algebraic homotopy

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## Whitehead's algebraic homotopy programme

In his talk at the 1950 ICM in Harvard, [[Henry Whitehead]] introduced the idea of *algebraic homotopy theory* and said

_"The ultimate aim of algebraic homotopy is to construct a purely
  algebraic theory, which is equivalent to homotopy theory in the same sort of
  way that 'analytic' is equivalent to 'pure' projective geometry."_

A statement of the aims of 'algebraic homotopy' might thus include the
following homotopy classification problem (from the same source,
J.H.C.Whitehead, (ICM, 1950)):

_Classify the homotopy types of polyhedra, $X$, $Y$, $\ldots$ , by algebraic data._

_Compute the set of homotopy classes of maps, $[X,Y]$, in terms of the
classifying data for $X$, $Y$._

These aims are still valid, but, within the context of these webpages, with the  enlargement of the class of objects of study to
include many other types of spaces, and ultimately $\infty$-[[infinity-groupoid|groupoids]].

One may summarise them, optimistically,
by saying that one searches for a nice "algebraic" category $\mathbf{A}$ together
with a functor or functors

$$\mathbf{F} : \mathbf{Spaces }\rightarrow \mathbf{A}$$

and an algebraically defined notion of 'homotopy' in  $\mathbf{A}$ such that

a) if $X\simeq Y$ in $\mathbf{Spaces}$, then $F(X) \simeq F(Y)$ in  $\mathbf
A$;

b) if $f \simeq g$ in $\mathbf{Spaces}$, then $F(f)\simeq F(g)$ in  $\mathbf
A$,

and $F$ induces an equivalence of [[homotopy category|homotopy categories]]

$$Ho(\mathbf{Spaces}) \simeq Ho(\mathbf{A}).$$

(Here $\mathbf{Spaces}$ is a category, perhaps of topological spaces such as
polyhedra or CW-complexes, but it may be larger than this and may contain the
sort of 'generalised space', [[topos]], etc., used in other contexts such as algebraic
geometry, and, of course, $\infty$-[[infinity-groupoid|groupoids]].)

## More recent developments

*  [[Baues]] has developed an approach to Whitehead's basic programme using a mix of [[cofibration category|cofibration categories]] and categories with a particular type of [[cylinder functor]], that he calls [[I-category|I-categories]]. These are treated in separate entries. Cofibration categories are very similar to the dual of K.S. Brown's abstract homotopy theory, as discussed in [[category of fibrant objects]] and [[BrownAHT]].

*  [[Ronnie Brown]]'s [[nonabelian algebraic topology]] has developed Whitehead's theory of crossed complexes along the lines suggested by the original papers of Whitehead, but extending that, in particular, using generalisations of van Kampen's theorem. (This is discussed in detail in the entry: [[nonabelian algebraic topology]].)

##Is Algebraic Homotopy 'the same as' Homotopical Algebra?

Often the objects of study are the same,  and there is an enormous interaction between the two areas, but the aims and objectives seem to be different.  Perhaps, tentatively, one could say that 'algebraic homotopy' is 'combinatorial homotopical algebra'.

##References
*  [[Hans-Joachim Baues]], _[[Algebraic Homotopy]]_, Cambridge studies in advanced mathematics 15, Cambridge University Press, (1989);
*   [[Hans-Joachim Baues]], _[[Combinatorial Homotopy and 4-Dimensional Complexes]]_, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991);
*   [[Hans-Joachim Baues]], _[[Homotopy Types]]_, in I.M.James, ed., _Handbook of Algebraic Topology_, 1--72, Elsevier, (1995).
[[!redirects Algebraic Homotopy Theory]]


algebraic integer

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## Idea

In [[number theory]], the concept of _algebraic integer_ is a generalization of that of [[integer]] to more general base-[[number fields]]. These algebraic integers form what is called the _[[ring of integers]]_ and so in order to distinguish that from the standard integers $\mathbb{Z}$ these are sometimes called _rational integers_, since they are the algebraic integers in the ring of [[rational numbers]].

## Definition

Colloquially, an **algebraic integer** is a solution to an equation

$$x^n + a_1 x^{n-1} + \ldots + a_n = 0 \qquad (1)$$

where each $a_i$ is an [[integer]] (hence a [[root]] of the [[polynomial]] on the left). More precisely, an element $x$ belonging to an [[algebraic extension]] of the [[rational numbers]] $\mathbb{Q}$ is an (algebraic) integer, or more briefly is _integral_, if it satisfies an equation of the form (1). Equivalently, if $k$ is an algebraic extension of $\mathbb{Q}$ (e.g., if $k$ is a number field), an element $\alpha \in k$ is integral if the subring $\mathbb{Z}[\alpha] \subseteq k$ is finitely generated as a $\mathbb{Z}$-module.

This notion may be relativized as follows: given an [[integral domain]] in its [[field of fractions]] $A \subseteq E$ and a finite [[field extension]] $E \subseteq F$, an element $\alpha \in F$ is **integral** over $A$ if $A[\alpha] \subseteq F$ is finitely generated as an $A$-module.

If $\alpha, \beta$ are integral over $\mathbb{Z}$ (say), then $\alpha + \beta$ and $\alpha \cdot \beta$ are integral over $\mathbb{Z}$. For, if $\beta$ is integral over $\mathbb{Z}$, it is _a fortiori_ integral over $\mathbb{Z}[\alpha]$, hence

$$(\mathbb{Z}[\alpha])[\beta] = \mathbb{Z}[\alpha, \beta]$$

is finitely generated over $\mathbb{Z}[\alpha]$ and therefore, since $\alpha$ is integral, also finitely generated over $\mathbb{Z}$. It follows that the submodules $\mathbb{Z}[\alpha + \beta]$ and $\mathbb{Z}[\alpha \cdot \beta]$ are therefore also finitely generated over $\mathbb{Z}$ (since $\mathbb{Z}$ is a [[Noetherian ring]]). Thus the integral elements form a [[ring]]. In particular, the integral elements in a [[number field]] $k$ form a ring often denoted by $\mathcal{O}_k$, usually called the **[[ring of integers]]** in $k$. This ring is a [[Dedekind domain]].

## Examples

* The algebraic integers in the [[rational numbers]] are the ordinary [[integers]].

