algebraic stack

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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#### Geometry
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[[!include higher geometry - contents]]
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#Contents#

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## 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

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

## 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

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### Context
#### Functional analysis
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[[!include functional analysis - contents]]
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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 ...
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#### 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

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### Context
#### Categorical algebra
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#### Higher algebra
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[[!include higher algebra - contents]]
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## 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

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#### Homotopy theory
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[[!include homotopy - contents]]
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#### Higher algebra
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[[!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

**[[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

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

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=--
=--
=--

#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

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=--
#### Linear algebra
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[[!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

[[!redirects natural weak factorization system]]

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#### Category theory
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[[!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$2 → $K$3 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

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+-- {: .toc .clickDown tabindex="0"}
###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

+-- {: .rightHandSide}
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### 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 (doi:10.1016/0022-4049(92)90169-G)

* [[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

* 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

< [[Algebras]]
[[!redirects Algebras -- history]]


algebroid

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### Context
#### Enriched category theory
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[[!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

* [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 (doi:10.1016/0034-4877(74)90019-6)

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

+-- {: .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

+-- {: .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

a.k.a Alizter

[mathoverflow](https://mathoverflow.net/users/54401/ali-caglayan)


Ali Chamseddine

* [website](http://sites.google.com/site/achamseddine2/home)

category: people


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

* see _[[Ali Övgün]]_


Alice Rogers

Alice Rogers is emeritus professor of pure mathematics at King's college London.

## related $n$Lab entries

* [[supermanifold]]

category: people


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

* [website](http://myweb.lmu.edu/acrans/)

## related entries

* [[Lie 2-algebra]]

* [[2-vector space]]

category: people

[[!redirects Alissa S. Crans]]


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

< [[all arrows monic]]


all changes

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+-- {: .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]]

> 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:

> 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:

> 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