and categorial descriptions, based on [Johnstone](#Johnstone), §§3.7&8. Say that a category $C$ is:
* _small algebraic_ if it is given by a (small) set of operation symbols and equations;
* _[[algebraic category|algebraic]]_ if it is given by a monad on the category of (small) sets;
* _large algebraic_ if it is given by a (possibly proper) class of operation symbols and equations.
Then any small algebraic category is algebraic, and any algebraic category is large algebraic, but neither implication may be reversed.
#### Multi-sorted operations
Lawvere theories can also be generalized to handle multi-sorted operations. If $S$ is a set of sorts, then multisorted operations are of the form
$$\prod_{s \in S} s^{n_s} \to t$$
so that arities are functors $n: S \to Set$, where $S$ is seen as a discrete category. Thus, an infinitary multi-sorted Lawvere theory $T$ involves an essentially surjective product-preserving functor
$$(Set^S)^{op} \to T$$
and the development goes through very much as in the single-sorted case.
#### Generalized Algebraic Theories
See _[[generalized algebraic theory]]_.
## Definition
For the moment we discuss the single-sorted case. The many-sorted case should be a straightforward extension.
For any [[cardinal]] $n$, let $[n]$ be a set of that cardinality (sometimes we just use $n$).
+-- {: .un_def}
######Definition:
A **Lawvere theory** or **algebraic theory** is a [[locally small category]] $C$ with small products that is equipped with an object $x$ such that the (unique-up-to-isomorphism) product-preserving functor
$$i: Set^{op} \to C: [1] \mapsto x$$
is essentially surjective.
=--
### Variations
Algebraic theories can be extended or specialized in various directions. Here are a few variations on the theme.
#### Essentially algebraic theories
_[[essentially algebraic theory|Essentially algebraic theories]]_ allow for partially-defined operations. Just as finitary algebraic theories can be understood as Lawvere theories, which live in the [[doctrine]] of [[cartesian monoidal category|cartesian monoidal categories]], so finitary essentially algebraic theories can be understood by a generalisation to [[finitely complete category|finitely complete categories]].
#### Multisorted algebraic theories
_[[multi-sorted theory|Multi-sorted theories]]_ allow for more than one sort or type in the theory.
Let $S$ be a set whose elements are called _sorts_. There is a canonical map
$$i: S \to Ob(Set/S)$$
which sends $s \in S$ to the object $s: 1 \to S$ in $Set/S$. Each object $U \to S$ of $Set/S$ may be thought of as a monomial term $\prod_s x_{s}^{U_s}$ where $\{x_s\}$ is a set of variables indexed by $S$, although it makes better sense to think of it that way when it is regarded as an object of $(Set/S)^{op}$.
Thus, objects of $(Set/S)^{op}$ are pairs $(n, x: n \to S)$, where $n$ is any set, and morphisms $(n, x) \to (m, y)$ are functions $f: [m] \to [n]$ such that $y = x \circ f$, or $y_i = x_{f(i)}$ for all $i \in [m]$. Clearly, $(Set/S)^{op}$ has small products. In fact, any object $(n, x)$ of $(Set/S)^{op}$ is a product of objects of the form $i(s)$.
+-- {: .num_prop}
######Proposition
$(Set/S)^{op}$ is the free category with small products generated by the set $S$.
=--
+-- {: .proof}
######Proof
Let $C$ be a category with small products and let $\Phi: S \to Ob(C)$ be any function. Define a functor
$$\Pi: (Set/S)^{op} \to C$$
so that $(n, x: n \to S)$ is taken to $\prod_{i \in n} \Phi(x(i))$. It is immediate that $\Pi$ is a product-preserving functor and is, up to unique isomorphism, the unique product-preserving functor that extends $\Phi$.
=--
+-- {: .num_def}
######Definition
A **multi-sorted algebraic theory** over the set of sorts $S$ consists of a locally small category with small products, $C$, together with a sort assignment $\Phi: S \to C$ such that the product-preserving extension
$$\Pi: (Set/S)^{op} \to C$$
is essentially surjective. An **operation of arity** $x_1, \ldots, x_n \to y$ in $C$ is a morphism of the form $\Pi(n, x) \to \Phi(y)$ in $C$. If $D$ has small products, a **model** of $C$ in $D$ is a product-preserving functor $M: C \to D$. A **homomorphism of models** is simply a natural transformation between product-preserving functors.
=--
It violates the [[principle of equivalence]], but is nevertheless harmless and sometimes convenient, to suppose $\Pi$ is an isomorphism on objects, since we can define $C'$ to have the same objects as $Set/S$ and define hom-sets by $C'(x, y) = C(\Pi(x), \Pi(y)$. Then, the functor $(Set/S)^{op} \to C$ evidently factors as
$$(Set/S)^{op} \stackrel{\Pi}{\to} C' \to C$$
where the second functor $C' \to C$ is an equivalence, so we may as well work with the functor $\Pi: (Set/S)^{op} \to C'$.
#### Commutative theories
_[[commutative algebraic theory|Commutative algebraic theories]]_ are (single-sorted) algebraic theories for which each operation is an algebra homomorphism. These form an important subclass. Their categories of models are [[closed monoidal category|closed]]: the [[hom sets]] have a natural model-structure (algebra-structure), and the enriched Hom-functor has a [[left adjoint]], _[[tensor product]]_.
The theory of $R$-modules for a fixed commutative ring $R$ is perhaps the most familiar example. The theory of complete lattices and supremum-preserving functions is an interesting non-finitary example.
## Relation to monads
{#RelationToMonads}
We flesh out the relationships between algebraic theories and [[monads]], starting from the most general situation and then adding conditions to cut down on the size of theories. The term "[[Lawvere theory]]" as used here will mean
a large (but locally small) [[infinitary Lawvere theory]].
(Under this relation ordinary finitary Lawvere theories correspond to _[[finitary monads]]_.)
### The monad of a locally small Lawvere theory
Suppose $C$ is a (locally small, multi-sorted) Lawvere theory, so we have a product-preserving functor
$$\Pi: (Set/S)^{op} \to C$$
which we may assume to be the identity on objects. We define an adjoint pair between the category of models $Mod(C, Set)$, consisting of product-preserving functors $C \to Set$ and transformations between them, and the category $Set/S$. We also denote this model category by $Prod(C, Set)$.
+-- {: .num_remark}
###### Remark
Observe that $(Set/S)^{op}$ is a Lawvere theory which is the theory of $S$-multi-sorted sets,
$$Prod((Set/S)^{op}, Set) \stackrel{- \circ i}{\simeq} Set^S \simeq Set/S,$$
where the first equivalence obtains precisely because $(Set/S)^{op}$ is the free category with products generated by $S$.
=--
Let $y: C^{op} \to Mod(C, Set)$ be the Yoneda embedding, taking $c$ to the product-preserving functor $hom(c, -): C \to Set$.
+-- {: .un_thm}
######Theorem 1
The functor
$$Set/S \stackrel{\Pi^{op}}{\to} C^{op} \stackrel{y}{\to} Mod(C, Set)$$
is left adjoint to the functor
$$Prod(C, Set) \stackrel{Prod(\Pi, Set)}{\to} Prod((Set/S)^{op}, Set) \simeq Set/S$$
(using the remark above).
=--
+-- {: .proof}
######Proof
We must exhibit a natural isomorphism
$$Nat((y(\Pi^{op} (n \stackrel{x}{\to} S)), G) \cong Set/S(n \stackrel{x}{\to} S, G \circ \Pi \circ i)$$
where $Nat(-, -)$ indicates the hom-functor on the functor category $Mod(C, Set)$. The left side is naturally isomorphic to
$$G(\Pi(n \stackrel{x}{\to} S))$$
by the [[Yoneda lemma]]. The right side is isomorphic to
$$Set/S(n \stackrel{x}{\to} S, G\Pi i)$$
where $i: S \to (Set/S)^{op}$ is the canonical embedding. Now both $G \circ \Pi$ and $Set/S(-, G\Pi i)$ are product-preserving functors $(Set/S)^{op} \to Set$, so to check these functors are isomorphic, it suffices (by the universal property of $(Set/S)^{op}$ to check they give isomorphic results when restricted along $i$:
$$G \Pi i \cong Set/S(i-, G\Pi i)$$
However, because $i: S \to (Set/S)^{op}$ is itself a Yoneda embedding $y^{op}: S \to (Set^S)^{op}$, the last isomorphism is just an instance of the Yoneda lemma, and this concludes the proof.
=--
The **monad of a Lawvere theory $C$** is the monad $T: Set/S \to Set/S$ associated with this adjunction.
### Large Lawvere theory of a monad
Now let $T: Set/S \to Set/S$ be a [[monad]] on $Set/S$, with unit $u: 1 \to T$ and multiplication $m: T T \to T$.
+-- {: .un_def}
###### Definition
The **large Lawvere theory** $Th(T)$ **of $T$** is the [[opposite category|opposite]] of the [[Kleisli category]], $Kl(T)^{op}$.
=--
The left adjoint $Set/S \to Kl(T)$ is coproduct-preserving, so we have a product-preserving functor
$$(Set/S)^{op} \to Kl(T)^{op}$$
which is a bijection on the classes of objects. Therefore $Kl(T)^{op}$ is indeed a (large, multi-sorted) Lawvere theory.
+-- {: .un_thm}
###### Theorem 2
Let $T$ be a monad on $Set/S$. The monad associated with the theory $Th(T)$ is isomorphic to $T$.
=--
+-- {: .proof}
###### Proof
In other words, we claim the monad of the adjunction
$$Set/S \stackrel{\Pi^{op}}{\to} Kl(T) \stackrel{y}{\to} Prod(Kl(T)^{op}, Set)) \dashv (Prod(Kl(T)^{op}, Set) \stackrel{Prod(\Pi, 1)}{\to} Prod((Set/S)^{op}, Set) \simeq Set/S$$
is isomorphic to $T$. Now the functor $\Pi^{op}: Set/S \to Kl(T)$ is left adjoint to the underlying functor $U: Kl(T) \to Set/S$, and the underlying monad there is of course $T$. It is obvious that the composite
$$Set/S \stackrel{\Pi^{op}}{\to} Kl(T) \stackrel{y}{\to} Prod(Kl(T)^{op}, Set) \stackrel{Prod(\Pi, 1)}{\to} Prod((Set/S)^{op}, Set)$$
takes an object $f: X \to S$ to
$$Kl(T)(\Pi^{op}-, \Pi(f)) \cong Set/S(-, U \Pi(f)) \cong Set/S(-, T(f))$$
and since the equivalence $Prod((Set/S)^{op}, Set) \to Set/S$ is adjoint to the yoneda embedding, it takes $Set/S(-, T(f))$ to $T(f)$. This proves the claim.
=--
In the other direction, we have
+-- {: .un_thm}
###### Theorem 3
Let $C$ be an $S$-sorted Lawvere theory. Then the Lawvere theory of the monad of $C$ is equivalent to $C$.
=--
We assume for convenience that the product-preserving functor $\Pi: (Set/S)^{op} \to C$ is the identity on the class of objects.
+-- {: .proof}
###### Proof
We need to exhibit a comparison functor $Kl(T)^{op} \to C$, where $T$ is the monad of $C$. Such a comparison functor exists provided that $\Pi: (Set/S)^{op} \to C$ has a left adjoint whose associated monad is isomorphic to $T$. Now the composite
$$C^{op} \stackrel{y}{\to} Prod(C, Set) \stackrel{Prod(\Pi, 1)}{\to} Prod((Set/S)^{op}, Set)$$
sends an object $c$ of $C^{op}$ to the product-preserving functor $C(c, \Pi-): (Set/S)^{op} \to Set$ which, by the remark above, is represented by an object of $Set/S$ which we denote as $U^{op} c$. In other words we have a natural isomorphism
$$C(c, \Pi-) \cong (Set/S)^{op}(U^{op} c, -)$$
and by the usual Yoneda yoga, we obtained a functor $U^{op}: C \to (Set/S)^{op}$ which is left adjoint to $\Pi$. The monad $T$ is, by definition (see theorem 1) the monad associated with the adjoint pair $(\Pi^{op}: Set/S \to C) \dashv (U: C \to Set/S)$.
We thus obtain the comparison functor $Kl(T)^{op} \to C$, and it is the identity on objects. On hom-sets it is given by the natural isomorphism
$$Kl(T)^{op}(f, g) \cong (Set/S)^{op}(f, T(g)) \cong (Set/S)^{op}(f, U\Pi^{op}(g)) \cong C(\Pi(f), \Pi(g))$$
and hence the comparison functor is an equivalence.
=--
### Algebras and models
Each [[algebra over a monad|algebra]] $X$ of the monad $T$ gives rise to a model $M_X$ of the Lawvere theory:
$$Kl(T)^{op} \hookrightarrow Alg(T)^{op} \stackrel{\hom(-, X)}{\to} Set$$
and similarly a morphism of algebras $f: X \to Y$ gives rise to a homomorphism $M_f: M_X \to M_Y$, so that we have a functor $M: Alg(T) \to Mod(Th(T), Set)$. This functor is an equivalence.
It is convenient to proceed as follows. By Theorem 2, the underlying functor
$$Prod(Kl(T)^{op}, Set) \to Set/S$$
has a left adjoint such that the associated monad is $T$, and this yields a comparison functor
$$A: Prod(Kl(T)^{op}, Set) \to Alg(T)$$
+-- {: .un_thm}
###### Theorem 4
$A$ is an equivalence.
=--
+-- {: .proof}
###### Proof
In outline, this proceeds as follows:
* $A$ is essentially surjective, because if $X$ is a $T$-algebra, then $M_X: Kl(T)^{op} \to Set$ is a product-preserving functor such that $A(M_X) \cong X$.
* $A$ is full, because any algebra map $f: X \to Y$ gives rise to a model homomorphism $M_f: M_X \to M_Y$.
* $A$ is faithful. For this it suffices to prove that the underlying functor
$$U: Prod(Kl(T)^{op}, Set) \to Set/S$$
is faithful. Let $f: X \to Y$ be a morphism of $Prod(Kl(T)^{op}, Set)$. Now every object of $Kl(T)^{op}$ is a product $\prod_i s_i$ of objects in the image of
$$S \stackrel{i}{\to} (Set/S)^{op} \to Kl(T)^{op}$$
From the naturality of the diagram
$$\array{
X(\prod_i s_i) & \overset{f(\prod_i s_i)}{\to} & Y(\prod_i s_i) \\
\mathllap{X(\pi)} \downarrow & & \downarrow \mathrlap{Y(\pi_i)} \\
X(s_i) & \underset{f(s_i)}{\to} & Y(s_i)
}$$
and the fact that $Y$ preserves products, we see that the component of $f$ at $\prod_i s_i$ is uniquely determined from the components $f(s): X(s) \to Y(s)$ as $s$ ranges over the image of $\Pi i: S \to Kl(T)^{op}$, in other words that the functor $U$ defined by $U(X) = X \Pi i$ is faithful.
Thus $A$ is an equivalence, with essential inverse $M$.
=--
## Metaphor
Ring theory is a branch of mathematics with a well-developed terminology. A ring $A$ determines and is determined by an algebraic theory, whose models are left $A$-modules and whose $n$-ary operations have the form
$$(x_1,\ldots ,x_n) \to a_1 x_1 + \cdots + a_n x_n$$
for some n-tuple $(a_1,\ldots ,a_n)$ of elements of $A$. We may call such an algebraic theory **annular**. The pun _model/module_ is due to [[Jon Beck]]. The notion that an algebraic theory is a generalized ring is often a fertile one, that automatically provides a slew of suggestive terminology and interesting problems. Many fundamental ideas of ring/module-theory are simply the restriction to annular algebraic theories of ideas that apply more widely to algebraic theories and their models.
Let us denote the category of models and homomorphisms (in $Set$) of an algebraic theory $A$ by $A Mod$. Then compare the following to their counterparts in ring theory:
* [[tensor product theory|Tensor Product of Theories]]
* [[matrix theory|Matrix Theories]]
* [[bimodel|Bimodels]]
## Related concepts
* [[essentially algebraic theory]]
* **algebraic theory** / [[Lawvere theory]] / [[2-Lawvere theory]] / [[(∞,1)-algebraic theory]]
* [[algebraic side effect]]
* [[generalized algebraic theory]]
* [[globular theory]]
* [[monad]] / [[(∞,1)-monad]]
* [[operad]] / [[(∞,1)-operad]]
* [[finitely complete category]], [[cartesian functor]], [[cartesian logic]], [[cartesian theory]]
* [[regular category]], [[regular functor]], [[regular logic]], [[regular theory]], [[regular coverage]], [[regular topos]]
* [[coherent category]], [[coherent functor]], [[coherent logic]], [[coherent theory]], [[coherent coverage]], [[coherent topos]]
* [[geometric category]], **geometric functor**, [[geometric logic]], [[geometric theory]]
## References
* {#Manes76} [[Ernest G. Manes]], *Algebraic Theories*, Springer (1976) [[doi:10.1007/978-1-4612-9860-1](https://doi.org/10.1007/978-1-4612-9860-1)]
* [[Jiří Adámek]], [[Jiří Rosický]], [[Enrico Vitale]], *Algebraic theories*, Cambridge University Press (2011) [[doi:10.1017/CBO9780511760754](https://doi.org/10.1017/CBO9780511760754), [pdf](https://perso.uclouvain.be/enrico.vitale/gab_CUP2.pdf)]
* {#Johnstone} [[Peter Johnstone]], _[[Stone Spaces]]_
* B. Badzioch, "Algebraic Theories in Homotopy Theory", Annals of Mathematics, 155, 895--913 (2002).
* {#keml-diagrams} [[Andreas Nuyts]], _Understanding Universal Algebra Using Kleisli-Eilenberg-Moore-Lawvere Diagrams_, [note](https://anuyts.github.io/files/keml-diagrams.pdf)
For a framework to compare different notions of algebraic theory see
* [[Soichiro Fujii]], _A unified framework for notions of algebraic theory_, ([arXiv:1904.08541](https://arxiv.org/abs/1904.08541))
[[!redirects algebraic theory]]
[[!redirects algebraic theories]]
[[!redirects finitary algebraic theory]]
[[!redirects finitary algebraic theories]]
[[!redirects equational theory]]
[[!redirects equational theories]]
|
algebraic topology | https://ncatlab.org/nlab/source/algebraic+topology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Algebraic topology_ refers to the application of methods of [[algebra]] to problems in [[topology]]. More specifically,
the method of algebraic topology is to assign [[homeomorphism]]/[[homotopy]]-[[invariants]] to [[topological spaces]], or more systematically, to the construction and applications of [[functors]] from some [[category]] of topological objects (e.g. [[Hausdorff spaces]], topological [[fibre bundles]]) to some algebraic category (e.g. [[abelian groups]],
[[modules]] over the [[Steenrod algebra]]). Landing in an algebraic category aids to the computability,
but typically loses some information (say getting from
a topological spaces with a continuum or more points
to rather discrete algebraic structures).
### The idea of functorial invariants
The basic idea of the functorial method for the problem of existence of morphisms is the following:
If $F:A\to B$ is a [[functor]]
(we present here a general statement, but in the above context
$A$ is a category of topological objects
and $B$ some category of algebraic objects)
and $d:D\to A$ a [[diagram]] in $A$ then
$F\circ d$ is a diagram in $B$.
If one can fill certain additional
arrow $f$ in the diagram $d$ making the extended diagram commutative,
then $F(f)$ is a morphism between the corresponding vertices in $B$ extending $F\circ d$ to a commutative diagram.
Thus if we prove that there is no morphism extending $F\circ d$ then
there was no morphism extending $d$ in the first place.
Therefore, the functorial method is very suitable to prove
_negative_ existence for morphisms. Sometimes,
however, there is a theorem showing that some set of invariants
completely characterizes a problem hence being able to show
positive existence or uniqueness for maps or spaces.
For the uniqueness for morphisms, it is enough to show that $F$ is
faithful and that there is at most one solution for the existence
problem in the target category.
Faithful functors in this context are rare,
but it is sufficient for $F$ to be faithful on some subcategory $A_p$
of $A$ containing at least all morphisms which are
the possible candidates for the solution of
the particular existence problem for morphisms.
## Overview of methods
The archetypical example is the classification of [[surfaces]] via their [[Euler characteristic]]. But as this example already shows, algebraic topology tends to be less about [[topological spaces]] themselves as rather about the [[homotopy types]] which they [[homotopy hypothesis|present]]. Therefore the topological invariants in question
are typically homotopy invariants of spaces with some exceptions, like the [[shape theory|shape invariants]] for spaces with bad local behaviour.