* The algebraic integers in the [[Gaussian numbers]] are the [[Gaussian integers]].

* [[golden ratio]]

## Related concepts

* [[ring of integers]]

## References

Textbook account:

* {#Cassels86} [[J. W. S. Cassels]], Section 10.3 of: *Local Fields*, Cambridge University Press, 1986 (ISBN:9781139171885, [doi:10.1017/CBO9781139171885](https://doi.org/10.1017/CBO9781139171885))

Lecture notes:

* [[James Milne]], Chapter 2 of: *Algebraic number theory*, 2020 ([pdf](https://www.jmilne.org/math/CourseNotes/ANT.pdf))

[[!redirects algebraic integer]]
[[!redirects algebraic integers]]

[[!redirects rational integer]]
[[!redirects rational integers]]


algebraic K-theory

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## Idea

Algebraic K-theory is about natural constructions of [[cohomology theories]]/[[spectra]] from [[algebra|algebraic]] data such as [[commutative rings]], [[symmetric monoidal categories]] and various [[homotopy theory|homotopy theoretic]] refinements of these.

From a modern perspective, the algebraic K-theory [[spectrum]] $\mathbf{K}(R)$ of a [[commutative ring]] is simply the [[K-theory of a symmetric monoidal (∞,1)-category|∞-group completion]] of [[algebraic vector bundles]] on $Spec(R)$; this will be discussed in more detail [below](#AsTheKTheoryOfAlgebraicVectorBundles). In particular there is a natural concept of algebraic K-theory of "[[brave new rings]]", i.e. of [[ring spectra]]/[[E-∞ rings]].

Historically, the _algebraic K-theory_ of a [[commutative ring]] $R$ (what today is the "0th" algebraic K-theory group) was originally defined to be the [[Grothendieck group]] of its [[symmetric monoidal category]] of [[projective modules]] (under [[tensor product]] of modules). Under the [relation between modules and vector bundles](module#RelationToVectorBundlesInIntroduction), this is directly analogous to the basic definition of [[topological K-theory]], whence the common term. (In fact when applied to the _[[stack]]_ of [[vector bundles]] then algebraic K-theory subsumes [[topological K-theory]] and also [[differential K-theory]], see [below](#OnMonoidalStacks)).

There are canonical maps $K_0(R)\to Pic(R)$ from the 0th algebraic K-theory of a ring to its [[Picard group]] and $K_1(R)\to GL_1(R)$ from the first algebraic K-theory group of $R$ to its [[group of units]] which are given in components by the [[determinant]] functor. This fact is sometimes used to motivate algebraic K-theory as a "generalization of [[linear algebra]]" (see e.g. [this MO discussion](http://mathoverflow.net/a/171369/381)). This is also how the traditional [[regulator of a number field]] relates to [[Beilinson regulators]] of algebraic K-theory.

More generally, following the axiomatics of _[[Whitehead-generalized cohomology]]_ any _algebraic K-theory_ should be given by a sequence of [[functors]] $K_i$ from some suitable class of [[categories]] of "algebraic nature" to [[abelian groups]], satisfying some natural conditions. Moreover, following the [[Brown representability theorem]] these groups should arise as the [[homotopy groups]] of a [[spectrum]], the algebraic _[[K-theory spectrum]]_. Classical constructions producing this by combinatorial means are known as the _[[Quillen Q-construction]]_ defined on _[[Quillen exact categories]]_ and more generally the _[[Waldhausen S-construction]]_ defined on _[[Waldhausen categories]]_.

For more on the history of the subject see ([Arlettaz 04](#Arlettaz04), [Grayson 13](#Grayson13)) and see at at _[[Algebraic K-theory, a historical perspective]]_.

There are two ways to think of the traditional algebraic K-theory of a commutative ring more conceptually: on the one hand this construction is the [[group completion]] of the [[direct sum]] [[symmetric monoidal category|symmetric monoidal]]-structure on the [[category of modules]], on the other hand it is the group completion of the addition operation expressed by [[short exact sequences]] in that category. This leads to the two modern ways of expressing and viewing algebraic K-theory:

1. **monoidal.** The [[core]] of a [[symmetric monoidal category]] or more generally of a [[symmetric monoidal (∞,1)-category]] has a [[universal construction|universal]] completion to an [[abelian ∞-group]]/[[connective spectrum]] optained by universally adjoining [[inverses]] to the symmetric monoidal operation -- the [[K-theory of a symmetric monoidal (∞,1)-category|∞-group completion]]. This yields the concept of [[K-theory of a permutative category|algebraic K-theory of a symmetric monoidal category]] and more generally that of [[algebraic K-theory of a symmetric monoidal (∞,1)-category]];

1. **exact/stable.** Analogously, inverting the addition operation expressed by the [[exact sequences]] in an [[abelian category]] or more generally in a [[stable (∞,1)-category]] yields the [[algebraic K-theory of a stable (∞,1)-category]]. Explicit ways to express this are known as the  _[[Quillen Q-construction]]_ and the _[[Waldhausen S-construction]]_. This turns out to be a universal construction in the context of [[non-commutative motives]].

Here the second construction may be understood as first [[split exact sequence|splitting]] the [[exact sequences]] and then applying the first construction to the resulting [[direct sum]] monoidal structure. Typically the first construction here contains more information but is harder to compute, and vice versa (see also MO-discussion [here](http://mathoverflow.net/a/98602/381) and [here](http://mathoverflow.net/a/102583/381)).

Both of these constructions produce a [[spectrum]] (hence [[Brown representability theorem|representing]] a [[Whitehead-generalized cohomology theory]]) -- called the _[[K-theory spectrum]]_ -- and the algebraic K-theory groups are the [[homotopy groups]] of that spectrum.

The classical case of the algebraic K-theory of a commutative ring $R$ is a special case of this general concept of algebraic K-theory by either forming the [[symmetric monoidal category]] $(Mod(R), \oplus)$ and applying the [[abelian ∞-group]]-completion to that, or else forming the [[stable (∞,1)-category of chain complexes]] of $R$-modules and applyong the [[Waldhausen S-construction]] to that. In both cases the result is a [[spectrum]] whose degree-0 [[homotopy group]] is the ordinary algebraic K-theory of $R$ as given by the [[Grothendieck group]] and whose higher homotopy groups are its _higher algebraic K-theory_ groups.

## Constructions

### Symmetric monoidal K-theory

For a [[symmetric monoidal category]] $C$, K-theory may be defined by taking

* the [[maximal subgroupoid]] $i C \subset C$,
* the [[classifying space]] $B(i C)$ (a [[topological monoid]]),
* the [[group completion]] $\Omega B B (i C)$.

See at

* [[K-theory of a symmetric monoidal (∞,1)-category]]

* [[K-theory of a permutative category]]

* [[K-theory of a bipermutative category]]

### Quillen Q-construction (for exact categories)

Given an [[Quillen exact category]] $E$, one defines $K(E)$ by applying

* the [[Quillen Q-construction]] $Q(E)$,
* the [[group completion]] $\Omega B Q(E)$.

See at

* [[Quillen Q-construction]]

### Waldhausen $S_\bullet$-construction (for Waldhausen categories)

Given a [[Waldhausen category]] $(C, w C)$, one defines its $K$-theory by applying

* the [[Waldhausen S-construction]] $w S_\bullet C$ (a [[simplicial category]]),
* the [[nerve]] (a [[bisimplicial set]]),
* the [[colimit]] (a [[simplicial set]]),
* the [[geometric realization]] (a [[space]]).

There is also a [[Waldhausen S-construction]] for [[stable (infinity,1)-categories]] and, most generally, for [[Waldhausen (infinity,1)-categories]].

See at

* [[Waldhausen S-construction]]
* [[K-theory of a stable (infinity,1)-category]]

## Examples

### For rings

We recall several constructions of the [[algebraic K-theory]] of a [[ring]].
See ([Weibel, IV.4.8, IV.4.11.1](#Weibel)) for details.

#### Plus construction

Given an associative unital [[ring]] $R$, one may define the algebraic K-theory [[space]] $K(R) = BGL(R)^+$ by taking

* the [[general linear groups]] $GL_n(R)$ for $n \ge 0$,
* their [[classifying spaces]],
* the [[colimit]] $BGL(R) = colim_n BGL_n(R)$,
* the [[Quillen plus construction]].

#### Direct sum K-theory

Consider the category $P(X)$ of [[finitely generated]] [[projective]] (right) $R$-modules.
It has a [[symmetric monoidal structure]] given by [[direct sum]].
The algebraic K-theory $K(R)$ may be described as the [[K-theory of a symmetric monoidal (infinity,1)-category]] of $P(R)$.
That is, it is the [[group completion]]
  $K(R) = \Omega B B (i P(X))$
where $i P(X)$ denotes the [[maximal subgroupoid]].
See ([Weibel, IV.4.8, IV.4.11.1](#Weibel)).

#### Exact K-theory

Consider the category $P(X)$ of [[finitely generated]] [[projective module|projective]] (right) $R$-modules.
This is an [[exact category]] and the K-theory $K(R)$ may be described via the [[Quillen Q-construction]]:
  $$ K(R) = \Omega B (Q(P(R)). $$

### For schemes

For [[schemes]], there are two constructions which do not agree in full generality.
See [Thomason-Trobaugh 90](#ThomasonTrobaugh90).

#### Quillen K-theory

The Quillen K-theory of a [[scheme]] $X$ is defined as the algebraic K-theory of the [[exact category]] $Vect(X)$ of [[vector bundles]] on $X$ (using the [[Quillen Q-construction]]).

#### Thomason-Trobaugh K-theory

Let $Perf(X)$ be the category of [[perfect complexes]] on $X$.
This admits the structure of a [[Waldhausen category]], and the Thomason-Trobaugh K-theory of $X$ is defined via the [[Waldhausen S-construction]].

It may also be defined as the [[K-theory of a stable (infinity,1)-category]] of $Perf(X)$ viewed as a [[stable (infinity,1)-category]].

Thomason-Trobaugh K-theory coincides with Quillen K-theory for schemes that admit an ample family of [[line bundles]], but has the advantage of better global descent properties.

### For smooth manifolds

Discussion of algebraic K-theory as a [[smooth spectrum]] $SmoothMfd^{op} \longrightarrow Spectra$ via $X \mapsto K(C^\infty(X))$ is in ([Bunke-Nikolaus-Voelkl 13](#BunkeNikolausVoelkl13), [Bunke 14](#Bunke14)).

For more on this see at

* [[algebraic K-theory of smooth manifolds]]

* [[differential cohomology hexagon]], section _[Algebraic K-theory of smooth manifolds and the e-invariant](differential+cohomology+diagram#SoothVectorBundlesWithConnectionAndEInvariant)_

## Properties

### Chern characters

#### Regulators and relation to ordinary cohomology

See at _[[Beilinson regulator]]_.

#### Cyclotomic trace and relation to topological Hochschild homology

Given a ring $R$, then there is a natural morphism of [[spectra]]

\begin{tikzcd}
& TC(R) \arrow[d]\\
K(R) \arrow[ru] \arrow[r] & THH(R)\\
\end{tikzcd}

from the algebraic K-theory spectrum to the [[topological Hochschild homology]] spectrum and factoring through the [[topological cyclic homology]] spectrum called the _[[cyclotomic trace]]_ which much like a [[Chern character]] map for algebraic K-theory.

### Comparison map and Relation to topological K-theory

* [[comparison map between algebraic and topological K-theory]]

### Descent
 {#Descent}

See also

* Moritz Kerz, Florian Strunk, [[Georg Tamme]], _Algebraic K-theory and descent for blow-ups_ ([arXiv:1611.08466](https://arxiv.org/abs/1611.08466))

#### Zariski and Nisnevich descent

The algebraic K-theory spectrum $\mathbf{K}$ satisfies [[descent]] to give a [[sheaf]] of [[connective spectra]] on the [[Zariski site]].  For regular noetherian schemes this statement is due to ([Brown Gersten 73](#BrownGersten73)). The generalization to finite dimensional noetherian schemes is due to ([Thomason-Trobaugh 90](#ThomasonTrobaugh90)).

Moreover, $\mathbf{K}$ satisfies descent with respect to the [[Nisnevich site|Nisnevich topology]] (which lies between Zariski and étale).  This is due to ([Nisnevich 89](#Nisnevich89)) and was generalized in turn to finite dimensional noetherian schemes in the same paper of Thomason.