Hence modern algebraic topology is to a large extent the application of algebraic methods to [[homotopy theory]].
A general and powerful such method is the assignment of [[homology]] and [[cohomology]] [[groups]] to topological spaces, such that these [[abelian groups]] depend only on the [[homotopy type]]. The simplest such are [[ordinary homology]] and [[ordinary cohomology]] groups, given by [[singular simplicial complexes]]. This way algebraic topology makes use of tools of [[homological algebra]].
The [[axiom|axiomatization]] of the properties of such [[cohomology]] group assignments is what led to the formulation of the trinity of concepts of _[[category]]_, _[[functor]]_ and _[[natural transformations]]_, and algebraic topology has come to make intensive use of [[category theory]].
In particular this leads to the formulation of [[generalized (Eilenberg-Steenrod) cohomology]] theories which detect more information about classes of homotopy types. By the [[Brown representability theorem]] such are represented by [[spectra]] (generalizing [[chain complexes]]), hence [[stable homotopy types]], and this way algebraic topology comes to use and be about [[stable homotopy theory]].
Still finer invariants of [[homotopy types]] are detected by further refinements of these "algebraic" structures, for instance to [[multiplicative cohomology theories]], to [[equivariant homotopy theory]]/[[equivariant stable homotopy theory]] and so forth. The construction and analysis of these requires the intimate combination of algebra and homotopy theory to [[higher category theory]] and [[higher algebra]], notably embodied in the [[universal algebra|universal]] higher algebra of [[operads]].
The central tool for breaking down all this [[higher algebra|higher algebraic]] data into computable pieces are [[spectral sequences]], which are maybe the main heavy-lifting workhorses of algebraic topology.
## Related entries
* [[topology]], [[differential topology]]
* [[homology]]/[[cohomology]]
* [[homotopy theory]], [[shape theory]]
* [[rational homotopy theory]]
* [[nonabelian algebraic topology]]
* [[topological data analysis]]
* [[homotopy lifting property]], [[Hurewicz fibration]], [[Hurewicz connection]], [[Serre fibration]]
* [[homotopy extension property]], [[Hurewicz cofibration]], [[deformation retract]]
* [[suspension]], [[loop space]], [[mapping cylinder]], [[mapping cone]], [[mapping cocylinder]]
* [[cohomology]], [[spectrum]], [[Brown representability theorem]]
* [[fundamental group]], [[fundamental groupoid]]
* [[homotopy group]], [[Eckmann-Hilton duality]], [[H-space]], [[Whitehead product]]
* [[topological K-theory]], [[complex cobordism]], [[elliptic cohomology]], [[tmf]]
* [[CW complex]], [[CW approximation]], [[simplicial complex]], [[simplicial set]]
* [[model category]], [[model structure on topological spaces]], [[homotopy category]]
* [[fibration sequence]], [[cofibration sequence]]
* [[Freudenthal suspension theorem]], [[Whitehead theorem]]
## References
{#References}
[[!include homotopy theory and algebraic topology -- references]]
|
algebraic topology - contents | https://ncatlab.org/nlab/source/algebraic+topology+-+contents |
**[[algebraic topology]]** -- application of [[higher algebra]] and [[higher category theory]] to the study of ([[stable homotopy theory|stable]]) [[homotopy theory]]
* [[topological space]], [[homotopy type]]
* [[homotopy group|homotopy]]
[[cohomology]]
[[homology]]
* [[spectral sequence]]
|
algebraic triangulated category | https://ncatlab.org/nlab/source/algebraic+triangulated+category | A __triangulated category__ is called __algebraic__ (in the sense of [[B. Keller]]) if it is equivalent to the stable category of a [[Quillen exact category]] of a Frobenius category (a Quillen exact category is Frobenius if it has enough injectives and enough projectives and the two classes coincide).
* [[B. Keller]], _On differential graded categories_, In: Proc. ICM, Madrid, 2006. vol. II, pp. 151–190, Eur. Math. Soc., Zürich (2006) [pdf](http://www.math.jussieu.fr/~keller/publ/dgcatX.pdf)
* [[Stefan Schwede]], _Algebraic versus topological triangulated categories_, in Triangulated categories, 389--407, London Mathematical Society Lecture Notes __375__,
Cambridge Univ. Press 2010, [MR2681714](http://www.ams.org/mathscinet-getitem?mr=MR2681714), [pdf](http://www.math.uni-bonn.de/~schwede/algebraic_topological.pdf).
* Fernando Muro, Stefan Schwede, Neil Strickland, _Triangulated categories without models_, Invent. math. __170__, 231–241 (2007) [doi](http://dx.doi.org/10.1007/s00222-007-0061-2), [pdf](http://www.math.uni-bonn.de/~schwede/nomodel-InvMath.pdf)
Every algebraic triangulated category which is well generated in the sense of [[Amnon Neeman]] is triangle equivalent to a localization of the derived category of a small pretriangulated dg-category by a localizing subcategory generated by a set of objects:
* M. Porta, _The Popescu-Gabriel theorem for triangulated categories_, Adv. Math. __225__ (2010) 1669-1715 [doi](https://doi.org/10.1016/j.aim.2010.04.002)
|
algebraic variety | https://ncatlab.org/nlab/source/algebraic+variety |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[algebraic geometry]], __algebraic variety__ (not to be confused with [[variety of algebras]]) is a [[scheme]] which is [[integral scheme|integral]], [[separated]] and of [[finite type]] over an [[algebraically closed field]] $k$.
Classically, the term algebraic variety referred to a [[scheme]] as above which is further [[quasi-projective scheme|quasi-projective]], i.e. admits a locally closed embedding into [[projective space]]. Thus, these were objects which locally are cut out inside [[projective space]] as the geometric locus of zeros of a set of [[polynomial]] equations in finitely many variables. (The first example of an algebraic variety which is not quasi-projective was given by [[Nagata]].)
Historically, there were several formalisms of various schools including the Italian school of [[algebraic geometry]] in the early 20th century (Veronese, Castelnuovo, Severi, ...), the American school between the two wars ([[Oscar Zariski]]), [[Andre Weil]]), the abstract varieties of [[Jean-Pierre Serre]] and finally the language of [[schemes]] introduced by the [[Grothendieck]] school. One should note that in the case of (esp. [[projective variety|projective]]) varieties over complex numbers there is an additional possibility to work using complex-analytic tools and complex topology.
## Definition
Given an [[algebraically closed field]] $k$, an __algebraic $k$-variety__ usually means either a quasiprojective variety or an abstract variety (in the sense of Serre). 'Quasiprojective' unifies affine, quasiaffine, [[projective variety|projective]] and embedded quasiprojective $k$-varieties. Many modern sources by a variety mean a reduced separated scheme of finite type over a field, often requiring also irreducibility (that is integral = reduced and irreducible).
* An embedded affine $k$-variety (or an affine algebraic set) is a set of zeros of a locus of common zeros of a set of polynomial equations in the affine space $\mathbf{A}^n_k$. By the Hilbert [[Nullstellensatz]] there is a more invariant definition. __[[affine variety|Affine]]__ $k$-varieties are [[maximal spectrum|maximal spectra]] (= sets of [[maximal ideals]]) of finitely generated [[noetherian ring|noetherian]] (commutative unital) $k$-[[commutative algebra|algebras]] without [[nilpotent element|nilpotents]] with the [[Zariski topology]]; the algebra can be recovered as the coordinate ring of the variety; this correspondence is an equivalence of categories, if the morphisms are properly defined.
Affine varietes can be embedded as closed subvarieties into an [[affine space]] (in the sense of algebraic geometry). As topological spaces affine varieties are [[noetherian space|noetherian]].
* __[[projective variety|Projective]]__ $k$-varieties are obtained in a similar way from [[graded algebra|graded]] $k$-algebras, or, in embedded incarnation, as loci of zeros of a set of homogeneous polynomials in projective space $\mathbf{P}^n_k$.
* Embedded __quasiaffine__ $k$-varieties are Zariski-open subspaces of affine $k$-varieties.
* Embedded __quasiprojective__ $k$-varieties are Zariski-open subspaces of projective $k$-varieties. We can remove the embedding by equipping them with the sheaf of regular functions and therefore considering them as [[locally ringed space]]s. In the category of locally ringed spaces, projective, affine, and quasiaffine varieties are (isomorphic to) special cases of quasiprojective. Alternatively, we can put all 4 classes without sheaves into a category, by defining regular maps directly, and we get an isomorphic category of varieties.
In fact, by noticing that the affine $k$-space is Zariski open in a projective space of the same dimension, we see that the quasiprojective case includes all others.
Morphisms between varieties are sometimes called [[regular maps]].
Sometimes a smooth algebraic variety may also be called __algebraic manifold__.
An abstract $k$-prevariety in the sense of Serre is a locally ringed space which is locally isomorphic to affine $k$-variety. The category of $k$-prevarieties has a product which is obtained by locally gluing products in the category of affine $k$-varieties. This enables defining a diagonal $X\to X\to X$; a prevariety is separated, or an abstract $k$-variety if the diagonal is closed in Zariski topology (which is, of course, not a product of Zariski topologies of factors).
## Properties
### Relation to schemes
There is an [[equivalence of categories]] between the [[category]] of [[integral schemes]] of finite type over $Spec\,k$, where $k$ is an [[algebraically closed field]], and the category of (irreducible) algebraic $k$-varieties.
Of course, given a variety the corresponding [[scheme]] and variety have different sets of points; the points in common are the closed points of the scheme. The remaining points are the generic points of subvarieties. Generic points were often used, without proper foundations, in other language, already in the works of the Italian school.
Some modern algebraic geometers mean, by varieties, objects of certain slightly bigger categories of relative $S$-schemes of finite type (where $S$ is not necessarily $Spec\,k$ for $k$ a field); typically they are required to be [[separated scheme|separated]] [[reduced scheme|reduced]] $S$-[[relative scheme|schemes]] [[morphism of finite type|of finite type]].
## Related concepts
* [[geometric point]]
* [[singular point of an algebraic variety]]
* [[complete algebraic variety]]
* [[semialgebraic manifold]]
## References
* Igor Shafarevich, _Basic algebraic geometry_, vol. I
* J. S. Milne, _Algebraic geometry_, 2017 [pdf](http://www.jmilne.org/math/CourseNotes/AG.pdf)
* Joe Harris, _Introductory algebraic geometry_
* chapter I of Robin Hartshorne, _Algebraic geometry_, Springer
An amusing discussion on the differences between schemes and varieties can be found at _Secret blogging seminar_: [algebraic geometry without prime ideals](http://sbseminar.wordpress.com/2009/08/06/algebraic-geometry-without-prime-ideals).
category: algebraic geometry
[[!redirects algebraic varieties]]
[[!redirects affine algebraic variety]]
[[!redirects affine algebraic varieties]]
[[!redirects algebraic manifold]]
[[!redirects algebraic manifolds]]
[[!redirects quasiaffine variety]]
[[!redirects quasiaffine varieties]]
[[!redirects quasiaffine algebraic variety]]
[[!redirects quasiaffine algebraic varieties]]
[[!redirects quasi-projective variety]]
[[!redirects quasiprojective variety]]
[[!redirects quasi-projective varieties]]
[[!redirects quasiprojective varieties]]
[[!redirects quasi-projective algebraic variety]]
[[!redirects quasiprojective algebraic variety]]
[[!redirects quasi-projective algebraic varieties]]
[[!redirects quasiprojective algebraic varieties]]
[[!redirects quasi-projective scheme]]
[[!redirects quasi-projective scheme]]
[[!redirects Var]]
[[!redirects Varieties]]
|
algebraic vector bundle | https://ncatlab.org/nlab/source/algebraic+vector+bundle |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of [[vector bundle]] in [[algebraic geometry]].
Usually characterized in terms of its [[sheaf]] of [[sections]] on the [[locally ringed site]] (with [[structure sheaf]] $\mathcal{O}$) of the given [[scheme]]: a locally free $\mathcal{O}$-module of finite [[rank]].
## Properties
### Relation to analytic vector bundles
{#RelationToAnalyticVectorBundles}
Over the [[complex numbers]], [[GAGA]] relates algebraic to [[holomorphic vector bundles]] (e.g. [Neeman 07, theorem 1.1.4](#Neeman07)).
## Related concepts
* [[topological vector bundle]]
* [[differentiable vector bundle]]
* [[shtuka]]
## References
* {#Neeman07} [[Amnon Neeman]], _Algebraic and analytic geometry_, London Math. Soc. Lec. Note Series __345__, 2007 ([publisher](http://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/algebraic-and-analytic-geometry))
[[!redirects algebraic vector bundles]]
|
algebraic weak factorization system | https://ncatlab.org/nlab/source/algebraic+weak+factorization+system | [[!redirects natural weak factorization system]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##Basic idea##
**Algebraic weak factorization systems** (AWFS) are algebraizations of [[weak factorization system]]s (WFS). The elements in the left and right classes of morphisms are replaced by coalgebras and algebras, respectively, for a certain [[comonad]] and [[monad]] on the [[arrow category]]. This comonad and monad also determine the functorial factorization and give natural coalgebra and algebra structures to the left and right factors.
Algebraic weak factorization systems were originally called **natural weak factorization systems** by Grandis and Tholen.
##Preliminaries##
Recall a **functorial factorization** on a category $K$ is a functor $E : K$<sup>2</sup> → $K$<sup>3</sup> that is a section to the composition functor $d_1$, induced by the inclusion functor $d^1 : 2 \rightarrow 3$ between the ordinal categories.
Explicitly, $E$ factors a morphism $(u,v) : f \Rightarrow g$ in $K^{2}$ as
$$
\array{
\cdot &\stackrel{u}{\to}& \cdot
\\
\downarrow^f
&&
\downarrow^g
\\
\cdot &\stackrel{v}{\to}& \cdot
} \array{ & & } \stackrel{E}{\mapsto} \array{ &&}
\array{
\cdot &\stackrel{u}{\to}& \cdot
\\
\downarrow^{Lf}
&&
\downarrow_{Lg}
\\
\cdot & \stackrel{E(u,v)}{\to} & \cdot
\\ \downarrow^{Rf} && \downarrow_{Rg} \\
\cdot &\stackrel{v}{\to}& \cdot
}
$$
There are two other injective functors $d^0, d^2 : 2 \rightarrow 3$ whose image misses the object that appears as their superscript. When we compose $E$ with $d_2$ and $d_0$, we obtain endofunctors of $K^{2}$, which we call $L$ and $R$.
There are obvious natural transformations $1 \Rightarrow R$ and $L \Rightarrow 1$ whose components are given by the data of the functorial factorization $E$. We say $L$ and $R$ are **pointed** endofunctors, with these natural transformations in mind.
##Definition##
A AWFS on a category $K$ consists of a pair $(L,R)$ where $L$ is a comonad and $R$ is a monad, whose underlying pointed endofunctors arise from a functorial factorization $E$. Some authors (Garner) also require that the canonical natural transformation $L R \Rightarrow R L$, whose domain and codomain components are given by the comultiplication and multiplication maps, is a distributive law of the comonad over the monad. This amounts to the requirement that a pentagon involving the comultiplication and multiplication maps commutes.
We refer to the $L$-coalgebras as the **left class** of the AWFS and the $R$-algebras as the **right class**. When we forget the algebra structures, we obtain classes of maps in $K$. The retract closures of these classes form a WFS called the **underlying WFS** of this AWFS.
Given a lifting problem
$$
\array{
\cdot &\stackrel{u}{\to}& \cdot
\\
\downarrow^f
&&
\downarrow^g
\\
\cdot &\stackrel{v}{\to}& \cdot
}$$
where $f$ is a $L$-coalgebra and $g$ is an $R$-algebra, the functorial factorization, coalgebra, and algebra structures can be combined to define a solution, which proves that the left class has the left lifting property with respect to the right class. We leave the details as an exercise.
##Interesting features##
* The right class of a AWFS is closed under any limits that exist in $K^{2}$, because the forgetful functor to the underlying category of arrows creates all limits which exist. Note that it does not follow that the right class of the underlying WFS is closed under limits in the arrow category, because first, it is possible that some elements of the right class will not have an $R$-algebra structure, and second, not every map in the arrow category between $R$-algebras is necessarily an $R$-algebra map.
* Algebras for the monad of an AWFS can be composed canonically, as can the coalgebras for the comonad. The composition law for the algebras uses the comultiplication natural transformation, and dually for the coalgebras.
* Each AWFS $(L, R)$ on $K$ induces a [[right-connected double category]] $R$-$\mathbb{A}lg$, whose category of objects and horizontal morphisms is $K$, whose category of vertical morphisms and cells is $R$-$Alg$.
* A AWFS $(L,R)$ on $K$ induces a levelwise AWFS on any diagram category $K^A$. Note that its underlying WFS will not be similarly "levelwise". (Indeed, a WFS does not typically induce a levelwise WFS on a diagram category.)
* An AWFS can be detected as a [[functorial factorization]] that extends to a [[monad]] over $cod$ with a [[composition law for factorizations]].
##Small object argument##
The [[algebraic small object argument]], an enhancement of the [[small object argument]] due to Richard Garner, produces **cofibrantly generated** AWFS by adapting the construction of a [[free monad]] on an endofunctor. Importantly, Garner's small object argument allows the generators to be a small category over the arrow category $K^{[2]}$, rather than simply a set of arrows. As a result, there are WFS which are not cofibrantly generated in the classical sense, but which can be exhibited as the underlying WFS of a cofibrantly generated AWFS.
Every AWFS on a [[locally presentable category]] generated by such a small category of maps is in particular an [[accessible weak factorization system]], and every accessible WFS admits an algebraic enhancement generated by the algebraic small object argument. However, not every accessible AWFS is *itself* generated by the algebraic small object argument, unless we enhance it further to take a [[double category]] of generators; see [Bourke and Garner](#BourkeGarnerI).
## Related concepts
* [[algebraic model category]]
* [[accessible model category]]
* [[accessible weak factorization system]]
[[!include algebraic model structures - table]]
## References
* [[Marco Grandis]] and [[Walter Tholen]], _Natural weak factorization systems_, Arch. Math. (Brno) 42(4) (2006) 397–408. MR2283020 (2008b:18006), Zbl 1164.18300.
* [[Richard Garner]], _Understanding the small object argument_, [arXiv](http://arxiv.org/abs/0712.0724).
* [[Emily Riehl]], _Algebraic model structures_, ([arXiv:0910.2733](http://arxiv.org/abs/0910.2733)).
* [[Thomas Athorne]], _The coalgebraic structure of cell complexes_, [TAC](http://www.tac.mta.ca/tac/volumes/26/11/26-11abs.html)
* [[Tobias Barthel]] and [[Emily Riehl]], *On the construction of functorial factorizations for model categories*, Algebr. Geom. Topol. Volume 13, Number 2 (2013), 1089-1124, , [arxiv](https://arxiv.org/abs/1204.5427)
* {#BourkeGarnerI} [[John Bourke]] and [[Richard Garner]], _Algebraic weak factorisation systems I: Accessible AWFS_, Journal of Pure and Applied Algebra 220, 2016. [arXiv:1412.6559](http://arxiv.org/abs/1412.6559), [doi:10.1016/j.jpaa.2015.06.002](https://doi.org/10.1016/j.jpaa.2015.06.002)
* [[John Bourke]] and [[Richard Garner]], _Algebraic weak factorisation systems II: Categories of weak maps_, Journal of Pure and Applied Algebra 220, 2016. [arXiv:1412.6560](http://arxiv.org/abs/1412.6560), [doi:10.1016/j.jpaa.2015.06.003](https://doi.org/10.1016/j.jpaa.2015.06.003).