Further generalization of the descent result to finite dimensional quasi-compact quasi-separated schemes is due to ([Rosenschon 06](#Rosenschon06)).

#### Etale descent

The question of descent of $\mathbf{K}$ over the [[étale site]] is closely related to the [[Lichtenbaum-Quillen conjecture]], see also ([Thomason 85](#Thomason85)).  This is now a theorem of Rost and Voevodsky and it implies that K-theory does satisfy etale descent in sufficient large degrees.

> [MO comment](http://mathoverflow.net/a/180265/381)

#### Description of the K-theory sheaf via algebraic vector bundles
 {#AsTheKTheoryOfAlgebraicVectorBundles}

Let $Sch$ denote the gros [[Zariski site]] of regular, separated, noetherian schemes.
It is explained in ([Bunke-Tamme 12, section 3.3](#BunkeTamme12) that the presheaf of [[spectra]] on $Sch$ defined by algebraic K-theory admits the following description.

Regard the [[stack]] $\mathbf{Vect}^\oplus$ of [[algebraic vector bundles]]
on $Sch$ as taking values in [[symmetric monoidal (∞,1)-categories]], via the [[direct sum]] of vector bundles. Then apply the [[K-theory of a symmetric monoidal (∞,1)-category]]-construction $\mathcal{K}$ to this, yielding a [[sheaf of spectra]].
This identifies with the usual Thomason-Trobaugh K-theory sheaf,a fact that follows from

1. Zariski descent for Thomason-Trobaugh K-theory,

1. the Zariski-local equivalence between Thomason-Trobaugh K-theory, Quillen K-theory, and direct sum K-theory.

### Relation to non-commutative topology and non-commutative motives
 {#RelationToKKAndMotives}

[[!include noncommutative motives - table]]

### Red-shift conjecture

* [[red-shift conjecture]]

[[!include chromatic tower examples - table]]

## Examples

### On monoidal stacks
 {#OnMonoidalStacks}

Algebraic K-theory is traditionally applied to single [[symmetric monoidal (∞,1)-categories|symmetric monoidal]]/[[stable (∞,1)-category|stable]] [[(∞,1)-categories]], but to the extent that it is [[(∞,1)-functor|functorial]] it may just as well be applied to [[(∞,1)-sheaves]] with values in these.

Notably, applied to the monoidal [[stack]] of [[vector bundles]] (with [[connection on a bundle|connection]]) on the [[site]] of [[smooth manifolds]], the [[K-theory of a symmetric monoidal (∞,1)-category|K-theory of a monoidal category]]-functor produces a [[sheaf of spectra]] which is a form of [[differential K-theory]] and whose [[geometric realization]] is the [[topological K-theory]] spectrum. For more on this see at _[differential cohomology hexagon -- Differential K-theory](differential%20cohomology%20diagram#DifferentialKTheory)_.

## Related concepts

* [[Eilenberg swindle]]

* [[equivariant algebraic K-theory]]

* [[iterated algebraic K-theory]]

Types of categories for which a theory of algebraic K-theory exist include notably the notions

* [[Quillen exact category]],

* [[abelian category]],

* [[Waldhausen category]],

* [[triangulated category]].

Concrete examples of interest include for instance

* the category of finitely generated [[projective object]]s over a unital $k$-[[associative unital algebra|algebra]],

* the category of [[coherent sheaf|coherent sheaves]] over a [[noetherian ring|noetherian]] [[scheme]],

* the category of locally free sheaves over a scheme,

* [[Gersten resolution]]

* [[Milnor's K2]] ([[Steinberg group]], [[universal central extension]])

* [[higher algebraic K-theory]] [[Quillen exact category]], [[Quillen's Q-construction]], [[Waldhausen S-construction]], [[Volodin spaces]];

* [[topological cyclic homology]], [[algebraic K-theory of operator algebras]]

* [[Beilinson regulator]]

* [[differential algebraic K-theory]]

* [[non-connective algebraic K-theory]]

* [[real algebraic K-theory]]

* [[bivariant algebraic K-theory]]

* [[K-motive]]

* [[red-shift conjecture]]

* [[filtrations on algebraic K-theory]]

[[!include noncommutative motives - table]]

## References

### Introductions

Surveys with accounts of the historical development include

* {#Arlettaz04} [[Dominique Arlettaz]], _Algebraic K-theory of rings from a topological viewpoint_ ([pdf](http://www.math.uiuc.edu/K-theory/0420/Arlettaz-survey.pdf))

* {#Grayson13} [[Daniel Grayson]], _Quillen's work in algebraic K-theory_, J. K-Theory 11 (2013), 527–547 [pdf](http://www.math.uiuc.edu/~dan/Papers/qs-published-final.pdf)

An introductory textbook account is in

* {#Weibel} [[Charles Weibel]], _The K-Book: An introduction to algebraic K-theory_ ([web](http://www.math.rutgers.edu/~weibel/Kbook.html))

Further review includes

* {#Isely05} Olivier Isely, _Algebraic $K$-theory_, 2005-06 ([[IselyKTheory.pdf:file]])

* {#Gerhardt14} [[Teena Gerhardt]], _Computations in algebraic K-theory_, talk at [CUNY Workshop on differential cohomologies 2014](http://qcpages.qc.cuny.edu/~swilson/cunyworkshop14.html) ([video recording](http://videostreaming.gc.cuny.edu/videos/video/1800/in/channel/55/))

Review of the relation to [[Dennis trace]], [[topological cyclic homology]] and [[topological Hochschild homology]] is in

* {#DundasGoodwillieMcCarthy13} [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], *The local structure of algebraic K-theory*, Springer (2013) [[doi:10.1007/978-1-4471-4393-2](https://doi.org/10.1007/978-1-4471-4393-2)]

### Classical

Original articles include

* {#Quillen73} [[Daniel Quillen]], _Higher algebraic K-theory_, in Higher K-theories, pp. 85–147, Proc. Seattle 1972, Lec. Notes Math. 341, Springer 1973.
([pdf](http://math.mit.edu/~hrm/kansem/quillen-higher-algebraic-k-theory.pdf))

  also: [[Daniel Grayson]], _Higher algebraic K-theory II, [after Daniel Quillen]_ ([pdf](http://www.math.illinois.edu/~dan/Papers/HigherAlgKThyII.pdf))

* {#BrownGersten73} [[Kenneth Brown]], Stephen M. Gersten, _Algebraic K-theory as generalized sheaf cohomology_, Higher K-Theories, Lecture Notes in Mathematics Volume 341, 1973, pp 266-292.

* {#Waldhausen85} [[F. Waldhausen]], _Algebraic K-theory of spaces_, Alg. and Geo. Top., Springer Lect. Notes Math. 1126 (1985), 318-419, [pdf](http://www.maths.ed.ac.uk/~aar/surgery/rutgers/wald.pdf).

* {#Thomason85} [[R. W. Thomason]], _Algebraic K-theory and étale cohomology_, Ann. Sci. Ecole Norm. Sup. 18 (4), 1985, pp. 437–552.

* {#Nisnevich89} [[Yevsey Nisnevich]], _The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory_, Algebraic K-theory: connections with geometry and topology, 1989, pp 241-341.

* {#ThomasonTrobaugh90} [[R. W. Thomason]], Thomas Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_, _The Grothendieck Festschrift_, 1990, 247-435.

* {#Rosenschon06} Andreas Rosenschon, P.A. Ostvær, _Descent for K-theories_, Journal of Pure and Applied Algebra 206, 2006, pp 141–152.

For [[complex varieties]]:

* {#PedriniWeibel01} Claudio Pedrini, [[Charles Weibel]],  _The higher K-theory of complex varieties_, K-theory 21 (2001), 367-385 ([web](http://www.math.uiuc.edu/K-theory/0403/))

* Michael Paluch, _Algebraic K-theory and topological spaces_ ([pdf](http://www.math.uiuc.edu/K-theory/0471/alg-top.pdf))

For discussion of stable phenomena in algebraic K-theory, see section 4 of

* {#Cohen} [[Ralph Cohen]], _Stability phenomena in the topology of moduli spaces_ ([pdf](http://arxiv.org/PS_cache/arxiv/pdf/0908/0908.1938v2.pdf))

Discussion of the [[comparison map between algebraic and topological K-theory]] includes

* [[Jonathan Rosenberg]], _Comparison between algebraic and topological K-theory for Banach algebras and $C^\ast$-algebras_   ([pdf](http://www2.math.umd.edu/~jmr/algtopK.pdf))

For [[smooth manifolds]]:

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_, Journal of Homotopy and Related Structures October 2014 ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

* {#Bunke14} [[Ulrich Bunke]], _A regulator for smooth manifolds and an index theorem_ ([arXiv:1407.1379](http://arxiv.org/abs/1407.1379))

### Algebraic K-theory of quotient stacks
 {#ReferencesAlgebraicKTheoryForQuotientStacks}

Discussion of algebraic K-theory for [[algebraic stacks]] (generalizing algebraic [[equivariant K-theory]]) is in

* [[Robert Thomason]], _Algebraic K-theory of group scheme actions_, Algebraic Topology and Algebraic K-theory, Ann. Math. Stud., Princeton, 113, (1987), 539-563.

* [[Amalendu Krishna]], Charanya Ravi, _On the K-theory of schemes with group scheme actions_ ([arXiv:1509.05147](http://arxiv.org/abs/1509.05147))

See also at _[universal Chern-Simons 3-bundle -- For reductive groups](universal+Chern-Simons+line+3-bundle#ForReductiveAlgebraicGroups)_.

### Algebraic K-theory of ring spectra

The [[algebraic K-theory]] of [[ring spectra]]:

* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], chapter VI of _[[Rings, modules and algebras in stable homotopy theory]]_, AMS Mathematical Surveys and Monographs Volume 47 (1997) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

* {#BlumbergGepnerTabuada10} [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], Section 9.5 of: _A universal characterization of higher algebraic K-theory_, Geom. Topol. 17 (2013) 733-838 ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282), [doi:10.2140/gt.2013.17.733](http://dx.doi.org/10.2140/gt.2013.17.733))

* [[Jacob Lurie]], _Algebraic K-Theory of Ring Spectra_, Lecture 19 of _[Algebraic K-Theory and Manifold Topology](https://www.math.ias.edu/~lurie/281.html)_, 2014 ([pdf](http://people.math.harvard.edu/~lurie/281notes/Lecture19-Rings.pdf))

The  algebraic K-theory of specifically of [[suspension spectra]] of [[loop spaces]] (Waldhausen's _[[A-theory]]_) is originally due to

* [[Friedhelm Waldhausen]], _Algebraic K-theory of spaces_, In: A. Ranicki N.,  Levitt, F. Quinn (eds.), Algebraic and Geometric Topology, Lecture Notes in Mathematics, vol 1126. Springer, Berlin, Heidelberg (1985) ([doi:10.1007/BFb0074449](https://doi.org/10.1007/BFb0074449))

See also:

([Thomason-Trobaugh 90](#ThomasonTrobaugh90))

### Via stable $(\infty,1)$-categories

The [[stable (∞,1)-category]] theory picture is discussed in

* {#BlumbergGepnerTabuada10} [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _A universal characterization of higher algebraic K-theory_, Geom. Topol. 17 (2013) 733-838 ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282), [doi:10.2140/gt.2013.17.733](http://dx.doi.org/10.2140/gt.2013.17.733))

(in terms of [[noncommutative motives]]) and in

* [[Clark Barwick]], _On the algebraic K-theory of higher categories, I. The universal property of Waldhausen K-theory_ ([arXiv:1204.3607](http://de.arxiv.org/abs/1204.3607))

### Via symmetric monoidal $(\infty,1)$-categories

The perspective of [[algebraic K-theory of a symmetric monoidal (∞,1)-category]] is developed in

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], section 2.1 of _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451))

* {#Nikolaus13} [[Thomas Nikolaus]] _Algebraic K-Theory of $\infty$-Operads_ ([arXiv:1303.2198](http://arxiv.org/abs/1303.2198))

* {#BunkeTamme13} [[Ulrich Bunke]], [[Georg Tamme]], _Multiplicative differential algebraic K-theory and applications_ ([arXiv:1311.1421](http://arxiv.org/abs/1311.1421))

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], def. 6.1 in  _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

* {#GepnerGrothNikolaus13} [[David Gepner]], [[Moritz Groth]], [[Thomas Nikolaus]], _Universality of multiplicative infinite loop space machines_, [arXiv:1305.4550](http://arxiv.org/abs/1305.4550).