* J. Rosicky, _Accessible model categories_, [arxiv](https://arxiv.org/abs/1503.05010)
* [[Emily Riehl]], _Made-to-Order Weak Factorization Systems_, [doi](https://doi.org/10.1007/978-3-319-21284-5_17), [pdf](http://www.math.jhu.edu/~eriehl/made-to-order.pdf)
* Ignacio Lopez Franco, *Cofibrantly generated lax orthogonal factorisation systems* [[arxiv](https://arxiv.org/abs/1510.07131)]
* [[John Bourke]], _An orthogonal approach to algebraic weak factorisation systems_, Journal of Pure and Applied Algebra **227** 6 (2023) 107294 [[arxiv:2204.09584](https://arxiv.org/abs/2204.09584), [doi:10.1016/j.jpaa.2022.107294](https://doi.org/10.1016/j.jpaa.2022.107294)]
A new proof of the [[Strøm model structure]] using algebraic weak factorization systems:
* {#BarthelRiehl13} [[Tobias Barthel]], [[Emily Riehl]], _On the construction of functorial factorizations for model categories_, Algebr. Geom. Topol. 13 (2013) 1089-1124 ([arXiv:1204.5427](http://arxiv.org/abs/1204.5427), [doi:10.2140/agt.2013.13.1089](http://dx.doi.org/10.2140/agt.2013.13.1089), [euclid:agt/1513715550](https://projecteuclid.org/euclid.agt/1513715550))
Introductory texts:
* [[Introduction to Homotopy Theory]]
* {#Riehl2008} [[Emily Riehl]], [_Factorization Systems_](https://math.jhu.edu/~eriehl/factorization.pdf), 2008
* [[Emily Riehl]], §12.4 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]
The following dissertation section is entirely written after learning of [Riehl (2008)](#Riehl2008), but has complementary examples and may dive deeper into some proofs:
* {#Nuyts2020} [[Andreas Nuyts]], _Contributions to Multimode and Presheaf Type Theory, section 2.4: Factorization Systems_, [PhD thesis](https://lirias.kuleuven.be/retrieve/581985), KU Leuven, Belgium, 2020
[[!redirects natural weak factorisation system]]
[[!redirects natural weak factorisation systems]]
[[!redirects natural weak factorization system]]
[[!redirects natural weak factorization systems]]
[[!redirects algebraic weak factorisation system]]
[[!redirects algebraic weak factorisation systems]]
[[!redirects algebraic weak factorization systems]]
[[!redirects NWFS]]
[[!redirects nwfs]]
[[!redirects AWFS]]
[[!redirects awfs]]
[[!redirects algebraic wfs]]
[[!redirects algebraic WFS]]
|
algebraically closed field | https://ncatlab.org/nlab/source/algebraically+closed+field |
+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[field]] $k$ is __algebraically closed__ if every non-constant [[polynomial]] (with one variable and coefficients from $k$) has a root in $k$. It follows that every polynomial of degree $n$ can be factored uniquely (up to [[permutation]] of the factors) as
$$ p = c \prod_{i = 1}^n (\mathrm{x} - a_i) ,$$
where $c$ and the $a_i$ are elements of $k$.
An __algebraic closure__ of an arbitrary field $k$ is an algebraically closed field $\bar{k}$ equipped with a field homomorphism (necessarily an [[injection]]) $i: k \to \bar{k}$ such that $\bar{k}$ is an [[algebraic extension]] of $k$ (which means that every element of $\bar{k}$ is the root of some non-zero polynomial with coefficients only from $k$). For example, $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$. An algebraic closure of $k$ can also be described as a maximal algebraic extension of $k$. The [[axiom of choice]] proves the existence of $\bar{k}$ for any field $k$, as well as its uniqueness up to [[isomorphism]] over $k$. (See [[splitting field]] for a more refined result.) However, note that $\bar{k}$ need not be unique up to *unique* isomorphism, so it\'s not really appropriate to speak of [[the]] algebraic closure of $k$. For example, complex conjugation is a nontrivial [[automorphism]] of $\mathbb{C}$ over $\mathbb{R}$.
Without choice, the existence and uniqueness of algebraic closures may fail; see [Chow06](#Chow06), [Banaschewski92](#Banaschewski92), [Richman00](#Richman00).
Even with choice, algebraic closure is not [[functor|functorial]] in any reasonable sense. For example, it is very easy to demonstrate that there is no algebraic closure functor $F \mapsto \widebar{F}$ that renders the inclusion $i: F \to \widebar{F}$ natural:
+-- {: .num_example}
###### Example
Supposing there were such an algebraic closure functor $F \mapsto \widebar{F}$, consider its application to the (equalizer) diagram
$$\mathbb{R} \to \mathbb{C} \underoverset{conj}{id}{\rightrightarrows} \mathbb{C}.$$
We would have a commutative naturality diagram (meaning serially commutative on the right)
$$\array{
\mathbb{R} & \stackrel{i}{\to} & \mathbb{C} & \underoverset{conj}{id}{\rightrightarrows} & \mathbb{C} \\
\mathllap{i} \downarrow & & \mathllap{id} \downarrow & & \downarrow \mathrlap{id} \\
\widebar{\mathbb{R}} & \stackrel{\widebar{i}}{\to} & \widebar{\mathbb{C}} & \underoverset{\widebar{conj}}{\widebar{id}}{\rightrightarrows} & \widebar{\mathbb{C}}
}$$
where serial commutativity of the right square(s) forces $\widebar{id} \neq \widebar{conj}$, but functoriality applied to the equation $id \circ i = conj \circ i$ on the top forces $\widebar{id} = \widebar{conj}$ (no matter which isomorphism $\widebar{i}$ is taken to be, $id$ or $conj$).
=--
Thus, any two algebraic closures are isomorphic, but [[unnatural isomorphism|not naturally]] so.
## Classical invariants
Putting aside the concerns of constructive mathematics, and freely adopting the principle of the [[excluded middle]] and the [[axiom of choice]], algebraically closed fields are characterized (up to non-unique isomorphism) by just two cardinal invariants:
+-- {: .num_theorem}
###### Theorem
Two algebraically closed fields $K, K'$ are isomorphic iff they have the same characteristic $p$ (the nonnegative generator of the [[kernel]] of the unique [[ring]] map $\mathbb{Z} \to K$) and the same [[transcendence degree]] (the [[cardinality]] of any maximal set of algebraically independent elements).
=--
In outline, the proof is simple in structure. The "only if" statement is clear, provided we allow that transcendence degree is _well-defined_. For the "if" statement, $K$ contains a subring isomorphic to $\mathbb{Z}/(p)[S]$ where $S$ is a transcendence basis, and similarly $K'$ contains a subring isomorphic to $\mathbb{Z}/(p)[S']$. By hypothesis, there is a bijection $f: S \to S'$, which extends uniquely to an isomorphism of [[integral domains]] $\mathbb{Z}/(p)[S] \to \mathbb{Z}/(p)[S']$, which extends uniquely to an isomorphism of their fields of fractions $\mathbb{F}(S) \to \mathbb{F}(S')$. Then $K, K'$ are algebraic closures of these fields, and one applies a theorem that an isomorphism of fields $\mathbb{F}(S) \to \mathbb{F}(S')$ can be extended to an isomorphism $K \to K'$ of their algebraic closures.
The full details of such a proof carry some themes important in [[model theory]]:
* There is a notion of algebraic closure of a subset,
* There are prime models (algebraic closure of prime field $\mathbb{Z}/(p)$),
* There are notions of independence and basis, and well-defined degree or dimension,
* There are extensions of isomorphisms of independent sets to isomorphisms of their algebraic closures.
Perhaps the most subtle in the list is the notion of independence and well-definedness of (transcendence) degree, which notably involves verification of the Steinitz exchange axiom:
+-- {: .num_lemma}
###### Lemma
Let $K$ be an algebraically closed field, and let $cl: P(K) \to P(K)$ be the operator that takes a subset $S \subseteq K$ to the smallest algebraically closed subfield that contains $S$. Then $cl$ is a [[geometric stability theory|pregeometry]].
=--
+-- {: .proof}
###### Proof
For the moment, please consult Jacobson, Basic Algebra II, Theorem 8.34. This may be expanded upon a little later.
=--
Well-definedness of transcendence degree then follows from abstract considerations of pregeometries; see [this result](/nlab/show/matroid#welldefined).
## Examples
* The __[[fundamental theorem of algebra]]__ is, classically, the statement that the [[complex number]]s form an algebraically closed field $\mathbb{C}$.
Arguably, this theorem is not entirely algebraic; the algebraic portion is that $R[\mathrm{i}]$ is algebraically closed whenever $R$ is a [[real-closed field]]. Unusually, this algebraic portion is *not* (as stated) valid in [[constructive mathematics]], while the analytic result (that the [[real numbers]] form a real closed field $\mathbb{R}$) is constructively valid with the usual definitions.
* The algebraic closure $\overline{\mathbb{Q}}$ of the [[rational numbers]] $\mathbb{Q}$ is the [[algebraic numbers]].
## Related concepts
* [[Galois group]]
* [[separable closure]]
* [[geometric point]]
The algebraic closure of a field $F$ is the splitting field of the set of all [[monic polynomials]] over $F$. Thus for relevant material, see
* [[splitting field]]
## References
* {#Leinster21} [[Tom Leinster]], _[Algebraic closure](https://golem.ph.utexas.edu/category/2021/04/algebraic_closure.html)_, [[n-Category Café]]
* {#Chow06} _[Algebraic closure of Q](http://cs.nyu.edu/pipermail/fom/2006-May/010531.html)_, a thread on FOM started by [[Timothy Chow]];
be sure to check for improperly replied posts with the same subject in that and the next two months
* {#Banaschewski92} [[Bernhard Banaschewski]], *Algebraic closure without choice*, Mathematical Logic Quarterly, Volume 38, Issue 1, 1992, Pages 383-385, [[doi:10.1002/malq.19920380136](https://doi.org/10.1002/malq.19920380136)]
* {#Richman00} [[Fred Richman]], *The fundamental theorem of algebra: a constructive development without choice*, Pacific Journal of Mathematics **196** 1 (2000) 213–230 [[doi:10.2140/pjm.2000.196.213](http://dx.doi.org/10.2140/pjm.2000.196.213), [pdf](https://msp.org/pjm/2000/196-1/pjm-v196-n1-p10-p.pdf)]
* {#Ruitenberg91} Wim Ruitenberg, Constructing Roots of Polynomials over the Complex Numbers, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract, Vol. 84, Centre for Mathematics and Computer Science, Amsterdam, 1991, pp. 107–128. ([pdf](https://www.mscsnet.mu.edu/~wim/publica/roots_new.pdf))
[[!redirects algebraically closed field]]
[[!redirects algebraically closed fields]]
[[!redirects algebraic closure]]
[[!redirects algebraic closures]]
|
algebraically compact category | https://ncatlab.org/nlab/source/algebraically+compact+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
A [[category]] is called _algebraically compact_ if for every [[endofunctor]] on it the respective [[initial algebra of an endofunctor|initial algebra]] coincides with the [[final coalgebra]].
Under [[categorical semantics]] of [[programming languages]] this condition ensures the existence of [[inductive-recursive types]] (e.g. [Zamdzhiev 20](#Zamdzhiev20)). For that, recall:
In [[computer science]], a [[data type]] is often defined by an [[isomorphism]] of types $X\cong T(X)$ for some construction $T$ (an [[endofunctor]] on the [[category]] of types), namely as a [[fixed point]] of an endofunctor. (For example, the [[natural numbers]] may be defined -- see [there](inductive+type#NaturalNumbers) -- as being fixed $\mathbb{N} \cong \mathbb{N} + 1$ by the operation of [[disjoint union]] with a [[singleton]].) These are _[[inductive types]]_.
However, in a language with general [[recursion]] (including [[partial recursive functions]]), the data types have properties in addition to as the usual [[inductive type|inductive]] ones, which allow [[coinduction|coinductive]] reasoning. For example, in a lazy language such as [[Haskell]], there is an infinity element in $\mathbb{N}$, which is a [[fixed point]] for the [[successor]] operation.
In general, it is common to allow the construction $T$ to be mixed-variance. For example, $X\cong (X\to X)$ is a recursive data type whose inhabitants are expressions of the [[lambda-calculus#pure_lambda_calculus|untyped lambda calculus]]. Thus recursive data types are a generalization of [[reflexive object|reflexive objects]]. On the [[semantics|semantic]] side, recursive data types are sometimes called "recursive domain equations".
## Definition
+-- {: .num_defn}
###### Definition
A [[category]] is _algebraically complete_ if every [[endofunctor]] $F$ has an [[initial algebra of an endofunctor|initial algebra]] $F(A) \to A$. A category is _algebraically cocomplete_ if every endofunctor $F$ has a [[final coalgebra]] $Z\to F(Z)$.
By [[initial algebra of an endofunctor#LambeksTheorem|Lambek's lemma]], an initial algebra is an [[isomorphism]], and so is a final coalgebra, thus they can be regarded as [[coalgebra for an endofunctor|coalgebras]] and [[algebra for an endofunctor|algebras]] respectively. This gives rise to a canonical morphism from the initial algebra to the final coalgebra, $A\to Z$.
An algebraically complete and cocomplete category $C$ is _algebraically compact_ if this canonical morphism from the initial algebra to the final coalgebra is an [[isomorphism]].
=--
+-- {: .num_remark}
###### Remark
In [[classical mathematics|classical]] [[set theory]], very few categories are algebraically compact. Thus it is common to restrict attention to certain endofunctors.
One might then say that this class of endofunctors is algebraically compact.
A leading example is where $C$ is $V$-[[enriched category|enriched]], in which case we might restrict attention to $V$-endofunctors. For example, the $\mathbf{cpo}$-enriched category of pointed [[cpo]]'s and strict maps is $\mathbf{cpo}$-algebraically compact.
=--
Recall that a [[cpo]] is an $\omega$-chain-complete partial order.
The category $\mathbf{cpo}$ comprises cpo's and continuous maps.
A cpo is _pointed_ if it has a bottom element, and a continuous map is _strict_ if it preserves the bottom elements.
+-- {: .num_proposition}
###### Proposition
The category $\mathbf{cpo}_{\bot!}$ of pointed [[cpo]]'s and strict continuous maps is algebraically compact as a $\mathbf{cpo}$-enriched category.
=--
+-- {: .proof}
###### Proof sketch.
In a poset-enriched category, an embedding-projection pair is a retract
$e:X \to Y$, $p: Y\to X$ such that $p e=id$ and $e p\leq id$.
Let $F:\mathbf{cpo}_{\bot!}\to \mathbf{cpo}_{\bot!}$ be an endofunctor.
Consider the chain of projections
$$1 \xleftarrow{p} F(1) \xleftarrow{F(p)} F(F(1)) \xleftarrow{F(F(p))} \dots$$
We consider the limit $D$ of this sequence of projections as a poset, which is already a pointed cpo.
Because $1$ is also an initial object, each of these projections in the chain forms an embedding-projection pair, and we have a sequence
$$1 \xrightarrow{e} F(1) \xrightarrow{F(e)} F(F(1)) \xrightarrow{F(F(e))}\dots$$
One can show that $D$ can also be regarded as a colimit of this sequence of embeddings.
Now we use the universal properties of limits and colimits to show that $D$ is an initial algebra and final coalgebra of $F$.
=--
## Mixed variance domain equations and minimal solutions
Let $T:C^{op}\times C\to C$ be a [[bifunctor]]. A _solution_ for $T$ is an object $X$ together with an isomorphism
$$a:X \cong T(X,X)$$
In general there may be many solutions. One approach to comparing them is via retractions. If an object $Y$ is a [[retract]] of $X$, and $Y$ forms a solution $b:Y\cong T(Y,Y)$, then we can ask that the retraction respects the solution, i.e.
$$
\array{& Y & \overset{i}\rightarrow & X & \\
b & \downarrow &&\downarrow & a&\\
&T(Y,Y) & \underset{T(r,i)}\rightarrow& T(X,X) & \\
}
\quad and \quad
\array{& X & \overset{r}\rightarrow & Y & \\
a & \downarrow &&\downarrow & b&\\
&T(X,X) & \underset{T(i,r)}\rightarrow& T(Y,Y) & \\
}
$$
+-- {: .num_proposition}
###### Proposition ([Freyd](#Freyd))
If $C$ and $D$ are algebraically compact, so is $C\times D$. If $C$ is algebraically compact, so is the dual category $C^{op}$.
=--
As a result of this proposition, we solve a mixed-variance domain equation
$X \cong T(X,X)$
for $T:C^{op}\times C\to C$ on an algebraically compact category
by considering the initial algebra / final coalgebra of the endofunctor $\bar{T}:C^{op}\times C\to C^{op}\times C$
given by $\bar{T}(X,Y)=(T(Y,X),T(X,Y))$.
+-- {: .num_proposition}
###### Proposition ([Freyd](#Freyd), [Fiore](#fiore))
For a bifunctor $T:C^{op}\times C\to C$ on an algebraically compact category, the solution from the initial algebra / final coalgebra is minimal in the sense that it is uniquely a retract of every solution.
=--
## References
The notion is due to:
* [[Peter Freyd]], _Algebraically complete categories_, Category Theory in Como, 1990 ([dpi:10.1007/BFb0084215](https://doi.org/10.1007/BFb0084215)){#Freyd}
Barr proposed to look at certain functors as algebraically compact, and gave basic facts about building algebraically compact functors and numerous examples. Fiore and Plotkin looked enriched functors and proved "adequacy" results about data types in programming languages.
* [[Michael Barr]], _Algebraically compact functors_, Journal of Pure and Applied Algebra Volume 82, Issue 3, 26 October 1992, Pages 211-231 (<a href="https://doi.org/10.1016/0022-4049(92)90169-G">doi:10.1016/0022-4049(92)90169-G</a>)
* [[Marcelo Fiore]]. _Axiomatic domain theory in categories of partial maps_. CUP 1996. {#fiore}
* [[Marcelo Fiore]] and [[Gordon Plotkin]]. _An axiomatization of computationally adequate domain theoretic models of FPC_. In Proc. LICS, 1994. [pdf](http://homepages.inf.ed.ac.uk/gdp/publications/Ax_FPC.pdf).
Pitts gives a survey and discussion, together with further reasoning principles.
* [[Andrew Pitts]], _Relational properties of domains_, Information and Computation, 1996. [pdf](https://www.cl.cam.ac.uk/~amp12/papers/relpod/relpod.pdf).
See also:
* {#Zamdzhiev20} Vladimir Zamdzhiev, _Reflecting Algebraically Compact Functors_, EPTCS 323, 2020, pp. 15-23 ([arXiv:1906.09649](https://arxiv.org/abs/1906.09649))
|
algebraically injective object | https://ncatlab.org/nlab/source/algebraically+injective+object | * table of contents
{: toc}
## Definition
Let $J$ be a set of [[morphisms]] in a [[category]] $C$. An **algebraically $J$-injective object** is an object $X\in C$ equipped with the [[stuff, structure, property|structure]] of, for every morphism $i:A\to B$ in $J$ and every morphism $f:A\to X$, a specified morphism $g:B\to X$ such that $g \circ i = f$.
## Properties
### Algebras for a pointed endofunctor
Assuming that $C$ is [[locally small category|locally small]] and [[cocomplete]] (and $J$ is a small set), given an object $X$, let $F_J X$ be the following pushout:
$$
\array{
\coprod_{i:A\to B} (C(A,X) \cdot A) & \to & X \\
\downarrow & & \downarrow \\
\coprod_{i:A\to B} (C(A,X) \cdot B) & \to & F_J X
}
$$
where $\cdot$ represents the [[copower]] of $C$ over [[Set]]. Then $F_J$ is a [[pointed endofunctor]] of $C$, such that the (pointed) [[algebra for an endofunctor|endofunctor algebras]] of $F_J$ are precisely the algebraically $J$-injective objects.
### Monadicity
When $C$ is locally small and cocomplete as before, if the [[algebraically-free monad]] on the pointed endofunctor $F_J$ exists, then by definition the algebraically $J$-injective objects are its [[algebra for a monad|monad algebras]]. In particular, they are [[monadic functor|monadic]] over $C$.
### Solidity
...
## Related pages
* [[injective object]]
* [[algebraic small object argument]]
* [[algebraically fibrant object]]
## References
* {#Bourke17} [[John Bourke]], _Equipping weak equivalences with algebraic structure_, 2017, [arxiv](https://arxiv.org/abs/1712.02523)
* {#Bourke18} [[John Bourke]], _Iterated algebraic injectivity and the faithfulness conjecture_, 2018, [arxiv](https://arxiv.org/abs/1811.09532)
[[!redirects algebraically injective objects]]
[[!redirects algebraic injective]]
[[!redirects algebraic injectives]]
|
Algebras > history | https://ncatlab.org/nlab/source/Algebras+%3E+history | < [[Algebras]]
[[!redirects Algebras -- history]] |
algebroid | https://ncatlab.org/nlab/source/algebroid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Enriched category theory
+-- {: .hide}
[[!include enriched category theory contents]]
=--
=--
=--
# Algebroids (linear categories)
* table of contents
{: toc}
## Idea
A *linear category*, or *algebroid*, is a [[category]] whose [[hom-sets]] are all [[vector spaces]] (or [[modules]]) and whose [[composition]] operation is [[bilinear map|bilinear]]. This concept is a [[horizontal categorification]] of the concept of (unital associative) *[[unital associative algebra|algebra]]*.