### K-theory stacks

The system of [[infinite loop spaces]] of the algebraic K-theory spectrum regarded as an [[∞-stack]] on the [[Nisnevich site]] and the [[principal ∞-bundles]] over it is considered in

* {#Saito14} [[Sho Saito]], _Higher Tate central extensions via K-theory and infinity-topos theory_ ([arXiv:1405.0923](http://arxiv.org/abs/1405.0923))

implementing a suggestion stated in

* {#Drinfeld03} [[Vladimir Drinfeld]], _Infinite-dimensional vector bundles in algebraic geometry (an introduction)_ ([arXiv:math/0309155](http://arxiv.org/abs/math/0309155))

### Equivariant versions

Refinement to [[global equivariant stable homotopy theory]]:

* [[Stefan Schwede]], _Global algebraic K-theory_ ([arXiv:1912.08872](https://arxiv.org/abs/1912.08872))

### Examples

* {#Ananyevskiy} Alexey Ananyevskiy, _On the algebraic $K$-theory of some homogeneous varieties_ ([pdf](http://www.math.uni-bielefeld.de/lag/man/431.pdf))

[[!redirects algebraic K-theory of a stable (∞,1)-category]]
[[!redirects algebraic K-theory of a stable (infinity,1)-category]]


algebraic K-theory of smooth manifolds

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#### Differential cohomology
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=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The construction that sends a [[smooth manifold]] $X$ to the [[algebraic K-theory]] [[spectrum]] $K(C^\infty(X,\mathbb{C}))$ of its [[ring]] of [[smooth functions]] (with values in the [[complex numbers]]) presents (after [[infinity-stackification]]) a [[sheaf of spectra]] on the [[site]] of [[smooth manifolds]], hence a [[smooth spectrum]]

$$
  \mathbf{K} \in Stab(Smooth\infty Grpd) \simeq T_\ast Smooth \infty Grpd
  \,,
$$

i.e. an object of the [[tangent cohesive (∞,1)-topos]] of [[Smooth∞Grpd]]. (See also [this definition](differential+cohomology+diagram#KTheoryOfStackOfVectorBundles) at _[[differential cohomology hexagon]]_.)

## Properties

### Shape and relation to topological K-theory

The [[shape modality|shape]] of this $\mathbf{K}$ is the [[topological K-theory]] spectrum $ku$ ([Bunke-Nikolaus-Voelkl 13, lemma 6.3](#BNV13), [Bunke 14, (48) with def. 2.21](#Bunke14)):

$$
  ʃ \mathbf{K} \simeq ku
  \,.
$$

Hence $\mathbf{K}$ is a [[differential cohomology]] refinement of $ku$, a form of [[differential K-theory]].

### Regulator and relation to differential K-theory
 {#Regulator}

There is also the more standard [[differential K-theory]] refinement $\mathbf{ku}_{conn}$ of $ku$ ([[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 05]], [Bunke-Nikolaus-Voelkl 13](differential+K-theory#BunkeNikolausVoelkl13)) which is obtained by pulling back suitable sheaves of ($\mathbb{C}$-valued) [[differential forms]] $\mathbf{DD}^-$ along the usual  [[Chern character map]] $ch \colon ku \longrightarrow DD^{per}$. This Chern character lifts through the [[shape modality]] to a [[regulator]] map ([Bunke 14, (50)](#Bunke14))

$$
  \array{
    \mathbf{K} &\stackrel{\mathbf{reg}}{\longrightarrow}& \mathbf{DD}^-
    \\
    \downarrow^{\mathrlap{\eta^{ʃ}}} && \downarrow^{\mathrlap{\eta^{ʃ}}}
    \\
    ku &\stackrel{ch}{\longrightarrow}& DD^{per}
  }
$$

Moreover, this induces a differential [[regulator]] ([BNV 13, p.40 and example 6.9](#BNV13), [Bunke 14, def. 2.29](#Bunke14)):

$$
  \mathbf{reg}_{conn} \;\colon\; \mathbf{K} \longrightarrow \mathbf{ku}_{conn}
  \,.
$$

See also [this proposition](differential+cohomology+diagram#SmoothRegulator) at _[[differential cohomology diagram]]_.

### Moduli stacks

In ([Bunke 14](#Bunke14)) all this is generalized to the [[mapping spaces]] $[X,\mathbf{K}]$ out of a [[smooth manifold]] $X$, hence to the [[moduli stacks]] (here: moduli spectra) of algebraic K-theory cocycles on $X$.

The regulator from [above](#Regulator) induces a map

$$
  [X,\mathbf{reg}_{conn}]
  \;\colon\;
  [X,\mathbf{K}]
  \longrightarrow
  [X,\mathbf{ku}_{conn}]
  \,.
$$

(...)

## References

* {#BNV13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], section 6 of _Differential cohomology theories as sheaves of spectra_, Journal of Homotopy and Related Structures October 2014 ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

* {#Bunke14} [[Ulrich Bunke]], _A regulator for smooth manifolds and an index theorem_ ([arXiv:1407.1379](http://arxiv.org/abs/1407.1379))

* [[Ulrich Bunke]], _Smooth aspects of algebraic K-theory_ ([pdf](http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/smooth-asp.pdf))


algebraic K-theory spectrum

The term _algebraic K-theory spectrum_ refers to one of the following two notions:

* an object $K(C) \in \Spt$ in the [[stable (infinity,1)-category of spectra]], associated to a [[stable (infinity,1)-category]] or [[symmetric monoidal (infinity,1)-category]] $C$, whose [[homotopy groups]] give the [[algebraic K-theory]] of $C$;

* a [[motivic spectrum]] $KGL \in SH(S)$ in the [[stable motivic homotopy category]], representing an $\mathbf{A}^1$-[[homotopy invariant]] version of [[algebraic K-theory]], where $S$ is a scheme.

This entry is about the latter notion; for the former one, see [[algebraic K-theory]].