## Definitions
Fix a [[commutative ring]] $K$. (Often we want $K$ to be a [[field]], such as the field $\mathbb{C}$ of [[complex numbers]], but we could also choose more generally a [[rig|commutative rig]] for $K$.)
A __$K$-linear category__, or __$K$-algebroid__, is a [[enriched category|category enriched]] over $K\,$[[Mod]], the [[monoidal category]] of $K$-[[modules]] with the usual [[tensor product of modules]]. (Note that one usually speaks of $K\,$[[Vect]] instead of $K\,Mod$ when $K$ is a [[field]].)
Just as a $\mathbb{Z}$-algebra is the same thing as a [[ring]], so a $\mathbb{Z}$-algebroid is the same thing as a *[[ringoid]]*.
## Remarks
* An [[unital associative algebra|algebra]] is a [[pointed category|pointed]] algebroid with a single object, hence a one-object $K\,Mod$-enriched (or $K\,Vect$-enriched) category. Compare with similar '[[oidification|oidfied]]' concepts such as [[groupoid]] and [[ringoid]].
* Many linear categories are also assumed to be [[additive category|additive]]. A [[linear functor]] (that is, a $K\,Mod$-enriched or $K\,Vect$-[[enriched functor]]) between additive linear categories is automatically an [[additive functor]].
* A **symmetric monoidal $K$-linear category** is a category which is at the same time a $K$-linear category and a [[symmetric monoidal category]] and such that the composition and the tensor product on morphisms are bilinear.
* Beware that a [[Lie algebroid]] is not a special case of an algebroid in the above sense, just as a [[Lie algebra]] is not a [[unital associative algebra]]. The point is that there is a restrictive and a general sense of "algebra". In the restrictive sense an algebra is an associative unital algebra, hence a [[monoid]] in $Vect$, hence a one-object $Vect$-enriched category. But in a more general sense an algebra is an algebra over an [[operad]]. It is this more general sense in terms of which Lie algebras are special cases of algebras and [[Lie algebroid]]s their [[horizontal categorification]].
## Generalizations
* Replacing plain vector spaces with [[chain complexes]] of vector spaces leads to an $\infty$-version of algebroids: a category enriched in chain complexes, which following the above reasoning could justly be called a _DG algebroid_ is usually called a _[[DG-category]]_.
* Replacing plain vector spaces with [[Banach space]]s leads to a $C^*$-version of algebroids: a category enriched in Banach spaces with some extra structure (mimicing the extra structure of a $C^*$-[[C-star-algebra|algebra]]), which following the above reasoning could justly be call a _$C^*$-algebroid_ is usually called a $C^*$-category. See [[spaceoid]]s.
* [[vertex operator algebroid]]
[[!redirects algebroid]]
[[!redirects algebroids]]
[[!redirects linear category]]
[[!redirects linear categories]]
## References
- Gabriel & Roiter, *Representations of Finite-Dimensional Algebras*, 1992.
|
Algimantas Adolfas Jucys | https://ncatlab.org/nlab/source/Algimantas+Adolfas+Jucys |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element)
## Selected writings
Introducing the [[Jucys-Murphy elements]]:
* [[Algimantas Adolfas Jucys]], *Symmetric polynomials and the center of the symmetric group ring*, Rep. Mathematical Phys.5 (1974), pp. 107–112 (<a href="https://doi.org/10.1016/0034-4877(74)90019-6">doi:10.1016/0034-4877(74)90019-6</a>)
On expressing the [[Cayley distance kernel]] (not under that name, though) through [[Jucys-Murphy elements]]:
* {#Jucys71} [[Algimantas Adolfas Jucys]], *Factorization of Young projection operators for the symmetric group*, Lietuvos Fizikos Rinkinys, **11** (1) 9 (1971) [[journal content](http://www.itpa.lt/%7Elfd/Lfz/Turiniai/Turi1971.html#), [[Jucys-1971.pdf:file]]]
category: people
[[!redirects Algimantas Jucys]] |
algorithm | https://ncatlab.org/nlab/source/algorithm |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
(...)
## Applications
* [[effective homology]]
## Related concepts
* [[program]]
* [[programming language]]
* [[computation]]
* [[experimental mathematics]]
* [[long division]]
## References
* [[A. M. Turing]]. _On Computable Numbers, with an Application to the Entscheidungs problem_), Proceedings of the London Mathematical Society. 2 (1937) 42: 230–265. ([pdf](https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf))
* [[A. N. Kolmogorov]] and V. A. Uspénski. On the definition of an algorithm. Uspehi Mat. Nauk. 13 (1958), 3-28. English translation in American Mathematical Society Translations, Series II, Volume 29 (1963), pp. 217–245. ([math-net.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=7453&option_lang=eng)). Also see JSL review by Elliott Mendelson on [jstor](http://www.jstor.org/stable/2272011).
See also
* Wikipedia, _[Algorithm](http://en.wikipedia.org/wiki/Algorithm)_
[[!redirects algorithms]]
|
Algèbres Enveloppantes | https://ncatlab.org/nlab/source/Alg%C3%A8bres+Enveloppantes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
This pages provides links concerning the book:
* [[Jacques Dixmier]]:
\linebreak
**Algèbres Enveloppantes**
\linebreak
Cahiers scientifiques **37** (1974)
\linebreak
see also:
Proceedings of the International Congress of Mathematicians
Helsinki (1978)
[pdf](https://www.imj-prg.fr/wp-content/uploads/2020/prix/dixmier1978.pdf), [[Dixmier-AlgEnvelop-ICM78.pdf:file]]
\linebreak
English translation:
**Enveloping Algebras**
Graduate Studies in Mathematics **11**,
American Mathematical Society (1996)
[ams:gsm-11](https://bookstore.ams.org/gsm-11)
on [[universal enveloping algebras]].
category: reference
[[!redirects Algebres enveloppantes]]
|
Ali Caglayan | https://ncatlab.org/nlab/source/Ali+Caglayan | a.k.a Alizter
[mathoverflow](https://mathoverflow.net/users/54401/ali-caglayan) |
Ali Chamseddine | https://ncatlab.org/nlab/source/Ali+Chamseddine |
* [website](http://sites.google.com/site/achamseddine2/home)
category: people |
Ali Mozaffari | https://ncatlab.org/nlab/source/Ali+Mozaffari |
* [website](https://www.ucl.ac.uk/natural-sciences/dr-ali-mozaffari)
## Selected writings
On [[cosmic structure formation]] in view of [[MOND]]:
* [[Daniel Thomas]], [[Ali Mozaffari]], [[Tom Zlosnik]], *Consistent cosmological structure formation on all scales in relativistic extensions of MOND* ([arXiv:2303.00038](https://arxiv.org/abs/2303.00038))
category: people |
Ali Ovgun > history | https://ncatlab.org/nlab/source/Ali+Ovgun+%3E+history |
* see _[[Ali Övgün]]_
|
Alice Rogers | https://ncatlab.org/nlab/source/Alice+Rogers |
Alice Rogers is emeritus professor of pure mathematics at King's college London.
## related $n$Lab entries
* [[supermanifold]]
category: people |
Alisa Govzmann | https://ncatlab.org/nlab/source/Alisa+Govzmann |
* [institute page](https://wwwde.uni.lu/research/fstm/dmath/people/alisa_govzmann)
## Selected writings
On [[homotopy limits]], [[homotopy pullbacks]] and their [[pasting law for pullbacks|pasting law]] in/via [[model categories]]:
* [[Alisa Govzmann]], [[Damjan Pištalo]], [[Norbert Poncin]], *Indeterminacies and models of homotopy limits* [[arXiv:2109.12395](https://arxiv.org/abs/2109.12395)]
category: people
|
Alissa Crans | https://ncatlab.org/nlab/source/Alissa+Crans |
* [website](http://myweb.lmu.edu/acrans/)
## related entries
* [[Lie 2-algebra]]
* [[2-vector space]]
category: people
[[!redirects Alissa S. Crans]]
|
Alistair Savage | https://ncatlab.org/nlab/source/Alistair+Savage |
* [personal page](https://alistairsavage.ca/)
## Selected writings
Introduction to [[categorification]]:
* [[Alistair Savage]], *Introduction to categorification* [[arXiv:1401.6037](https://arxiv.org/abs/1401.6037), slides:[pdf](https://alistairsavage.ca/talks/2014-savage-intro-to-categorification.pdf)]
category: people |
all arrows monic > history | https://ncatlab.org/nlab/source/all+arrows+monic+%3E+history | < [[all arrows monic]] |
all changes | https://ncatlab.org/nlab/source/all+changes |
* [current](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
* [[2009 September changes|2009 September]]
* [[2009 August changes|2009 August]]
* [[2009 July changes|2009 July]]
* [[2009 June changes|2009 June]]
* [[2009 May changes|2009 May]]
* [[2009 April changes|2009 April]]
* [[2009 March changes|2009 March]]
* [[2009 February changes|2009 February]]
* [[2009 January changes|2009 January]]
* [[2008 changes|2008]]
<div markdown="1">[Edit this sidebar](/nlab/edit/all+changes)</div>
|
all horizontal weight systems are partitioned Lie algebra weight systems | https://ncatlab.org/nlab/source/all+horizontal+weight+systems+are+partitioned+Lie+algebra+weight+systems |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Knot theory
+-- {: .hide}
[[!include knot theory - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
All [[weight systems]] on [[horizontal chord diagrams]] may be realized as [[linear combinations]] of [[Lie algebra weight systems]] applied not necessarily to the given [[horizontal chord diagram]] itself, but to the result of regarding each of its strands as resolved by some [[finite number]] of strands.
We state this precisely as Prop. \ref{AllHorizontalWeightSystemsAreslNWeightSystems} below (due to [Bar-Natan 96](#BarNatan96)). First we introduce all the definitions that enter the statement:
## Ingredients
{#Ingredients}
Given any [[ground field]] $\mathbb{F}$ (or in fact just any [[commutative ring|commutative]] [[ground ring]])
1) write $\mathcal{A}^{pb}$ for the [[linear span]] of [[horizontal chord diagrams]] [[quotient vector space|modulo]] the [[2T relations]] and the [[4T relations]]
<center>
<img src="https://ncatlab.org/nlab/files/HorizontalChordDiagramsModulo2TAnd4T.jpg" width="900">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
regarded as a [[graded vector space]], graded by the number of chords,
and write
\[
\label{SpaceOfHorizontalWeightSystems}
\mathcal{W}_{pb}
\coloneqq
(\mathcal{A}^{pb})^\ast
\]
for its degreewise [[dual vector space]]: the space of _[[horizontal weight systems]]_;
2) write
$$
\mathcal{A}^{pb}
\overset{
\;\;\; p_n \;\;\;
}{\longrightarrow}
\mathcal{A}^{pb}_n
\overset{
\;\;\; i_n \;\;\;
}{\hookrightarrow}
\mathcal{A}^{pb}
$$
for projection onto and inclusion of the linear subspace spanned by [[horizontal chord diagrams]] with $n$ strands;
3) write
$$
\underset{\mathbb{N}}{\oplus} \mathbb{N}
\overset{
\;\;\;
\Delta^{(-)}
\;\;\;
}{\longrightarrow}
End( \mathcal{A}^{pb} \big)
$$
for the operation that reads in a [[finite number|finite]] [[tuple]] $k \coloneqq (k_1, \cdots, k_n)$ of [[natural numbers]], with [[sum]] $\left\vert k\right\vert \coloneqq \underset{i}{sum} k_i$, and produces the [[linear map]]
\[
\label{Delta}
\mathcal{A}^{pb}
\overset{
\;
p_n
\;
}{\to}
\mathcal{A}^{pb}_{n}
\overset{
\;\;\;
\Delta^k
\;\;\;
}{\longrightarrow}
\mathcal{A}^{pb}_{\left\vert k \right \vert}
\overset{
\;
i_{\left\vert k \right\vert}
\;
}{\hookrightarrow}
\mathcal{A}^{pb}
\]
which takes a [[horizontal chord diagram]] with $n$ strands to the [[linear combination]] of chord diagrams obtained by replacing its $i$-th strand by $k_i$ strands for all $i$ and then summing over all ways of re-attaching chords, with any vertex previously on some strand $i$ now to be put on one of the $k_i$ strands ([Bar-Natan 96, Def. 2.2](#BarNatan96)).
For example:
<center>
<img src="https://ncatlab.org/nlab/files/HorizontalChordDiagramPartitioning.jpg" width="600">
</center>
<center>
<img src="https://ncatlab.org/nlab/files/HorizontalChordDiagramPartitioningGenericII.jpg" width="700">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
\linebreak
Moreover, for $\mathfrak{g}$ a [[metric Lie algebra]]
1) write
$$
\mathfrak{g}Mod_{/\sim}
$$
for its [[set]] of [[isomorphism classes]] of [[finite dimensional vector space|finite dimensional]] [[Lie algebra representations]] ([[Lie modules]])
2) write
\[
\label{AssignLieAlgebraWeightSystem}
\mathfrak{g}Mod_{/\sim}
\overset{ w_{(-)} }{\longrightarrow}
\underset{n \in \mathbb{N}}{\oplus}
Hom_{\mathbb{F}}
\big(
\mathcal{A}^{pb}_n
,
End(C^{\otimes n})
\big)
\]
for the [[function]] that sends a [[Lie module]] $C$ over $\mathfrak{g}$ to the corresponding [[endomorphism ring]]-valued [[Lie algebra weight system]] $w_C$ on [[horizontal chord diagrams]].
Finally, for
1. $C \in \mathfrak{g} Mod$ a [[Lie algebra representation]] of $\mathfrak{g}$,
1. $n \in \mathbb{N}$ a [[natural number]],
1. $\sigma \in Sym(n)$ a [[permutation]] of $n$ elements
write
\[
\label{PermutationTraceOperation}
tr_\sigma
\;\colon\;
End
\big(
C^{\otimes n}
\big)
\longrightarrow
\mathbb{F}
\]
for the [[composition|composite]] operation of
1. [[composition|composing]] an [[endomorphism]] on the $n$-fold [[tensor product of vector spaces|tensor power]] of $C$ by the [[braiding]] according to the [[permutation]] $\sigma$;
1. forming the [[trace]] of the resulting endomorphism of $C^{\otimes n}$.
Then the [[composition]] of
1. the partitioning function (eq:Delta);
1. the assignment (eq:AssignLieAlgebraWeightSystem) of [[Lie algebra weight systems]];
1. the permuted [[trace]] operation (eq:PermutationTraceOperation)
yields a [[function]] from [[triples]] consisting of a [[Lie module]], a [[tuple]] of [[natural numbers]] and a [[permutation]] to [[horizontal weight systems]]:
\[
\label{MapAssigningPartitionedLieWeightSystemsToModules}
\array{
\big(
\mathfrak{g}Mod_{/\sim}
\big)
\;\times\;
\big(
\underset{\mathbb{N}}{\oplus} \mathbb{N}
\big)
\;
\underset{
\mathbb{N}
}{\times}
\;
\big(
\underset{n \in \mathbb{N}}{\sqcup} Sym(n)
\big)
&
\overset{
\;\;
tr_{(-)} \circ w_{(-)} \circ \Delta
\;\;
}{
\longrightarrow
}
&
\mathcal{W}_{pb}
\\
(C, \; k = (k_1, \cdots, k_n), \; \sigma)
&\mapsto&
\left(
D
\;\mapsto\;
tr_\sigma
\circ
w_C
\circ
\Delta^k (D)
\right)
}
\]
Finally, write also
\[
\label{LinearExtensionOfMapAssigningPartitionedLieWeightSystemsToModules}
Span
\big(
\mathfrak{g}Mod_{/\sim}
\;\times\;
\underset{\mathbb{N}}{\oplus} \mathbb{N}
\;\underset{\mathbb{N}}{\times}\;
\underset{n \in \mathbb{N}}{\mathbb{N}} Sym(n)
\big)
\overset{
tr_{(-)} \circ w_{(-)} \circ \Delta^{(-)} (-)
}{
\longrightarrow
}
\mathcal{W}_{pb}
\]
for the linear extension of this function (eq:MapAssigningPartitionedLieWeightSystemsToModules) to the [[linear span]] of its [[domain]] [[set]].
\linebreak
## Statement
+-- {: .num_prop #AllHorizontalWeightSystemsAreslNWeightSystems}
###### Proposition
([[all horizontal weight systems are partitioned Lie algebra weight systems]])
For $N \geq 2$ consider the [[general linear Lie algebra]] $\mathfrak{gl}(N)$, regarded as a [[metric Lie algebra]] not via its [[Killing form]], but via the fundamental trace $g(x,y) \,\coloneqq\, tr(x \circ y)$.
Then the space $\mathcal{W}_{pb} \coloneqq (\mathcal{A}^{pb})^\ast$ (eq:SpaceOfHorizontalWeightSystems) of [[weight systems]] on [[horizontal chord diagrams]] is [[linear span|spanned]] by partitioned $\mathfrak{gl}(N)$-[[Lie algebra weight systems]],
in that the linear extension (eq:LinearExtensionOfMapAssigningPartitionedLieWeightSystemsToModules) of the function (eq:MapAssigningPartitionedLieWeightSystemsToModules) assigning $\mathfrak{gl}(N)$-[[Lie algebra weight systems]] composed with partitioning (eq:Delta) is an [[epimorphism]]:
$$
\array{
Span
\Big(
\big(
\underset{
\mathclap{
\color{blue}
{Lie\;modules}
}
}{
\underbrace{
\mathclap{\phantom{\vert \atop \vert}}
\mathfrak{gl}(N) \, Mod_{/\sim}
}
}
\big)
\;
\times
\;
\big(
\underset{
\mathclap{
\color{blue}
tuples\;of\;numbers
}
}{
\underbrace{
\mathclap{ \phantom{\vert \atop \vert } }
\underset{\mathbb{N}}{\oplus} \mathbb{N}
}
}
\big)
\;
\underset{\mathbb{N}}{\times}
\;
\big(
\underset{
\color{blue}
permutations
}{
\underbrace{
\underset{n \in \mathbb{N}}{\sqcup}
Sym(n)
}
}
\big)
\Big)
&
\underoverset{\color{blue}epimorphism}{
\;\;\;
tr_{(-)}
\circ
w_{(-)}
\circ
\Delta^{(-)}
(-)
\;\;\;
}{\longrightarrow}
&
\overset{
\mathclap{
{\color{blue} horizontal\;weight\;systems}
\atop
{\phantom{a}}
}
}{
\mathcal{W}_{pb}
}
\\
(
C, \;\; k = (k_1, \cdots, k_n), \;\; \sigma
)
&\mapsto&
\left(
\;\;\;\;\;\;
\array{
\overset{
\mathclap{
\color{blue}
{
{horizontal}
\atop
{
{chord}
\atop
{diagram}
}
}
\atop
{\phantom{a}}
}
}{
D
}
\mapsto
&
\phantom{=\;}
\overset{
\mathclap{
\color{blue} \sigma\text{-}trace
}
}{
\overbrace{
tr_\sigma
}
}
\circ
\underset{
\mathclap{
{\color{blue}RT\;invariant}
}
}{
\underbrace{
W_{{}_{C^{\otimes k_1}, \cdots , C^{\otimes k_n} }}(D)
}
}
\;\;\;\;\;\;\;\;\;
\\
&
=
tr_\sigma
\circ
\underset{
\mathclap{
{\color{blue} End\text{-}valued\;Lie\;algebra\;weight\;system}
}
}{
\underbrace{
w_C
}
}
\circ
\overset{
\mathclap{
{\color{blue} partitioning}
}
}{
\overbrace{
\Delta^k
}
}
(D)
}
\right)
}
\,.
$$
=--
This is the statement of [Bar-Natan 96, Corollary 2.6](#BarNatan96).