## Idea

Suppose that $S$ is a [[regular scheme]]. Then there exists a [[motivic spectrum]] $KGL_S$ with the property that, for every $X\in Sm/S$,

$$ KGL^{p,q}(X) = K_{2q-p}(X), $$

where $K_*(X)$ are the [[algebraic K-theory]] groups of $X$ defined by [[Daniel Quillen|Quillen]]. In particular, $KGL$-cohomology is $(2,1)$-periodic: this is _Bott periodicity_ for algebraic K-theory.

The multiplicative structure of algebraic K-theory makes $KGL$ into a ring spectrum (up to homotopy), which comes from a unique structure of $E_\infty$-algebra (see [Naumann-Spitzweck-Ostvaer](#NSO11)).

Over non-regular schemes, the motivic spectrum $KGL$ is also defined and it represents Weibel's homotopy invariant version of [[algebraic K-theory]] (see [Cisinski13](#Cis13)).

## Related concepts

* [[algebraic K-theory]]
* [[motivic homotopy theory]]
* [[motivic cohomology]]
* [[algebraic cobordism]]

## References

* [[Fabien Morel]], [[Vladimir Voevodsky]], _$\mathbb{A}^1$-homotopy theory of schemes_ K-theory, 0305 ([web](http://www.math.uiuc.edu/K-theory/0305/) [pdf](http://www.math.uiuc.edu/K-theory/0305/nowmovo.pdf))
{#MorelVoevodsky}

* {#Voevodsky98} [[Vladimir Voevodsky]], _$\mathbf{A}^1$-Homotopy Theory_, Doc. Math., Extra Vol. ICM 1998(I), 417-442, [web](http://www.mathematik.uni-bielefeld.de/documenta/xvol-icm/00/Voevodsky.MAN.html).

* {#Cis13} [[Denis-Charles Cisinski]], _Descente par éclatements en K-théorie invariante par homotopie_, Ann. of Math. (2) 177 (2013), no. 2, pp. 425–448 ([pdf](http://hal.archives-ouvertes.fr/docs/00/52/56/40/PDF/descente3.pdf))

* {#NSO11} [[Niko Naumann]], [[Markus Spitzweck]], [[Paul Arne Østvær]], _Existence and uniqueness of E-infinity structures on motivic K-theory spectra_, ([arXiv](http://arxiv.org/abs/1010.3944))

* [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], _On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory_, Inventiones mathematicae, 2009, 175(2), 435-451, [DOI](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.244.7301), [arXiv](http://arxiv.org/abs/0709.4124).

* {#GepnerSnaith08} [[David Gepner]], [[Victor Snaith]], _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Documenta Math.  2008 ([arXiv:0712.2817](http://arxiv.org/abs/0712.2817)).

[[!redirects algebraic K-theory spectra]]
[[!redirects motivic algebraic K-theory spectrum]]
[[!redirects KGL]]


algebraic K-theory, a historical perspective

[[!redirects Algebraic K-theory, a historical perspective]]

## A little history:  the beginnings

[[algebraic K-theory|Algebraic K-theory]]  grew out of two apparently unrelated areas of algebraic geometry and algebraic
topology. The second of these, historically, was the development by Grothendieck of (geometric
and topological) K-theory based on projective modules over a ring, or finite dimensional vector
bundles on a space, that is the [[Grothendieck group]] of the given ring or of the ring of functions on the space. (The connection between these is that the space of global sections of a finite
dimensional vector bundle on a nice enough space, $X$ is a finitely generated projective module over
the ring of continuous real or complex functions on $X$ . This latter aspect is where the link with [[topological K-theory]] comes in.)

The second input is from  of [[simple homotopy theory]]. [[J. H. C. Whitehead]], following on from earlier ideas of [[Reidemeister]], looked at possible extensions of combinatorial
group theory, with its study of presentations of groups, to give a [[combinatorial homotopy theory]]. This would take the form of an 'algebraic homotopy theory' giving good algebraic models
for homotopy types, and would hopefully ease the determination of homotopy equivalences for instance of polyhedra. The 'combinatorial' part was exemplified by his two papers on 'Combinatorial
Homotopy Theory', but raised an interesting question.

_Could one give a 'combinatorial way of generating all homotopy equivalences, (up to homotopy), starting with some 'elementary expansions' and 'contractions'?_

He showed the answer was negative, and there was an invariant ([[Whitehead torsion]]) whose vanishing was a necessary and sufficient condition for a homotopy equivalence to be so constructible. That invariant was an element of a group constructed from the stable general linear group over the [[group ring]] of the [[fundamental group]] of the domain space. This was a quotient of what became known as $K_1(\mathbb{Z}\pi_1X)$, that is  the abelianisation of $GL(\mathbb{Z}\pi_1X)$.

This ties in, even at this basic level with the [[nPOV]] and the processes around categorification. For a ring, $R$, the [[Grothendieck group]], $K_0(R)$, looks at the [[core]] of the category of finitely generated projective modules, and takes its set of connected components, which is just the set of isomorphism classes. This becomes an abelian monoid under direct sum and then a group after group completion. For $K_1(R)$, you are taking the core again, but looking at the category of morphisms in it. If we take $K_0$ of that category we get $K_1$ of the ring. So $K_1$ looks at the loops whilst $K_0$ at the connected components.

## References

One of the best accounts of the history of K-theory is by [[Chuck Weibel]]:

* {#Weibel99} [[Charles A. Weibel]], [The development of algebraic K-theory before 1980](http://www.math.rutgers.edu/~weibel/papers-dir/khistory.pdf) published in _Algebra, K-theory, groups, and education:
on the occasion of Hyman Bass's 65th birthday_, a volume of _Contemporary
Mathematics, American Mathematical Soc, 1999.

Other historical accounts include

* {#Arlettaz04} [[Dominique Arlettaz]], _Algebraic K-theory of rings from a topological viewpoint_ ([pdf](http://www.math.uiuc.edu/K-theory/0420/Arlettaz-survey.pdf))

* {#Grayson13} [[Daniel Grayson]], _Quillen's work in algebraic K-theory_, J. K-Theory 11 (2013), 527–547 [pdf](http://www.math.uiuc.edu/~dan/Papers/qs-published-final.pdf)


algebraic Kan complex

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## Idea

The notion of _algebraic Kan complex_ is an [[algebraic definition of higher categories|algebraic definition of]] [[∞-groupoid]]s.