{#Example} For example:
<center>
<img src="https://ncatlab.org/nlab/files/PartitionedLieAlgebraWeightSystem.jpg" width="840">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
\linebreak
## Applications
[[!include chord diagrams as multi-trace observables in BMN matrix model -- example]]
## Related theorems
[[!include facts about chord diagrams and weight systems -- contents]]
## References
The theorem and its proof is due to:
* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))
[[!redirects all horizontal weight systems are Lie algebra weight systems]]
|
All Pages > history | https://ncatlab.org/nlab/source/All+Pages+%3E+history | |
Allan J. Silberger | https://ncatlab.org/nlab/source/Allan+J.+Silberger |
* [personal page](https://academic.csuohio.edu/silberger/)
## Related concepts
On [[L-parameters]] in the [[local Langlands correspondence]]
* {#SilbergerZink14} [[Allan J. Silberger]], [[Ernst-Wilhelm Zink]], Section 3 of: *Langlands Classification for L-Parameters*, Journal of Algebra Volume 511, 1 October 2018, Pages 299-357 ([arXiv:1407.6494](https://arxiv.org/abs/1407.6494), [doi:10.1016/j.jalgebra.2018.06.012](https://doi.org/10.1016/j.jalgebra.2018.06.012))
Talk notes:
* [[Ernst-Wilhelm Zink]], with [[Allan J. Silberger]], *Langlands classification for L-parameters* ([pdf](https://www.math.hu-berlin.de/~zyska/zink/Lvortrag1.pdf))
category: people
[[!redirects Allan Silberger]] |
Allan L. Edelson | https://ncatlab.org/nlab/source/Allan+L.+Edelson |
* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=1087)
## Selected writings
On [[Real vector bundles]] and their [[KR-theory]]:
* [[Allan L. Edelson]], *Real Vector Bundles and Spaces with Free Involutions*, Transactions of the American Mathematical Society **157** (1971) 179-188 [[doi:10.2307/1995841](https://doi.org/10.2307/1995841), [jstor:1995841](https://www.jstor.org/stable/1995841)]
category: people |
Allan Merino | https://ncatlab.org/nlab/source/Allan+Merino |
* [personal page](http://allanmerino.com/)
## Selected writings
On [[reductive dual pair|reductive dual pairs]]/[[Howe duality]] for [[super Lie algebras]]:
* [[Allan Merino]], Hadi Salmasian, *Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup* [[arXiv:2208.09746](https://arxiv.org/abs/2208.09746)]
category: people |
allegorical set theory | https://ncatlab.org/nlab/source/allegorical+set+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
There are two general approaches to [[structural set theory]]; those that attempt to axiomatise the [[category]] [[Set]] of [[sets]] and [[functions]], and those that attempt to axiomatise the [[allegory]] [[Rel]] of [[sets]] and [[relations]]. The latter is called **allegorical set theory**.
## Examples
* [[SEAR]] is an example of a dependently sorted allegorical set theory with primitives of [[sets]], [[elements]], and [[relations]]
## See also
* [[structural set theory]]
* [[categorical set theory]], [[allegorical set theory]]
[[!redirects allegorical set theory]]
[[!redirects allegorical set theories]] |
allegory | https://ncatlab.org/nlab/source/allegory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea ##
In [[category theory]], an _allegory_ is a [[category]] with properties meant to reflect properties that hold in a category [[Rel]] of [[relations]]. The notion was first introduced (as far as we know) and certainly first made famous in the book _[[Categories, Allegories]]_ ([Freyd-Scedrov](#FreydScedrov)).
Freyd and Scedrov argue that a categorical calculus of relations is an alternative and often more amenable framework for developing concepts traditionally couched in "functional" language (i.e., concepts which apply to [[sets]] and [[functions]]); for instance, a principal _raison d'etre_ for [[regular categories]] is precisely that one can do [[relational calculus]] in them (as had been long known, e.g., [[Saunders MacLane]] showed how to calculate with relations to do [[diagram chasing|diagram chases]] in [[abelian categories]]). Allegories, and correlative notions such as [[bicategory of relations|bicategories of relations]], also offer a smooth approach to [[regular and exact completion|regular and exact completions]], as used for example in the construction of [[realizability topos|realizability toposes]].
A signal feature of allegories is emphasis on the _modular law_ (see def. \ref{Allegory} below), which generalizes the [[modular lattice|modular law]] in lattices to more general relations, and which generalizes also so-called [[Frobenius reciprocity]] in [[categorical logic]].
## Definition
+-- {: .num_defn #Allegory}
###### Definition
An **allegory** is a [[locally posetal 2-category]] $A$ equipped with an [[involution]] $(-)^o \colon A^{op} \to A$ which is the [[identity-on-objects]], such that
1. the involution is order preserving and distributes over composition, i.e. $ (\psi\phi)^o = \phi^o\psi^o $,
1. each [[hom-object|hom-poset]] $A(x,y)$ has binary [[meet|meets]], and
1. the *[[modular lattice|modular law]]* holds: for $\phi\colon x\to y$, $\psi\colon y\to z$, and $\chi\colon x\to z$, we have $\psi \phi \cap \chi \le \psi (\phi \cap \psi^o \chi)$.
=--
From these properties we immediately get that
\[
(\phi \cap \psi)^o = \phi^o \cap \psi^o
\]
and
\[
(\phi \cap \psi) \chi \leq \phi\chi \cap \psi\chi
\quad\text{and}\quad
\chi (\phi \cap \psi) \leq \chi\phi \cap \chi\psi.
\]
\begin{proof}
The first claim follows from the observation that
$$
\begin{aligned}
\chi \leq (\phi \cap \psi)^o
&\iff
\chi^o \leq \phi \cap \psi
&\quad&(.)^o\;\text{monotone and involutive}
\\
&\iff
\chi^o \leq \phi \; \text{ and } \; \chi^o \leq \psi
&&\text{meet properties}
\\
&\iff
\chi \leq \phi^o \; \text{ and } \; \chi \leq \psi^o
&&(.)^o\;\text{ monotone and involutive}
\\
&\iff
\chi \leq \phi^o \cap \psi^o
&&\text{meet properties.}
\end{aligned}
$$
For the claim $(\phi \cap \psi) \chi \leq \phi\chi \cap \psi\chi$ we use the [[horizontal composition]] in a [[locally posetal 2-category]], i.e. the fact that composition is monoton. Due to $\phi \cap \psi \leq \phi$ we get $(\phi \cap \psi)\chi \leq \phi\chi$. Analogously, we get that $(\phi \cap \psi)\chi \leq \psi\chi$. Hence $(\phi \cap \psi) \chi \leq \phi\chi \cap \psi\chi$. The claim $\chi (\phi \cap \psi) \leq \chi\phi \cap \chi\psi$ follows the the same arguments or by applying involution and the first claim.
\end{proof}
## Examples
* If $C$ is a [[regular category]] and $Rel(C)$ is the [[locally posetal 2-category|locally posetal]] [[bicategory]] of [[internal relation|internal relations]], then $Rel(C)$ is an allegory.
* Any [[first-order hyperdoctrine with equality]] similarly gives rise to an allegory, as does any abstract [[bicategory of relations]] in the sense of Carboni-Walters.
* Any [[modular lattice]] can be regarded as a one-object allegory if we take composition to be union and the involution to be the identity.
## Maps, tabulations, and units
A __map__ $ r\colon x \to y $ in an allegory is a morphism that has a [[right adjoint]]. If $r \dashv s$, then $s = r^o$ (hint: use the modular law to show $r \dashv s \cap r^o$ and $r \cap s^o \dashv s$).
The [[unit of an adjunction|unit of the adjunction]] $ id_x \leq r^o r $ entails that the morphism is __[[entire relation|entire]]__ (sometimes also called _total_) while the [[unit of an adjunction|counit of the adjunction]] $ r r^o \leq id_y $ states the fact hat the morphism is __[[functional relation|functional]]__ (sometimes also called _univalent_).
Any 2-category has a [[bicategory of maps]]. In an allegory, the ordering between maps is discrete, meaning that if $f \leq g$ then $f = g$. Consequently, the bicategory of maps of an allegory is a [[category]].
A __tabulation__ of a morphism $\phi$ is a pair of maps $f,g$ such that $\phi = g f^o$ and $f^o f \cap g^o g = 1$. An allegory is **tabular** if every morphism has a tabulation, and **pretabular** if every morphism is contained in one that has a tabulation.
Every [[regular category]], and indeed every [[locally regular category]], has a tabular allegory of internal binary relations. Conversely, by restricting to the morphisms with left adjoints ("maps") in a tabular allegory, we obtain a locally regular category. These constructions are inverse, so tabular allegories are equivalent to locally regular categories.
A locally regular category has finite products if and only if its tabular allegory of relations has top elements in its hom-posets.
Finally, a **unit** in an allegory is an object $U$ such that $1_U$ is the greatest morphism $U\to U$, and every object $X$ admits a morphism $\phi\colon X\to U$ such that $1_X\le \phi^o\phi$. A locally regular category has a terminal object (hence is regular) if and only if its tabular allegory of relations has a unit.
Thus, regular categories are equivalent to unital (or unitary) tabular allegories. For more details, see [[Categories, Allegories]] ([Freyd-Scedrov](#FreydScedrov)), the [[Elephant]] ([Johnstone](#Johnstone)), or [[toddtrimble:Theory of units and tabulations in allegories]].
## Division allegories
A **union allegory** is an allegory whose hom-posets have finite joins that are preserved by composition. Thus a union allegory is locally a [[lattice]]. If additionally it is locally a [[distributive lattice]], it is called a **distributive allegory**. The category of maps in a unitary tabular union allegory is a [[coherent category]] (a "pre-logos"), and conversely the bicategory of relations in a coherent category is a (unitary tabular) distributive allegory. In particular, every unitary tabular union allegory is distributive, but in the non-tabular case this can fail: for instance, any [[modular lattice]] can be regarded as a one-object union allegory and need not be distributive.
A **division allegory** is a distributive allegory in which composition on one (and therefore the other) side has a right adjoint (left or right division). That is: given $r: A \to B$ and $s \colon A \to C$, there exists $s/r: B \to C$ such that
$$t \leq s/r \in \hom(B, C) \Leftrightarrow t \circ r \leq s \in \hom(A, C)$$
(so that $- \circ r$ has right adjoint $-/r$: an example of a right [[Kan extension]]). Composition on the other side, $r \circ -$, has a right adjoint (an example of a right Kan lift) given by
$$r\backslash u \coloneqq (u^o/r^o)^o.$$
In the bicategory of sets and relations, with notation as above, we have
$$(s/r)(b, c) \dashv \vdash \forall_{a \colon A} r(a, b) \Rightarrow s(a, c)$$
where $r(a, b)$ is shorthand for "$(a, b)$ belongs to $r$".
The category of maps (functional relations) of a unitary/unital tabular division allegory is a [[Heyting category]] (a "[[logos]]"), and conversely the bicategory of relations in a Heyting category is a unitary tabular division allegory. ([Freyd-Scedrov](#FreydScedrov), 2.32, p. 227.)
## Power allegories
A **power allegory** is, *more or less*, an allegory $\mathcal{A}$ such that the inclusion functor $i: Map(\mathcal{A}) \to \mathcal{A}$ has a right adjoint $P$. The idea is that $P$ assigns to an object $A$ a power object $P(A)$, as in topos theory; if we summarize the notion of topos as a regular category $\mathbf{E}$ for which the inclusion $i: \mathbf{E} \to Rel(\mathbf{E})$ has a right adjoint $P$, then it becomes apparent that the notion of power allegory is similar except that it takes the "relation side" as primary and derives the "function side" as $Map(\mathcal{A})$, whereas in topos theory it's just the other way around. But in either case the adjunction $i \dashv P$ is fundamental.
Since the inclusion $i$ is the identity on objects, the counit of the adjunction $i \dashv P$ at an object $B$ may be written
$$\ni_B: P(B) \to B$$
and we have a kind of [[comprehension scheme]] that to each $r: A \to B$ there is a unique map $\chi_r: A \to P(B)$, the *characteristic map* of $r$, such that $r = \ni_B \circ \chi_r$. (If $A$ and $B$ are related by a property $r$, then for each $a$ there is a subobject $\chi_r(a)$ of $B$ consisting of elements $b$ so related to $a$.) In the case $r = 1_A: A \to A$, the characteristic map $\chi_{1_A}: A \to P(A)$ is called the *singleton map* of $A$, and more general $\chi_r$ may be defined as $P(r) \sigma_A$.
### Freyd-Scedrov definition
The exact definition of power allegory is a matter for consideration. One can get a certain distance just by adopting the naive definition suggested above, that a power allegory is nothing more than an allegory for which the inclusion $Map(\mathcal{A}) \to \mathcal{A}$ has a right adjoint $P$. (Below we introduce a similar notion that we call a $P$-allegory.) But it seems hard to develop a theory from the naive notion that rises to a level comparable to topos theory.
Freyd and Scedrov start with a structure of division allegory (thus packing in a good amount of internal logic from the start) and introduce the fundamental adjunction $i \dashv P$ in terms of that structure. For them, a *power allegory* is defined to be a division allegory which associates to each object $B$ a morphism $\ni_B \colon P(B) \to B$ such that for all $r \colon A \to B$
* $1_A \leq (r \backslash \ni_B) \circ (\ni_B \backslash r)$
which expresses the truth of the formula $\forall_{a \colon A} \exists_{S: P(B)} \forall_{b \colon B} S \ni_B b \Leftrightarrow r(a, b)$, and
* $(\ni_B \backslash \ni_B) \wedge (\ni_B \backslash \ni_B)^o \leq 1_{P(B)}$
which internalizes an axiom of extensionality, which reads $\forall_{b \colon B} (S \ni_B b) \Leftrightarrow (T \ni_B b) \vdash S = T$. Given those axioms, and given $r: A \to B$, one may define
$$\chi_r \coloneqq (\ni_B \backslash r) \wedge (r\backslash \ni_B)^o,$$
which internalizes the formula-definition $\chi_r(a, S) \coloneqq \forall_b S \ni_B b \Leftrightarrow r(a, b)$, and then show $\chi_r$ is a map. ([Freyd-Scedrov](#FreydScedrov), pp. 235-236.)
The bicategory of relations in a [[topos]] is a power allegory; conversely, the category of maps in a unitary tabular power allegory is a topos.
### Variant notion
Nevertheless, the spare elegance of the naive definition gives one something to shoot for. It appears that quite a decent theory can be developed just by adding the assumption of coproducts in an allegory: a reasonable and fairly mild assumption. (Most allegories don't admit many colimits; for example having coequalizers is pretty rare. But the standard examples do have finite coproducts, coinciding with coproducts on the maps/functional side.)
+-- {: .num_defn}
###### Definition
A $P$-allegory is an allegory $\mathcal{A}$ with finite coproducts[^1] for which the inclusion $i: Map(\mathcal{A}) \to \mathcal{A}$ has a right adjoint $P$.
=--
[^1]: We mean coproducts as certain conical colimits *qua* [[locally posetal 2-category]].
As before, the counit is denoted $\ni: i P \to 1_{\mathcal{A}}$. It is perhaps surprising that the notion of $P$-allegory is at least as strong as power allegory in the Freyd-Scedrov sense:
+-- {: .num_theorem}
###### Theorem
Any $P$-allegory is a division allegory in which the Freyd-Scedrov conditions on $\ni$ are satisfied.
=--
+-- {: .proof}
###### Proof (sketch)
For now we content ourselves with a description of the division structure.
Let $r: A \to C$ and $s: B \to C$ be morphisms; we construct a right Kan lift $s \backslash r$ of $r$ through $s$. This means that for all $t: A \to B$ we have $s \circ t \leq r$ if and only if $t \leq s \backslash r$. We begin by constructing the right Kan lift for the case $r = s = \ni_C: P C \to C$.
Define the internal union $\bigcup_C$ by $\bigcup_C = P(\ni_C): P P C \to P C$, and define an internal order relation $[\Rightarrow]_C: P C \to P C$ by $P C \stackrel{\in_{P C}}{\to} P P C \stackrel{\bigcup_C}{\to} P C$. (Here $\in \coloneqq \ni^o$.) Define $[\Leftarrow]_C \coloneqq [\Rightarrow]_C^o$.
We now show $[\Leftarrow]_C = \ni_C \backslash \ni_C$. That is, for $R: P C \to P C$, we show $\ni_C R \leq \ni_C$ is equivalent to $R \leq [\Leftarrow]$. The backward implication is easy; for the forward implication, one may prove it first for *reflexive* $R$ where we have an actual equation $\ni_C R = \ni_C$. It follows that $P(\ni_C) P(R) = P(\ni_C)$, whence $P(R) \leq \bigcup_C^o \bigcup_C$. A short calculation then yields $R = \ni_C P(R) \sigma_{P C} = \ni_C \bigcup_C^o \bigcup_C \sigma_{P C} = \ni_C \bigcup_C^o = [\Leftarrow]$. The case for general $R$ reduces to the reflexive case: form the reflexive completion $1 \vee R$ as the composite
$$C \stackrel{\langle 1_{P C}, R \rangle}{\to} C + C \stackrel{\nabla}{\to} C$$
which is where coproducts come in; here $\nabla$ is the codiagonal and we use the fact that $C + C$ in an allegory is a biproduct to form the pairing for the first arrow. It is easy to show that $\ni_C(1 \vee R) = \ni_C \vee \ni_C R = \ni_C$, and then we derive $R \leq 1 \vee R \leq [\Leftarrow]$ from before. Thus we have shown $[\Leftarrow]_C = \ni_C \backslash \ni_C$.
For general $r: A \to C, s: B \to C$, we claim the right Kan lift $s \backslash r: A \to B$ is given by $\chi_s^o (\ni_C \backslash \ni_C) \chi_r$. For, we have $r = \ni_C \chi_r$, whence
$$
\begin{aligned}
s t &\leq r &
\;&\iff & \ni_C \chi_s t &\leq \ni_C \chi_r
\\
&&& \iff & \ni_C \chi_s t \chi_r^o &\leq \ni_C
\\
&&& \iff & \chi_s t \chi_r^o &\leq \ni_C \backslash \ni_C
\\
&&& \iff & t &\leq \chi_s^o (\ni_C \backslash \ni_C) \chi_r
\end{aligned}
$$
thus proving the claim.
Further details may be found [here](https://ncatlab.org/toddtrimble/published/Note+on+power+allegories).
=--
Thus the notion of $P$-category is just as strong as the Freyd-Scedrov notion of power allegory, and one can then piggy-back on their further developments.
## Syntactic allegories
Let $T$ be a [[regular theory]]. There is then an allegory $\mathcal{A}_T$ given as follows:
* the objects are finite strings of [[sorts]] of $T$;
* a morphism $\vec X \to \vec Y$ is a predicate $P(\vec x, \vec y)$ of sort $\vec X, \vec Y$ (or rather a provable-equivalence class of such predicates);
* the identity $\vec X \to \vec X$ is (named by) $x_1 = x_1 \wedge x_2 = x_2 \wedge \cdots \wedge x_n = x_n$;
* the composite of $R \colon \vec X \to \vec Y$ and $S \colon Y \to \vec X$ is named by $\exists \vec y. R(\vec x, \vec y) \wedge S(\vec y, \vec z)$.
That $\mathcal{A}_T$ is an allegory is B.311 in [Freyd--Scedrov](#FreydScedrov); that it is in fact unitary and pre-tabular is B.312.
Further structure on $T$ gives rise to further structure on $\mathcal{A}_T$ (B.313): if $T$ is a [[coherent theory|coherent]], [[first-order theory|first-order]] or [[higher-order logic|higher-order]] theory, then $\mathcal{A}_T$ will be a distributive, division or power allegory respectively.
Every pre-tabular allegory has a tabular completion, given by splitting its coreflexive morphisms (i.e. those endomorphisms $R$ such that $R \subset id$). The category of maps in the coreflexive splitting of $\mathcal{A}_T$ is precisely the [[syntactic category]] of $T$.
### The existential quantifier
There are two possible ways to interpret a regular formula of the form $\exists y. R(x,y) \wedge S(y,z)$ in a unitary pre-tabular allegory, if $R$ and $S$ are interpreted as $r \colon X \to Y$ and $s \colon Y \to Z$ respectively: as the composite $s \cdot r$, or more 'literally' by:
* pulling $r$ and $s^o$ back to the same hom set and taking their intersection: $(p_1^o r p_1) \cap (p_2^o s^o p_2) \colon X \times Z \to Y \times Y$;
* then forcing the two $Y$s to be equal by post-composing with $\Delta_Y^o \colon Y \times Y \to Y$, applying the existential quantifier by post-composing with the unique map $!_Y$ to the unit, and post-composing with $!_Z^o$ to get a morphism into $Z$;
* then forcing the $Z$ in the domain to be equal to the $Z$ in the codomain by taking the meet with $p_2 \colon X \times Z \to Z$;
* and finally pulling back along (precomposing with the right adjoint of) $p_1 \colon X \times Z \to X$ to get a morphism $X \to Z$.
We would like to know that these morphisms are equal, so that an existential formula will have a unique interpretation:
+-- {: .num_prop}
###### Proposition
$$ s r = (p_2 \cap !_Z^o !_Y \Delta_Y^o ((p_1^o r p_1) \cap (p_2^o s^o p_2))) p_1^o $$
=--
+-- {: .proof}
###### Proof
We show inclusion in each direction.
Firstly, $s r = s r \cap p_2 p_1^o$, because the product projections tabulate the top morphism. Notice also that the RHS above is equal to
$$ (\top_{Y Z} (r p_1 \cap s^o p_2) \cap p_2) p_1^o $$
where $\top_{Y Z}$ is the top morphism $Y \to Z$.
Now we can calculate:
$$
\begin{aligned}
s r \cap p_2 p_1^o
& = (s r p_1 \cap p_2) p_1^o &\quad& \text{modular law} \\
& = (s r p_1 \cap p_2 \cap p_2) p_1^o \\
& \leq (s(r p_1 \cap s^o p_2) \cap p_2) p_1^o && \text{modular law} \\
& \leq (\top_{Y Z} (r p_1 \cap s^o p_2) \cap p_2) p_1^o
\end{aligned}
$$
In the other direction, we have
$$
\begin{aligned}
(\top_{Y Z} (r p_1 \cap s^o p_2) \cap p_2) p_1^o
& \leq (\top_{Y Z} s^o (s r p_1 \cap p_2) \cap p_2) p_1^o
&\quad& \text{modular law} \\
& \leq (\top_{Y Y} (s r p_1 \cap p_2) \cap p_2) p_1^o
&& \top_{Y Z} s^0 \leq \top_{Y Y}\\
& = (p_2 (p_1^o s r p_1 \cap p_1^o p_2) \cap p_2) p_1^o
&& \top{Y Y} = p_2 p_1^o \\
& = p_2 (p_1^o s r p_1 \cap p_1^o p_2 \cap p_2^o p_2) p_1^o
&& \text{modular law} \\
& = p_2 (p_1^o s r p_1 \cap (p_1^o \cap p_2^o) p_2) p_1^o
&& \text{maps distribute} \\
& = p_2 (p_1^o s r p_1 \cap \Delta p_2) p_1^o
&& \text{see below} \\
& = p_2 \Delta (\Delta^o p_1^o s r p_1 \cap p_2) p_1^o
&& \text{modular law} \\
& = (s r p_1 \cap p_2) p_1^o
&& p_1 \Delta = p_2 \Delta = id \\
& = s r \cap p_2 p_1^o
&& \text{modular law}
\end{aligned}
$$
and we are back to where we started. In the fourth-last step we used the fact that if $p_1, p_2 \colon Z \times Z \to Z$ are the projections, then $p_1 \cap p_2 = \Delta^o$. But $\Delta = \langle id, id \rangle$, and from lemma 1 [[finnlawler:allegory|here]] we have that $\langle id, id \rangle = p_1^o \cap p_2^o$.
=--
## Related entries
* [[dagger 2-poset]]
Other attempted axiomatizations of the same idea "something that acts like the category of relations in a regular category" include:
* [[bicategory of relations]] (a special sort of [[cartesian bicategory]])
* [[1-category equipped with relations]]
Discussion of the relation between pretabular unitary allegories and bicategories of relations, and also between tabular unitary allegories and regular categories is in
* [[toddtrimble:Theory of units and tabulations in allegories]]
## References
The standard monograph is
* {#FreydScedrov}[[Peter Freyd]] and Andre Scedrov, _[[Categories, Allegories]]_, Mathematical Library Vol 39, North-Holland (1990). ISBN 978-0-444-70368-2.
The notion is discussed also in chapter A3 of
* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]]_
In
* [[Bob Walters]], _Categorical algebras of relations_ ([blog post](http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html) )
it is shown that any [[bicategory of relations]] is an allegory.
See also
* Wikipedia, _[Allegory](http://en.wikipedia.org/wiki/Allegory_%28category_theory%29)_
An introduction headed towards applications in computer science and, in particular, fuzzy controllers can be found in
* Michael Winter, _Goguen Categories_ (2007)
[[!redirects tabular allegory]]
[[!redirects union allegory]]
[[!redirects division allegory]]
[[!redirects power allegory]]
[[!redirects allegories]]
[[!redirects tabular allegories]]
[[!redirects union allegories]]
[[!redirects division allegories]]
[[!redirects power allegories]]
|
Allen Hatcher | https://ncatlab.org/nlab/source/Allen+Hatcher |
* [website](https://math.cornell.edu/allen-hatcher)
## Selected writings
On [[algebraic topology]]:
* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]
## Related entries
* [[topology]], [[algebraic topology]]
* [[vector bundles]], [[topological K-theory]]
* [[Whitehead tower]]
category: people
[[!redirects Alan Hatcher]] |
Allen Knutson | https://ncatlab.org/nlab/source/Allen+Knutson |
* [webpage](http://www.math.cornell.edu/m/People/Faculty/knutson)
category: people
|
almost connected topological group | https://ncatlab.org/nlab/source/almost+connected+topological+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn #AlmostConnected}
###### Definition
A [[locally compact topological space|locally compact]] [[topological group]] $G$ is called **[[almost connected topological group|almost connected]]** if the underlying [[topological space]] of the [[quotient]] topological group $G/G_0$ (of $G$ by the [[connected component]] of the [[neutral element]], also called the [[identity component]]) is [[compact topological space|compact]].
=--
See for instance ([Hofmann-Morris, def. 4.24](#HofmannMorris)).
\begin{remark}
Since the connected component $G_0$ is [[closed subspace|closed]], the [[neutral element]] in $G/G_0$ is a [[closed point]]. It follows that $G/G_0$ is [[T1-space|$T_1$]] and therefore (because it is a [[uniform space]]) also $T_{3 \frac1{2}}$ (a [[Tychonoff space]]; see at *[[uniform space]]* for details). In particular, $G/G_0$ is a [[compact Hausdorff space]].
\end{remark}
+-- {: .num_defn }
###### Example
Every [[compact topological space|compact]] and every [[connected topological space|connected]] [[topological group]] is almost connected.
Also every [[quotient]] of an almost connected group is almost connected.
=--
## Related concepts
* [[connected topological space]]
* [[maximal compact subgroup]]
## References
Textbooks with relevant material:
* {#Stroppel} M. Stroppel, _Locally compact groups_, European Math. Soc., (2006)
* {#HofmannMorris} [[Karl Hofmann]] Sidney Morris, _The Lie theory of connected pro-Lie groups_, Tracts in Mathematics 2, European Mathematical Society, (2000)
Original articles:
* Chabert, Echterhoff, Nest, _The Connes-Kasparov conjecture for almost connected groups and for linear $p$-adic groups_ ([pdf](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_2003__97_/PMIHES_2003__97__239_0/PMIHES_2003__97__239_0.pdf))
[[!redirects almost connected group]]
[[!redirects almost connected topological groups]]
[[!redirects almost connected groups]]
|
almost equality | https://ncatlab.org/nlab/source/almost+equality |
In [[measure theory]], two [[functions]] $f, g\colon X \to Y$ are __almost-everywhere equal__, or __almost equal__, if their [[equaliser]]
$$ \{ a\colon X \;|\; f(a) = g(a) \} $$
is a [[full set]]. We are usually only interested when the functions $f$ and $g$ are [[measurable function|measurable]], although this is not technically necessary. We also may consider when $f$ and $g$ are [[almost functions]] ([[partial functions]] whose domains are full).
As we need to know what a full set is, there is no notion of almost equality on an arbitrary classical [[measurable space]]. However, if the measurable space is equipped with such a notion (as is always the case with a [[Cheng measurable space]] or a [[localisable measurable space]]), then we have almost functions. Of course, a [[measure space]] also has plenty of structure for this. The morphisms between [[measurable locales]] are also inherently considered only up to almost equality.
Besides measure theory, the concept applies whenever we have a notion of something being true (in this case, that two functions are equal) [[almost everywhere]].
[[!redirects almost equal]]
[[!redirects almost equality]]
[[!redirects almost-everywhere equal]]
[[!redirects almost-everywhere equality]]
|
almost function | https://ncatlab.org/nlab/source/almost+function |
In [[measure theory]], an __almost-everywhere-defined function__, or __almost function__, is a [[partial function]] whose domain is a [[full set]]. We are usually only interested in *[[measurable function|measurable]]* almost functions. Typically, we consider almost functions up to the [[equivalence relation]] of almost equality, whereby two almost functions are __[[almost equality|almost equal]]__ if their [[equaliser]] is also a full set, that is if they are equal almost everywhere.
As we need to know what a full set is, there is no notion of almost function on an arbitrary classical [[measurable space]]. However, if the measurable space is equipped with such a notion (as is always the case with a [[Cheng measurable space]] or a [[localisable measurable space]]), then we have almost functions. Of course, a [[measure space]] also has plenty of structure for this. The morphisms between [[measurable locales]] also inherently correspond to measurable almost functions (up to almost equality).
It is a commonplace that one really only cares about measurable functions up to almost equality. That one only needs measurable functions to be defined almost everywhere is the same idea. However, in [[classical mathematics]], every almost function may be extended (using [[excluded middle]]) to an actual (everywhere-defined) [[function]], and this extension is unique up to almost equality. Accordingly, the notion of almost function is only necessary in [[constructive mathematics]]. However, even classically, using them from the start can avoid annoying but trivial technicalities. (For example, one does not need the notion of [[essentially bounded function]]; [[bounded function|bounded]] almost functions will do.)
Besides measure theory, the concept applies whenever we have a notion of something being true (in this case, that a partial function is defined) [[almost everywhere]].
[[!redirects almost function]]
[[!redirects almost functions]]
[[!redirects almost-everywhere-defined function]]
[[!redirects almost-everywhere-defined functions]]
[[!redirects almost-everywhere defined function]]
[[!redirects almost-everywhere defined functions]]
[[!redirects almost everywhere-defined function]]
[[!redirects almost everywhere-defined functions]]
[[!redirects almost everywhere defined function]]
[[!redirects almost everywhere defined functions]]
[[!redirects almost-everywhere-defined]]
[[!redirects almost-everywhere defined]]
[[!redirects almost everywhere-defined]]
[[!redirects almost everywhere defined]]
|
almost Hermitian structure | https://ncatlab.org/nlab/source/almost+Hermitian+structure |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An _almost Hermitian structure_ a [[reduction of the structure group]] along the inclusion $U(n) \hookrightarrow GL(n,\mathbb{C})$ of the [[unitary group]] into the [[complex numbers|complex]] [[general linear group]].
Under further embedding $U(n) \hookrightarrow GL(n,\mathbb{C}) \hookrightarrow GL(2n,\mathbb{R})$ an almost hermitian structure on the [[frame bundle]] of a [[smooth manifold]], hence a [[G-structure]] for $G = U(n)$, is first of all the choice of an [[almost complex structure]] and then an [[almost Hermitian manifold]] structure.
An [[integrability of G-structure|first-order intgrable]] $U(n)$-structure (almost Hermitian manifold) structure is _[[Kähler manifold]]_ structure.
By the fact (see at _[unitary group -- relation to orthogonal, symplectic and general linear group](unitary+group#RelationToOrthogonalSymplecticAndGeneralLinearGroup)_) that $U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})$ this means that an almost Hermitian structure is precisely a joint [[orthogonal structure]], [[almost symplectic structure]] and [[almost complex manifold]].
## Properties
### Relation to almost complex structure
Since the inclusion $U(n) \hookrightarrow GL(2n,\mathbb{R})$ factors through the [[symplectic group]] via the [[maximal compact subgroup]] inclusion
$$
U(n) \hookrightarrow Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})
$$
an almost Hermitian manifold structure is in particular an [[almost complex structure]]. Conversely, since the [[maximal compact subgroup]] inclusion is a [[homotopy equivalence]], there is no [[obstruction]] to lifting an almost complex structure to an almost Hermitian structure.
### Relation to Kähler manifolds
An [[integrable G-structure|first-order integrable]] almost Hermitian structure is a _[[Kähler manifold]] structure_.
## Related concepts
* [[Kähler manifold]]
* [[Hermitean Yang-Mills connection]]
[[!redirects almost Hermitian structures]]
[[!redirects Hermitean structure]]
[[!redirects Hermitean structures]]
[[!redirects almost Hermitean structure]]
[[!redirects almost Hermitean structures]] |
almost Kähler geometric quantization | https://ncatlab.org/nlab/source/almost+K%C3%A4hler+geometric+quantization |
#Contents#
* table of contents
{:toc}
## Idea
A [[Kähler structure]] on a [[symplectic manifold]] induces a [[polarization]], and [[geometric quantization]] with respect to such [[Kähler polarizations]] works particularly well and has numerous examples (e.g. the [[orbit method]] and [[quantization of Chern-Simons theory]]).
An _[[almost Kähler structure]]_ does not induce a [[polarization]] in the usual sense, unless it is actually a Kähler structure. Nevertheless, via [[geometric quantization by push-forward]] there is still a concept of [[geometric quantization]] of almost Kähler structures (in the case of finite-dimensional manifolds), which is "as good" as Kähler quantization ([Borthwick-Uribe 96](#BorthwickUribe96)).
Examples of almost-Kähler structures on [[phase spaces]] include the phase space of the [[scalar field]] with reasonable [[interaction]] terms, such as the [[free field]] or the [[phi^4 theory|phi^4 interaction]] ([Collini 16, section 3.2.2](#Collini16)).
## Related concepts
* [[pseudo-holomorphic vector bundle]]
* [[Fedosov deformation quantization]] naturally applies to almost-Kähler structures, too
## References
A discussion of a kind of [[geometric quantization via push-forward]] on finite-dimensional almost Kähler manifolds is in
* {#BorthwickUribe96} [[David Borthwick]], [[Alejandro Uribe]], _Almost complex structures and geometric quantization_ ([arXiv:dg-ga/9608006](https://arxiv.org/abs/dg-ga/9608006))
Discussion of the almost-Kähler structure on the [[phase space]] of the [[scalar field]] is in
* {#Collini16} [[Giovanni Collini]], chapter 3 of _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))
On [[symplectic reduction]] in the almost-Kähler case:
* Martin Otto, _A reduction scheme for phase spaces with almost Kähler symmetry: Regularity results for momentum level sets_, Journal of Geometry and Physics
Volume 4, Issue 2, 1987, Pages 101-118
[[!redirects almost Kähler quantization]]
|
almost module | https://ncatlab.org/nlab/source/almost+module |
#Contents#
* table of contents
{:toc}
_Under construction: Extracted from a series of tweets by Syzygay_
## Idea
Let $R$ be a (commutative unital) [[ring]]. Suppose $I$ is a flat idempotent $R$-ideal. To be a flat ideal means $-\otimes I$ is an [[exact functor]]. The tensor product is always right exact, so in particular, $-\otimes I$ preserves injections. To be idempotent, $I^2 = I$.
Fixing a pair $(R,I)$ like this, we construct the category of "$I$-almost $R$-modules", $alMod R$.
We take the full subcategory of $R$-modules spanned by $M$ so that $I\otimes M \equiv M$. (This is not the actual definition, but this is an equivalent category. The correct definition is the quotient of the category of $R$-modules by the category of $I$-almost $0$ modules. See below.)
the category structure (it's the essential image of the functor ModR→ModR via M↦I⊗M, it's an Abelian category, and it's a localization of ModR).
There is an inclusion of categories j_! : alModR → ModR.
There is an exact right adjoint of j_!, denoted j* : ModR → alModR which sends M to I⊗M.
And j* has a right adjoint j_* : alModR → ModR via
j_*(M) = Hom(I, j_!(M)).
j* does legitimately map into alModR. To prove this, we must show that I⊗j*M≅j*M.
I⊗j*M = I⊗(I⊗M).
Since I is a flat ideal, ≅ I²⊗M.
Since I is idempotent, ≅ I⊗M = j*M.
we can think of j* as sending a module to the nearest almost module.
In some sense, the ideal I acts like an identity (under the tensor product) for almost modules. So the idea here is that I doesn't really do anything to the (I-)almost R modules.
An R-module M is (I-)almost 0 if IM = 0.
Since I is idempotent, if IM = 0, then j*M = I⊗M = 0 as well.
So the nearest "almost module" to an "almost 0" module is 0 itself. The only defect to being 0 is killed by the identity-like ideal.
Let f : M → N be an R-linear map.
f is almost injective if ker f is almost 0.
f is almost surjective if coker f is almost 0.
f is an almost isomorphism if ker f and coker f are almost 0.
An "almost" property of a map gets sent to the actual property by j*.
M is almost flat if j*M is flat in alModR; equivalently, if Torᵢ(M,N) is almost 0 for all i > 0 and for all N.
The problem with alModR is that the projective objects behave strangely, so instead, we define "almost Hom", alHom.
For two almost modules M and N,
alHom(M,N) = j*(Hom(M,N)).
Note: alHom(j*M,j*N) ≅ j*(Hom(M,N)).
Note: For L, M, N in alModR,
Hom(L⊗M,N) ≅ Hom(L, alHom(M,N)) where Hom's are taken in the category alModR.
So we do also get a form of "almost" hom-tensor adjunction.
A module M is almost projective if alHom(j*M,-) is exact in alModR, or equivalently, if Extⁿ(M,N) is almost 0 for all N and for all n > 0.
What are examples of rings R with flat idempotent ideals I?
It turns out that if R is a perfectoid ring and I is generated by a pseudo-uniformizer, then (R,I) satisfies these conditions!
One application is proving the tilting equivalence of perfectoid algebras. Given a perfectoid field K whose pseudo-uniformizer ϖ satisfies |p|≤|ϖ|≤1 for some prime p, there is a perfectoid field K♭ of char p (the tilt of K).
If K is a perfectoid field, then the category of perfectoid K-algebras is equivalent to the category of perfectoid K♭-algebras. Hence, we can reduce some problems in mixed char to char p.
The category of I-almost 0 modules is a Serre subcategory of Mod R. The "correct" definition of the category of I-almost R-modules is the quotient of the category of R-modules by the category of I-almost 0 modules.
## Related entries
* [[almost scheme]]
* [[p-adic Hodge theory]]
* [[perfectoid space]]
## References
* [[Bhargav Bhatt]], _Almost Ring Theory I_, 2014 ([[BhattAlmostRing.pdf:file]])
* [[Ofer Gabber]], [[Lorenzo Ramero]], _Almost ring theory_, Springer 2003 ([arXiv:math/0002064](https://arxiv.org/abs/math/0002064), [doi:10.1007/b10047](https://link.springer.com/book/10.1007/b10047))
[[Ofer Gabber]], [[Lorenzo Ramero]], *Foundations for almost ring theory* ([arXiv:math/0409584](https://arxiv.org/abs/math/0409584))
* [[Gerd Faltings]], _Almost étale extensions_, Cohomologies p-adiques et applications arithmétiques (II), Astérisque no. 279 (2002), p. 185-270 ([numdam:AST_2002__279__185_0](http://www.numdam.org/item/AST_2002__279__185_0))
See also
* Wikipedia, _[Almost ring](https://en.wikipedia.org/wiki/Almost_ring)_
[[!redirects almost modules]]
[[!redirects almost ring]]
[[!redirects almost rings]]
[[!redirects almost ring theory]]
[[!redirects almost mathematics]]
|
almost open subspace | https://ncatlab.org/nlab/source/almost+open+subspace |
A [[subspace]] $A$ of a [[space]] $X$ is __almost open__ if it is [[open subspace|open]] modulo the $\sigma$-[[sigma-ideal|ideal]] of [[meagre subspace|meagre subspaces]]. We also say that $A$ has the __Baire property__.
Explicitly, $A$ is almost open if there exist an open subspace $G$ and an [[infinite sequence]] $N_1, N_2, \ldots$ of [[nowhere dense subspace|nowhere dense subspaces]] (meaning that their [[topological closure|closures]] have [[empty subspace|empty]] [[topological interior|interiors]]) such that
$$ A \cup \bigcup_i N_i = G \cup \bigcup_i N_i .$$
That every subspace of the [[real line]] is almost open follows from the [[axiom of determinacy]] but contradicts the [[axiom of choice]]. In the absence of choice, it is a convenient assumption to make and is one of the axioms of [[dream mathematics]].
category: foundational axiom
[[!redirects almost open set]]
[[!redirects almost open sets]]
[[!redirects almost open subset]]
[[!redirects almost open subsets]]
[[!redirects almost open subspace]]
[[!redirects almost open subspaces]]
[[!redirects baire property]]
[[!redirects Baire property]]
|
almost scheme | https://ncatlab.org/nlab/source/almost+scheme |
#Contents#
* table of contents
{:toc}
## Motivation
Gabber and Ramero have developed a chapter in abstract algebra and [[algebraic geometry]] which should, in the style of Grothendieck's theory of [[schemes]], capture some of the ideas of
* [[Gerd Faltings]], _Almost étale extensions_, Cohomologies p-adiques et applications arithmétiques (II), Astérisque no. 279 (2002), p. 185-270 ([numdam:AST_2002__279__185_0](http://www.numdam.org/item/AST_2002__279__185_0))
To this aim they define a notion of almost ring/algebra, almost module, and finally, __almost scheme__. The framework is akin to many ideas in [[noncommutative algebraic geometry]] and in particular uses localization theory (compare also the generalized algebraic geometry of [[Nikolai Durov]]).
## Definitions
One starts with a "basic setup" consisting of a fixed _commutative_ unital ("[[base ring|base]]") ring $V$ containing an ideal $I$ such that $I^2 = I$. Usually one also assumes that $\tilde{I} := I \otimes_V I V$ is a flat $V$-module. A $V$-module $M$ is __almost zero__ if $IM=0$. A morphism $f:M\to N$ of $V$-modules is __almost isomorphism__ if $Ker f$ and $Coker f$ are almost zero, or equivalently if $\tilde{I}\otimes_V f : \tilde{I}\otimes_V M\to \tilde{I}\otimes_V N$ is an isomorphism. Almost isomorphisms form a category of fractions in $V-Mod$. The full subcategory $\Sigma$ of $V-Mod$ of $V$-modules which are almost isomorphic to $0$ is a Serre subcategory of $V-Mod$, hence one can construct the quotient category $V^a-Mod := V-Mod/\Sigma$ with the localization functor $Q^* : V-Mod\to V-Mod/\Sigma$; the image of $V$ in $V-Mod/\Sigma$ is denoted by $V^a$. The usual tensor product on $V-Mod$ induces a tensor product on $V^a-Mod$ making it into an abelian symmetric monoidal category. An __almost algebra__ is a commutative monoid in this tensor category; almost algebras form a category $V^a-Alg$; in the usual way one defines the (categories of) left and right (unital or not) modules over any given almost algebra.
An __affine almost scheme__ is just an object of an opposite category to $V^a-Alg$. A number of Grothendieck topologies are available on the category of affine almost schemes $V^a-Aff$; leading to the notions of locally affine spaces in these topologies; or better in relative version $R^a-Aff = (R^a-Alg)^{op}$ where $R$ is a $V$-algebra. By convention, the default topology is an appropriate version of flat topology, leading to the site $(R^a-Aff)_{fpqc}$. An __almost $R$-scheme__ is by the definition a sheaf of sets on $(R^a-Aff)_{fpqc}$.
More general "basic setups" in terms of topoi are also considered in the book.
##Literature##
* [[Ofer Gabber]], Lorenzo Ramero, _Almost Ring Theory_, Lec. Notes in Math. 1800, Springer 2003.
Abstract of the first arXiv release (51 pages [math.0002064](http://arxiv.org/abs/math/0002064))
>The categories of almost modules and almost algebras are introduced as a convenient setting for the development of Faltings' method of almost etale extensions. After some preliminaries of general "almost homological algebra" we construct the almost version of the cotangent complex and we use it to generalise some results of Faltings on the lifting of almost etale morphisms and almost etale algebras over nilpotent extensions. We also study the "almost trace" of an almost flat and almost finitely presented morphism, in particular we show that the almost trace is (almost) perfect if and only if the morphism is almost etale. Finally we study some cases of non-flat descent for almost rings, and establish the invariance of almost etale morphisms under Frobenius.
Abstract of 6th public release at [arXiv:math/0201175] (http://arxiv.org/abs/math/0201175) (about 230 pages)
>We develop almost ring theory, which is a domain of mathematics somewhere halfway between ring theory and category theory (whence the difficulty of finding appropriate MSC-class numbers). We apply this theory to valuation theory and to p-adic analytic geometry. You should really have a look at the introductions (each chapter has one).
In the introduction (6th public release), the authors heuristically place their work in the general field of _abstract algebraic geometry_:
>The purpose of the game is to reconstruct in this new framework as much as possible (and useful) of classical linear and commutative algebra. Essentially, this is the same as the ideology informing Deligne's paper [[Catégories Tannakiennes]], which sets out to develop algebraic geometry in the context of abstract tannakian categories. We could also claim an even earlier ancestry, in that some of the leading motifs resonating throughout our text, can be traced as far back as Gabriel's memoir [[Des catégories abéliennes]].
>In evoking Deligne's and Gabriel's works, we have unveiled another source of motivation whose influence has steadily grown throughout the long gestation of our paper. Namely, we have come to view almost ring theory as a contribution to that expanding body of research of still uncertain range and shifting boundaries, that we could call "abstract algebraic geometry". We would like to encompass under this label several heterogeneous developments: notably, it should include various versions of non-commutative geometry that have been proposed in the last twenty years, but also the relative schemes of M.HAKIM, _Topos annelés et schémas relatifs_, Springer Ergebnisse Math. Grenz. 64 (1972), as well as Deligne's ideas for algebraic geometry over symmetric monoidal categories.
> The common thread loosely unifying these works is the realization that "geometric spaces" do not necessarily consist of set-theoretical points, and - perhaps more importantly -functions on such "spaces" do not necessarily form (sheaves of) commutative rings. Much effort has been devoted to extending the reach of geometric intuition to non-commutative algebras; alternatively, one can retain commutativity, but allow "structure sheaves" which take values in tensor categories other than the category of rings. As a case in point, to any given almost ring $A$ one can attach its spectrum $Spec A$, which is just $A$ viewed as an object of the opposite of the category $V^a-Alg$. $Spec A$ has even a natural flat topology, which allows to define more general almost schemes by gluing (i.e. taking colimits of) diagrams of affine spectra; all this is explained in section 5.7, where we also introduce quasi-projective almost schemes and investigate some basic properties of the smooth locus of a quasi-projective almost scheme. By way of illustration, these generalities are applied in section 5.8 in order to solve a deformation problem for torsors over affine almost group schemes; let us stress that the problem in question is stated purely in terms of affine objects (i.e. almost rings and "almost Hopf algebras"), but the solution requires the introduction of certain auxiliary almost schemes that are not affine.
## Related concepts
* [[almost module]]
[[!redirects almost schemes]] |
almost surely | https://ncatlab.org/nlab/source/almost+surely |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
In [[probability theory]] an [[event (probability theory)|event]] of [[probability]] 1 is said to happen *almost surely*.
In [[measure theory]], such subsets are also known as [[full subsets]].
Their [[complements]] are known as [[null subsets]] or [[negligible subsets]].
## Related concepts
* [[null subset]]
* [[full subset]]
* [[deterministic random variable]]
## References
See also:
* Wikipedia, *[Almost surely](https://en.wikipedia.org/wiki/Almost_surely)*
|
Alonso Botero | https://ncatlab.org/nlab/source/Alonso+Botero |
* [Institute page](https://fisica.uniandes.edu.co/en/professors/alonso-botero-mejia)
* [GoogleScholar page](https://scholar.google.com/citations?user=e06A7mUAAAAJ&hl=en)
## Selected writings
Relating [[quantum information theory]] to the [[representation theory of the symmetric group]]:
* [[Alonso Botero]], *Quantum Information and the Representation Theory of the Symmetric Group*, Rev. colomb. mat. vol.50 no. 2 Bogotá July/Dec. 2016 ([doi:10.15446/recolma.v50n2.62210](https://doi.org/10.15446/recolma.v50n2.62210), [pdf](http://www.scielo.org.co/pdf/rcm/v50n2/v50n2a05.pdf))
category: people |
Alonso Perez-Lona | https://ncatlab.org/nlab/source/Alonso+Perez-Lona |
doctoral student at Virginia Tech in the group of [[Eric Sharpe]].
* [MO page](https://mathoverflow.net/users/172910/alonso-perez-lona)
## Selected writings
On [[orbifolds]] by [[2-groups]] in view of [[sigma-models]] inspired from [[string theory]]:
* [[Alonso Perez-Lona]], [[Eric Sharpe]], *Three-dimensional orbifolds by 2-groups* [[arXiv:2303.16220](https://arxiv.org/abs/2303.16220)]
category: people
[[!redirects Alonso Perez Lona]]
[[!redirects perezl.alonso]]
|
Alonzo Church | https://ncatlab.org/nlab/source/Alonzo+Church | * [Wikipedia entry](http://en.wikipedia.org/wiki/Alonzo_Church)
## Selected writings
On [[simple type theory]]:
* {#Church40} [[Alonzo Church]], §5 of: *A Formulation of the Simple Theory of Types*, The Journal of Symbolic Logic **5** 2 (1940) 56-68 [[doi:10.2307/2266170](https://doi.org/10.2307/2266170)]
## Related entries
* [[judgement]]
category:people |
alpha particle | https://ncatlab.org/nlab/source/alpha+particle | [[!redirects alpha particles]]
## Idea
The [[atomic nucleus]] of [[Helium]] is also called the _alpha-particle.
## References
* Wilipedia, _[Alpha particle](https://en.m.wikipedia.org/wiki/Alpha_particle)
|
alpha-equivalence | https://ncatlab.org/nlab/source/alpha-equivalence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--
\tableofcontents
## Idea
In [[logic]] and [[type theory]], $\alpha$-equivalence is the principle that two syntactic expressions ([[types]], [[terms]], [[propositions]], [[contexts]], whatever) are equivalent for all purposes if their only difference is the renaming of bound [[variables]].
Depending on the technicalities of how variables are managed, $\alpha$-equivalence may be a necessary axiom, a provable theorem, or entirely trivial. In any case, it is often (usually? always?) seen as a technicality devoid of conceptual interest. However, it can be a technically nontrivial task to implement $\alpha$-conversion (which is necessary to avoid capture of free variables upon [[substitution]]) when programming logic or type theory into a computer.
## Definition
We work in a [[dependent type theory]] which acts as the metatheory for our object type theory.
The type $\mathbb{A}$ is a [[countable set]] (and thus a [[type]] with [[decidable equality]]) whose terms are called *names* or *atoms*. The [[type]] $\mathrm{Exp}$ of *syntactic expressions* is an [[h-set]] which is inductively defined by binary functions $P:\mathrm{Exp} \times \mathrm{Exp} \to \mathrm{Exp}$ and $B([(-)](-)):\mathbb{A} \times \mathrm{Exp} \to \mathrm{Exp}$. There are dependent types $E:\mathrm{Exp} \vdash \mathrm{occ} \; \mathrm{type}$ representing the occuring atoms, $E:\mathrm{Exp} \vdash \mathrm{free} \; \mathrm{type}$ representing the free atoms, and $E:\mathrm{Exp} \vdash \mathrm{bnd} \; \mathrm{type}$ representing the bound atoms, such that for each expression $E:\mathrm{Exp}$, there are [[embeddings]]
$$i_\mathrm{occ}(E):\mathrm{occ}(E) \hookrightarrow \mathbb{A}$$
$$i_\mathrm{free}(E):\mathrm{free}(E) \hookrightarrow \mathbb{A}$$
$$i_\mathrm{bnd}(E):\mathrm{bnd}(E) \hookrightarrow \mathbb{A}$$
which may be defined by recursion on $\mathrm{Exp}$.
We define the following dependent types $a:\mathbb{A}, E:\mathrm{Exp} \vdash a \lhd E \; \mathrm{type}$ as
$$a \lhd E \coloneqq \sum_{x:\mathrm{occ}(E)} i_\mathrm{occ}(E)(x) =_\mathbb{A} a$$
representing that $a$ occurs in $E$ at least once,
$$a \lhd_\mathrm{free} E \coloneqq \sum_{x:\mathrm{free}(E)} i_\mathrm{free}(E)(x) =_\mathbb{A} a$$
representing that $a$ freely occurs in $E$ at least once, and
$$a \# E \coloneqq (a \lhd_\mathrm{free} E) \to \emptyset$$
representing that $a$ is fresh (does not freely occurs in $E$ at least once)
...
## See also
* [[equality]], [[syntactic equality]], [[definitional equality]]
* [[substitution]]
* [[rewriting]]
## References
* [[Roy L. Crole]], *Alpha equivalence equalities*, Theoretical Computer Science, Volume 433, 18 May 2012, Pages 1-19, ([doi:10.1016/j.tcs.2012.01.030](https://doi.org/10.1016/j.tcs.2012.01.030))
The distinction between $\alpha$-equivalence and syntactic equality of expressions is briefly discussed in:
* [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))
[[!redirects alpha equivalent]]
[[!redirects alpha-equivalent]]
[[!redirects α-equivalent]]
[[!redirects alpha equivalence]]
[[!redirects alpha-equivalence]]
[[!redirects α-equivalence]]
[[!redirects alpha conversion]]
[[!redirects alpha-conversion]]
[[!redirects α-conversion]]
|
alternating group | https://ncatlab.org/nlab/source/alternating+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An [[alternating group]] $A_n$ is a [[subgroup]] of a [[symmetric group]] $S_n$ consisting of the _even_ [[permutations]].
The alternating group $A_n$ is to the [[symmetric group]] $S_n$ as the [[special orthogonal group]] $SO(n)$ is to the [[orthogonal group]] $O(n)$. See also at _[symmetric group -- Whitehead tower](https://ncatlab.org/nlab/show/permutation#WhietheadTowerAndSupersymmetry)_
## Examples
* The alternating group $A_4$ on four elements is [[isomorphism|isomorphic]] to the orientation-preserving [[tetrahedral group]].
* The alternating group $A_5$ on five elements, of order $60$, is the smallest [[nonabelian group|nonabelian]] [[simple group]]. Geometrically, it may be realized as [[finite subgroup of SO(3)]] which carries a regular [[icosahedron]] into itself: the [[icosahedral group]].
* For all $n \geq 5$, the alternating group $A_n$ is simple. This is true even if $n$ is infinite: define $Alt(X)$ for any set $X$ to consist of all permutations of $X$ each of which fixes all but finitely elements, and which is an even permutation on that finite subset.
## Related concepts
* [[finite group]]
* [[alternating representation]]
## References
* Wikipedia, _[Alternating group](https://en.wikipedia.org/wiki/Alternating_group)_
[[!redirects alternating groups]]
|
alternating multifunction | https://ncatlab.org/nlab/source/alternating+multifunction |
# Alternating functions
* table of contents
{: toc}
## Idea
In [[linear algebra]], alternating [[multilinear functions]] are well known, and are in many cases (over the [[real numbers]], for example) are equivalent to [[antisymmetric multifunction|antisymmetric functions]]. In cases where they differ (such as in [[characteristic]] $2$), it is often the alternating functions that behave better.
Actually, being alternating is not, in itself, really about linearity, and we can abstract away to a nonlinear concept of _alternating function_. (That said, there is one bit of very mild linear structure that is needed: a [[basepoint]] in the [[target]] set.)
The property of being alternating is called _alternation_ (rather than alternatingess), although in principle one could also use _alternating_ as a noun.
## Definitions
Let $X$ be a [[set]], and let $(Y,0)$ be a [[pointed set]] (so $Y$ is a set and $0$ is one of its elements). Let $n$ be a [[natural number]] (or indeed any [[cardinal number]]). Recall that a [[multifunction]] of [[arity]] $n$ to $Y$ from $X$ is the same thing as a [[function]] to $Y$ from the $n$-fold [[cartesian power]] $X^n$.
An __alternating multifunction__ (or simply _alternating function_) of arity $n$ from $X$ to $(Y,0)$ is a multifunction of arity $n$ from $X$ to $Y$ such that, whenever two of the function\'s arguments are equal, the value of the function is $0$. In arity $0$ or $1$, every multifunction is trivially alternating; in arity $2$, we can write this as the [[equational law]] $f(a,a) = 0$; in arity $3$, we have the equational laws $f(a,a,b) = 0$, $f(a,b,a) = 0$, and $f(a,b,b) = 0$; etc.
## Properties
There are many nice properties of alternating *[[multilinear function|multilinear]]* functions. So suppose that $X$ and $Y$ are [[modules]] over a [[base rig]] $K$ and that $f$ is a multilinear function from $X$ to $Y$; use the usual [[zero]] element of the module $Y$ as the basepoint $0$. In the case where $Y$ is $K$ itself, we speak of an __alternating form__ (a phrase which is usually taken to include multilinearity).
We will sometimes want to assume that scalar multiplication by $2$ is cancellable in $Y$ (which for example is always the case when $2$ is invertible in $K$, in particular when $K$ is a [[field]] of [[characteristic]] other than $2$), but only when stated. Since alternation requires looking at two arguments of $f$, we will often, when this leads to no loss of generality, assume that these are the first two arguments, writing $\vec{z}$ to represent all of the other arguments.
+-- {: .num_prop #alterationImpliesAntisymmetry}
###### Proposition
An alternating [[multilinear function]] is [[antisymmetric function|antisymmetric]].
=--
+-- {: .proof}
By multilinearity,
$$ f(x+y,x+y,\vec{z}) = f(x,x,\vec{z}) + f(x,y,\vec{z}) + f(y,x,\vec{z}) + f(y,y,\vec{z}) .$$
Applying alternation, most of these terms vanish:
$$ 0 = 0 + f(x,y,\vec{z}) + f(y,x,\vec{z}) + 0 .$$
Therefore,
$$ f(x,y,\vec{z}) + f(y,x,\vec{z}) = 0 ,$$
which is antisymmetry.
=--
+-- {: .num_prop #antisymmetryImpliesAlternation}
###### Proposition
If multiplication by $2$ is cancellable in $Y$, then an [[antisymmetric function|antisymmetric]] function to $Y$ is alternating.
=--
+-- {: .proof}
###### Proof
By antisymmetry,
$$ f(x,x,\vec{z}) + f(x,x,\vec{z}) = 0 ,$$
or equivalently
$$ 2 f(x,x,\vec{z}) = 2 \cdot 0 .$$
Cancelling $2$,
$$ f(x,x,\vec{z}) = 0 ,$$
which is alternation.
=--
+-- {: .num_remark #antisymmetryWarning}
###### Remark
It is false in both directions to state in general that alternating functions and antisymmetric functions are the same, but for different reasons. An alternating function must be antisymmetric *if* it is multilinear, regardless of the behaviour of $2$, but not when it is nonlinear; an antisymmetric function must be alternating *if* multiplication by $2$ is cancellable in the target, regardless of linearity, but not when $2$ is noncancellable.
The simplest strongest-possible counterexamples are
$$ (x,y \mapsto |y-x|)\colon \mathbb{R}^2 \to \mathbb{R} ,$$
which is alternating but not antisymmetric, and
$$ (x,y \mapsto x y)\colon \mathbb{F}_2^2 \to \mathbb{F}_2 ,$$
which is antisymmetric but not alternating.
Of course, alternating and antisymmetric functions *are* the same in the context of multilinear functions to a module in which $2$ is cancellable, in particular for multilinear functions between [[vector fields]] over the [[real numbers]].
=--
+-- {: .num_remark}
###### Remark
The [[alternating groups]] are really about antisymmetric functions rather than alternating functions as such. (Whereas a [[symmetric function]] is preserved by the application of any element of the [[symmetric group]], an antisymmetric function is preserved by and only by the elements of the alternating group.) Nevertheless, this precise distinction between 'alternating' and 'antisymmetric' is well established in the theory of vector spaces over a field of [[characteristic]] $2$ (in which multiplication by $2$ is as uncancellable as possible).
=--
## Constructive aspects
In [[constructive mathematics]], we usually assume that the arity $n$ of $f$ has [[decidable equality]], which is true if $n$ is a [[natural number]] (which is most common) or even a (possibly infinite) [[extended natural number]]. However, as long as the arity is equipped with an [[inequality]], then we can state the definition: whenever equal arguments have inequal indices, the value of the multifunction $f$ there is zero. If $X$ and $Y$ are also equipped with inequalities, then $f$ is __strongly alternating__ if, whenever its value is inequal to $0$ in $Y$, then arguments with inequal indices must be inequal in $X$. (In arity $2$, for example, if $f(a,b) \ne 0$, then $a \ne b$.) If the inequality on $Y$ is [[tight relation|tight]] (so that its [[negation]] is [[equality]] in $Y$), then every strongly alternating function is alternating, but the reverse requires [[excluded middle]] in general.
[[!redirects alternating multifunction]]
[[!redirects alternating multifunctions]]
[[!redirects alternating multimap]]
[[!redirects alternating multimaps]]
[[!redirects alternating function]]
[[!redirects alternating functions]]
[[!redirects alternating map]]
[[!redirects alternating maps]]
[[!redirects alternating multilinear function]]
[[!redirects alternating multilinear functions]]
[[!redirects alternating multilinear map]]
[[!redirects alternating multilinear maps]]
[[!redirects alternating form]]
[[!redirects alternating forms]]
|
alternating representation | https://ncatlab.org/nlab/source/alternating+representation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $S_n$ a [[symmetric group]], its _alternating representation_ is the 1-[[dimension|dimensional]] [[linear representation]]
$$
alt \;\; S_n \longrightarrow GL(1)
$$
which sends even [[permutations]] to $+1$ and odd permutations to $-1$.
More generally, for $G$ any [[finite group]] and $H \subset G$ a [[subgroup]] of [[index of a subgroup|index]] 2, then the corresponding alternating representation of $G$ sends elements of $H$ to $+1$ and all other elements to -1.
This reduces to the previous special case by setting $G \coloneqq S_n$ and $H \coloneqq A_n \subset S_n$ the [[alternating group]].
## Related concepts
* [[trivial representation]]
* [[regular representation]]
* [[permutation representation]]
## References
For instance
* João Pedro Martins dos Santos, Def. 3 in _Representation Theory of Symmetric Groups_, 2012 ([pdf](https://www.math.tecnico.ulisboa.pt/~ggranja/joaopedro.pdf))
[[!redirects alternating representations]]
[[!redirects sign representation]]
[[!redirects sign representations]]
|
alternative algebra | https://ncatlab.org/nlab/source/alternative+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definitions
+-- {: .num_defn #AlternativeAlgebra}
###### Definition
Consider the following equational laws of a [[binary operation]] (written multiplicatively):
* __left alternativity__: $(x x) y = x (x y)$;
* __flexibility__: $(x y) x = x (y x)$;
* __right alternativity__: $(y x) x = y (x x)$.
An operation satisfying one of these conditions is __left-alternative__, __flexible__, or __right-alternative__, respectively, and it is __alternative__ if it is both left-alternative and right-alternative. A [[magma]] is so if its binary operation is so, and a [[nonassociative algebra]] (or [[nonassociative ring]]) is so if its multiplication operation is so.
=--
## Properties
A [[commutative operation|commutative]] operation/magma/algebra must be flexible, and it is left-alternative iff it is right-alternative (and so simply alternative). An [[associative operation|associative]] operation/magma/algebra is both alternative and flexible.
Among algebras (but not magmas), any algebra with two of these three properties must have the third. In particular, an alternative algebra must be flexible. This follows from the characterization in terms of the associator below.
### In terms of the associator
+-- {: .num_prop #InTermsOfAssociator}
###### Proposition
For a [[nonassociative algebra]] $A$, alternativity according to def. \ref{AlternativeAlgebra} is equivalent to the condition that the [[associator]], i.e. the [[multilinear map|tri-linear map]]
$$
[-,-,-] \;\colon\; A \otimes A \otimes A \longrightarrow A
$$
given by
$$
[a,b,c] \coloneqq (a b) c - a (b c)
$$
is [[alternating map|alternating]], in that whenever two of the three arguments are equal, the result is zero.
=--
+-- {: .num_cor #skewsymmetry}
###### Corollary
Alternativity implies that the associator is [[skew-symmetric function|skew-symmetric]], in that for any [[permutation]] $\sigma$ of three elements then $[a_{\sigma_1}, a_{\sigma_2}, a_{\sigma_3}] = (-1)^{\vert \sigma\vert} [a_1, a_2, a_3]$ for ${\vert\sigma\vert}$ the [[signature of a permutation|signature of the permutation]]. Over a [[field]] whose [[characteristic]] is different from $2$, or more generally over any [[commutative ring]] in which $2$ is invertible or even cancellable, alternativity is equivalent to skew-symmetry of the associator.
=--
+-- {: .proof}
###### Proof
In one direction, left alternativity says directly that the associator is alternating in the first two arguments:
$$
[x,x,y] = (x x) y - x (x y) = (x x) y - (x x) y = 0 ,
$$
and right alternativity says the same thing in the last two arguments:
$$
[x,y,y] = (x y) y - x (y y) = (x y) y - (x y) y = 0 .
$$
To be fully alternating, we then argue using multi-linearity of the associator:
$$
[x,y,x] = [x,x,x] + [x,y,x] + [y,x,x] + [y,y,x] = [(x+y),(x+y),x] = 0 .
$$
Multilinearity also proves that the associator is skew-symmetric, in adjacent arguments:
$$
0 = [(x+y),(x+y),z] = [x,y,z] + [y,x,z]
$$
$$
0 = [z,(x+y),(x+y)] = [z,x,y] + [z,y,x]
$$
and hence in all arguments.
In the other direction, skew-symmetry of the associator implies alternativity by
$$
\begin{aligned}
& [x,x,y] = -[x,x,y]
\\
\Leftrightarrow & 2 [x,x,y] = 0
\\
\Leftrightarrow & [x,x,y] = 0
\end{aligned}
$$
using the assumption that $2$ is cancellable in $A$, and similarly for $[y,x,x] = 0$.
=--
+-- {: .num_prop #ArtinTheorem}
###### Proposition
A [[nonassociative algebra]] is alternative, def. \ref{AlternativeAlgebra}, prop. \ref{InTermsOfAssociator}, precisely if the [[subalgebra]] generated by any two elements is an [[associative algebra]].
=--
This is due to [[Emil Artin]], see for instance ([Schafer 95, p. 18](#Schafer95)).
+-- {: .num_prop #ZornTheorem}
###### Proposition
The only alternative [[division algebras]] over the [[real numbers]] are the [[real numbers]] themselves, the [[complex numbers]], the [[quaternions]] and the [[octonions]].
=--
This is due to ([Zorn 30](#Zorn30)).
## Examples
Every [[associative algebra]] is alternative and flexible. Every [[Lie algebra]] or [[Jordan algebra]] is flexible.
Every [[Cayley–Dickson algebra]] over a [[commutative ring]] $R$ is flexible. The first three (corresponding, if we start with the [[real numbers]], to the real numbers, [[complex numbers]], and [[quaternions]]) are associative and hence alternative. The next one (corresponding to the [[octonions]]) is still alternative despite not being associative (unless $R$ has characteristic $2$). After that (corresponding to the [[sedenions]] and above), they are not even alternative (unless $R$ has characteristic $2$).
## Related concepts
* [[Cayley-Dickson construction]]
* [[composition algebra]]
## References
* {#Schafer95} R. D. Schafer, chapter III of _Introduction to Non-Associative Algebras_ Dover, New York, 1995. ([web](https://archive.org/details/anintroductionto25156gut))
* {#Zorn30} [[Max Zorn]], _Theorie der alternativen Ringe_, Abhandlungen des Mathematischen Seminars der Universität Hamburg 8 (1930), 123-147
* Wikipedia, _[Alternative algebra](https://en.wikipedia.org/wiki/Alternative_algebra)_, _[Flexible algebra](https://en.wikipedia.org/wiki/Flexible_algebra)_
[[!redirects alternativity]]
[[!redirects alternative binary operation]]
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[[!redirects alternative magmas]]
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[[!redirects left alternativity]]
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|
alternative experimental definition of commutative diagram | https://ncatlab.org/nlab/source/alternative+experimental+definition+of+commutative+diagram | ##Discussion
[[Eric]]: Here is a degenerate loop:
$$\array{
X & \stackrel{e}{\to} & X \\
{} & \mathllap{\scriptsize{e}}{\nwarrow} & \darr\scriptsize{e} \\
{} & {} & X
}$$
To say this loop commutes, is to say $e^3 = 1_X$. Or is it?
>[[David Roberts]]: That is correct.
[[Eric]]: Since this loop is degenerate, we may or may not have (depending on convention?), parallel identity arrows running between each vertex.
If there are hidden identity morphisms, then to say the triangle commutes is to say $e = 1_X$.
>[[David Roberts]]: No: the above version $e^3 = id$ is correct.
[[Eric]]: Here is a degenerate triangle:
$$\array{
X & \stackrel{e}{\to} & X \\
{} & \mathllap{\scriptsize{e}}{\searrow} & \darr\scriptsize{e} \\
{} & {} & X
}$$
To say this triangle commutes is to say $e^2 = e$. Or is it?
>[[David Roberts]]: That is correct.
[[Eric]]: Since this triangle is degenerate, we may or may not have (depending on convention?), parallel identity arrows running between each vertex.
If there are hidden identity morphisms, then to say the triangle commutes is to say $e = 1_X$.
[[David Roberts]]: No: the above version $e^2 = e$ is correct. For both of these wrong statements, you are assuming that arrows somehow cancel. Especially the first one, think of it in terms of dimensional analysis. If you have a quantity in physics with dimensions of something else cubed, it cannot have the dimensions of that original thing (unless both are dimensionless). But this is only a rough analogy, please don't read anything deep into it. In my comment in reply to Harry's $e^2 = e$ being a loop, then I said I would draw this as a commuting triangle. I wouldn't draw a loop (i.e. an endomorphism) (resp. a degenerate loop) as a triangle (resp. commuting triangle) in any circumstances, because we are not given that the loop factors into a composite of other arrows.
[[Eric]]: Thanks David. I think I've managed to boil it down to the basic disagreement. It is about "shape dependence". I reject the notion of shape dependence. The morphism $f:X\to X$ is a loop and this loop is the same as $f:X\righttoleftarrow$. I [explain this](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=332&Focus=7368#Comment_7368) on the n-Forum. The point is, you can accept shape dependence, in which case I agree with everything you say. Or you reject shape dependence and work with semidiagrams instead.
+-- {: .query}
I don\'t think that there\'s any doubt that $e^2 = e$ is the intended interpretation of the second diagram by everybody who uses commutative diagrams. The fact that the same object $X$ shows up twice is irrelevant; the whole point of putting $X$ in there twice is because you *don\'t* want parallel identity arrows. Of course, you can invent an alternative interpretation if you want, but I don\'t think that you\'ll find any usage to match it in the wild. ---Toby
=--
##Main Article
A stub for now.
The basic goal of this page is to turn Domenico's and Urs' statements about connections into statements about commutative diagrams.
The rough is idea is:
$$\text{Higher Connection Flat}\quad\quad\simeq\quad\quad\text{Higher Diagram Commutes}$$
+--{.standout}
[Domenico](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=723&Focus=5015#Comment_5015): I guess this is well known, but let me try writing it here and see what happens.
If a connection on a principal G-bundle is locally represented by the 1-form $\omega$ with values in $\mathfrak{g}$, then the connection is flat if and only if the curvature 2-form $F=d\omega+\frac{1}{2}[\omega,\omega]$ vanishes, that is, if $\omega$ is a solution of the Maurer-Cartan equation.
Now, what is interesting is that one can see curvature only on 2-dimensional paths (it is a 2-form): if one restricts $\omega$ to a 1-dimensional submanifold, then there $\omega$ is clearly a solution to the Maurer-Cartan equation. so, if I think of an infinitesimal 1-simplex and look at my connection there, I could say that my connection is 1-flat. then, moving to an infinitesimal 2-simplex I see that the connection is (generally) not 2-flat: holonomy along two sides of the 2-simplex is not the same thing as holonomy along the third side. not the same, but in a very precise way: the curvature $FF$ exactly measures the gap to go from a horn of the 2-simplex to the third edge. this is very 2-categorical, and suggests I could cure the lack of flatness of my original connection by adding a copy of $\mathfrak{g}$ in degree -1 and cooking up a 2-Lie algebra $\mathfrak{l}$. Maurer-Cartan equation for this 2-Lie algebra would coincide with the original one one the 1-simplex (since only degree 1 elements are 1-forms with values in $\mathfrak{l}_0$. but on the 2-simplex we would have, in addition to these elements, also 2-forms with coefficients in $\mathfrak{l}_{-1}$, and the Maurer-Cartan equation on teh simplex would look like $d\omega+\frac{1}{2}[\omega,\omega]+\delta F=0$ where $\delta\colon \mathfrak{l}_{-1}\to \mathfrak{l}_{0}$ is the differential of the $L_\infty-algebra \mathfrak{l}$ (and it should be induced by the identity of $\mathfrak{g}$, thought as a degree 1 map from $\mathfrak{g}[-1]$ to $\mathfrak{g}$). So the original equation telling that $\omega$ had curvature $F$ is now equivalent to say that $\omega+F$ is flat.
in other words, what seemed a non-flat connection was so since I was not seeing the 2-bundle, but only a 1-bundle approximation. and on a 1-bundle I can only clearly see up to 1-simplices, wher my connection was actually flat.
once curvature has come in, we can repeat the argument: now we have a 2-flat connection and test it on the 3-simplex. if it has 3-curvature, that will presumibly be because we are not seeing the 3-bundle, yet. so I find it natural to wonder (to conjecture) whether any connection on a principal bundle (and more generally any $n$-flat connection on an n-bundle) can be seen as a flat connection on an $\infty$-bundle.
=--
+--{.standout}
[Domenico](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=723&Focus=5050#Comment_5050): I think I could take the task of working a bit on oo-Lie algebroid valued forms in the direction sketched above. yet, I'd like to still discuss it a bit here, sicne I'm now thinking of some more radical change. what I'm thinking is that the one-step definition as it is now is too strict. I mean, we are now defining flat $\mathfrak{a}$-connections as a morphism $\prod^{inf}(X)\to\mathfrak{a}$, and then general connections as something which becomes flat in a single step, i.e., going from $\mathfrak{a}$ to $cone(\mathfrak{a}$). why should we restrict to this? it would be natural to consider thigs that become flat in 2 sterps, 3, steps, and so on. eventually we could have things which "become flat after infinity steps" (and by the way this vague notion seems to me to better fit with classical $\mathfrak{g}$-connections).
let us see things the other way round: when we say flat, we mean flat in an $\infty$-categorical sense, so let us stress this by saying that a morphism $\prod^{inf}(X)\to\mathfrak{a}$ is an $\infty$-flat connection with values in $\mathfrak{a}$. then, what we are currently calling non-necessarily-flat $\mathfrak{a}$-connections would be the $(\infty-1)$-flat connections with values in $\mathfrak{a}$. but starting from $\infty$ is not too practical.., let us start from 0 instead.
the one-step definition of non-flat connections is reflected in the one step going from the 0-groupoid $X$ to the $\infty$-groupoid $\prod^{inf}(X)$. but we have a whole tower of higher and higher groupoids in between: the $n$-skeleta of $\prod^{inf}(X)$. so we have
$$X=\prod^{inf}(X)_{(0)}\hookrightarrow \prod^{inf}(X)_{(1)}\hookrightarrow \prod^{inf}(X)_{(2)}\hookrightarrow \cdots \hookrightarrow \prod^{inf}(X)$$
and we can talk of an $n$-flat connection with values in $\mathfrak{a}$ as a morphism $\prod^{inf}(X)_{(n)}\to \mathfrak{a}$. this will not lift to a morphism $\prod^{inf}(X)_{(n+1)}\to \mathfrak{a}$ unless the connection is $(n+1)$-flat, but will lift to a morphism $\prod^{inf}(X)_{(n+1)}\to cone(\mathfrak{a})$. the limit of this procedure will produce a "true connection" with values in $cone(cone(\cdots(cone(\mathfrak{a}))\cdots)$.
I have a vague feeling that starting with a 0-falt $\mathfrak{o}(n)$-connection one will this way meet the series $\mathfrak{so}(n), \mathfrak{spin}(n), \mathfrak{string}(n), \mathfrak{fivebrane}(n), \mathfrak{uniray}(n)$,..., but I have to think more carefully to this.
=--
##References
* n-Forum: [Is every connection flat?](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=723&page=1)
* n-Forum: [functor](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=332&Focus=7304#Comment_7304)
* Eric\'s web: [[ericforgy:Shape Dependence in Commutative Diagrams]]
[[!redirects Shape Dependence in Commutative Diagrams]]
|
alternative magmoid | https://ncatlab.org/nlab/source/alternative+magmoid | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Just as a [[groupoid]] is the [[oidification]] of a [[group]] and a [[ringoid]] is the oidification of a [[ring]], an alternative magmoid should be the oidification of a [[alternative magma]].
## Definition
An __alternative magmoid__ is a [[magmoid]] $Q$ such that for every two objects $a,b \in Ob(Q)$ and for every morphism $f:a \to b$ and endomorphisms $g:a \to a$ and $h:b \to b$,
$$
f \circ (g \circ g) = (f \circ g) \circ g
\,
$$
and
$$
h \circ (h \circ f) = (h \circ h) \circ f
\,
$$
## Examples
* Every [[category]] is an alternative magmoid.
* A one-object alternative magmoid is a [[alternative magma]].
* A [[Mod]]-enriched alternative magmoid is called a alternative [[linear magmoid]].
## Related concepts
[[!include oidification - table]] |
Alvaro Restuccia | https://ncatlab.org/nlab/source/Alvaro+Restuccia |
* [InSpire page](https://inspirehep.net/authors/991923)
* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=194082)
## Selected writings
On the [[super 2-brane in 4d]]-[[D=4 supergravity]]:
* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], _$\mathcal{N}=1$ 4D Supermembrane from $11D$_, JHEP0807 039 (2008) [[arXiv:0709.4632](http://arxiv.org/abs/0709.4632)]
Lift of T-duality from [[string theory]] to a [[SL(2,Z)]]-[[U-duality]] acting on the [[M2-brane]]-[[Green-Schwarz action functional|sigma-model]]:
* [[Maria P. Garcia del Moral]], I. Martin, [[Alvaro Restuccia]], *Nonperturbative $SL(2,\mathbb{Z})$ $(p,q)$-strings manifestly realized on the quantum M2* [[arXiv:0802.0573](https://arxiv.org/abs/0802.0573)]
* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], *Aspects of the T-duality construction for the Supermembrane theory*, J. Phys.: Conf. Ser. **720** (2016) 012025 [[arXiv:1504.06907](https://arxiv.org/abs/1504.06907), [doi:10.1088/1742-6596/720/1/012025](https://doi.org/10.1088/1742-6596/720/1/012025)]
On [[gauged supergravity|gauged]] [[D=9 supergravity]] from [[SL(2,Z)]]-[[U-duality]] acting on the [[M2-brane]]:
* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], _Supermembrane origin of type II gauged supergravities in 9D_, JHEP 1209 (2012) 063 [[arXiv:1203.2767](http://arxiv.org/abs/1203.2767)]
* [[Maria P. Garcia del Moral]], C. Las Heras, P. Leon, J. M. Pena, [[Alvaro Restuccia]], _Fluxes, Twisted tori, Monodromy and $U(1)$ Supermembranes_, J. High Energ. Phys. **2020** 97 (2020) [[arXiv:2005.06397](https://arxiv.org/abs/2005.06397), <a href="https://doi.org/10.1007/JHEP09(2020)097">doi:10.1007/JHEP09(2020)097</a>]
On [[massive type IIA supergravity]] via [[M2-branes]]:
* [[Maria P. Garcia del Moral]], [[Alvaro Restuccia]], *10D Massive Type IIA Supergravities as the uplift of Parabolic M2-brane Torus bundles*, Forthsch. d. Phys. **64** 4-5 (2016) [[arXiv:1511.04784](https://arxiv.org/abs/1511.04784), [doi:10.1002/prop.201500087](https://doi.org/10.1002/prop.201500087)]
* [[Maria P. Garcia del Moral]], P. Leon, [[Alvaro Restuccia]], *Wordsheet description of a* massive *type IIA superstring in 10D*, J. High Energ. Phys. **2023** 104 (2023) [[arXiv:2306.16620](https://arxiv.org/abs/2306.16620), <a href="https://doi.org/10.1007/JHEP11(2023)104">doi:10.1007/JHEP11(2023)104</a>]
category: people
|