It builds on the classical [[geometric definition of higher categories|geometric definition]] of $\infty$-groupoids in terms of [[Kan complex]]es. A Kan complex is like an algebraic $\infty$-groupoid in which we have forgotten what precisely the [[composition]] operation and what the [[inverse]]s are, and only know that they do exist. This becomes an _algebraic model_ for $\infty$-groupoids by adding the specific choices of composites back in.

The nontrivial aspect of the definition of algebraic Kan complexes is that they do still [[presentable (∞,1)-category|present]] the full [[(∞,1)-category]] [[∞Grpd]]. Notably the [[homotopy hypothesis]] is true for algebraic Kan complexes.

## Definition

An **algebraic Kan complex** is a [[Kan complex]] equipped with a _choice_ of [[horn]] fillers for all horns.

A morphism of algebraic Kan complexes is a morphism of the underlying Kan complexes that sends chosen fillers to chosen fillers.

This defines the category $Alg Kan$ of algebraic Kan complexes.

For more see the section [Algebraic  fibrant models for higher categories](http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects#AlgebaicHigherCategories) at [[model structure on algebraic fibrant objects]].

A slight variant of this definition is that of a [[simplicial T-complex]].

## Properties

### Monadicity

The category $Alg Kan$ is the category of algebras over a [[monad]]

$$
  sSet \stackrel{\leftarrow}{\to}  Alg sSet
  \,.
$$

This means that algebraic Kan complexes are formally an _algebraic model_ for higher categories.

See [[model structure on algebraic fibrant objects]] for details.

### Homotopy hypothesis-theorem

The [[homotopy hypothesis]] is true for algebraic Kan complexes:

there is a [[model category]] structure on $Alg Kan$ -- the [[model structure on algebraic fibrant objects]] -- and a [[Quillen equivalence]] to the standard [[model structure on simplicial sets]].

Moreover, there is a direct [[Quillen equivalence]]

$$
  \Pi_\infty : Top \stackrel{\leftarrow}{\to} AlgKan : |-|_r
  \,,
$$

to the standard [[model structure on topological spaces]], where the [[left adjoint]] $|-|_r$ is a quotient of the [[geometric realization]] of the underlying Kan complexes and $\Pi_\infty$ is a version of the [[fundamental ∞-groupoid]]-functor with values in algebraic Kan complexes.

See homotopy hypothesis -- for algebraic Kan complexes for details.

### Algebraicization

If we assume the [[axiom of choice]], then any [[Kan complex]] can be made into an algebraic Kan complex by making a simultaneous choice of a filler for every horn.

In the absence of AC, one might argue that algebraic Kan complexes are a better model of $\infty$-groupoids than non-algebraic ones.  For instance, an algebraic Kan complex always has the right lifting property with respect to all [[anodyne morphisms]], whereas for a non-algebraic Kan complex this fact requires choice.

## References

* {#Nikolaus} [[Thomas Nikolaus]], _Algebraic models for higher categories_, Indagationes Mathematicae
Volume 21, Issues 1–2, July 2011, Pages 52-75
 ([arXiv/1003.1342](http://arxiv.org/abs/1003.1342), [doi:10.1016/j.indag.2010.12.004](https://doi.org/10.1016/j.indag.2010.12.004))

[[!redirects algebraic Kan complex]]
[[!redirects algebraic Kan complexes]]


algebraic lattice

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# Contents
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{: toc}

## Definition {#Def}

+-- {: .num_defn}
###### Definition

An **algebraic lattice** is a [[lattice]] which is

* a [[complete lattice]];

* such that every element is a [[join]] of [[compact elements]].

=--

An __algebraic lattice__ is a [[complete lattice]] (equivalently, a [[suplattice]], or in different words a [[poset]] with the [[extra property|property]] of having arbitrary [[colimits]] but with the [[structure]] of [[directed colimits]]/[[directed joins]]) in which every element is the [[supremum]] of the [[compact element]]s below it (an element $e$ is compact if, for every subset $S$ of the lattice, $e$ is less than or equal to the supremum of $S$ just in case $e$ is less than or equal to the supremum of some finite subset of $S$).

Here is an alternative formulation:
+-- {: .num_defn}
###### Definition
An algebraic lattice is a [[poset]] which is [[locally finitely presentable category|locally finitely presentable]] as a category.
=--

This formulation suggests a useful way of viewing algebraic lattices in terms
of [[Gabriel-Ulmer duality]] (but with regard to enrichment in [[truth values]], instead of in $Set$).

As this last formulation suggests, algebraic lattices typically arise as [[subobject lattices]] for objects in locally finitely presentable categories. As an example, for any (finitary) [[Lawvere theory]] $T$, the subobject lattice of an object in $T$-$Alg$ is an algebraic lattice (this class of examples explains the origin of the term "algebraic lattice", which is due to Garrett Birkhoff). In fact, all algebraic lattices arise this way (see Theorem \ref{GS} below).

It is trivial that every finite lattice is algebraic.

## Properties

### The category of algebraic lattices

The [[morphisms]] most commonly considered between algebraic lattices are the [[finitary functors]] between them, which is to say, the [[Scott topology|Scott-continuous]] functions between them; i.e., those functions which preserve directed joins (hence the parenthetical remarks [above](#Def)).

The resulting category __AlgLat__ is [[cartesian closed]] and is dually equivalent to the category whose objects are [[meet semilattices]] (construed as categories with [[finite limits]] [[enriched category|enriched]] over [[truth values]]) and whose morphisms are meet-preserving [[profunctors]] between them (using the convention that a $V$-enriched profunctor from $C$ to $D$ is a functor $D^{op} \times C \rightarrow V$; of course, with an opposite convention, one could similarly state a covariant equivalence).

There is a _full_ [[full embedding|embedding]]

$$i \colon AlgLat \to Top_0$$

to the category of $T_0$-[[separation axioms|spaces]], taking an algebraic lattice $L$ to the space whose points are elements of $L$, and whose [[open sets]] $U$ are defined by the property that their [[characteristic maps]]

$$\chi_U: L \to \mathbf{2}$$

($\chi_U(a) = 1$ if $a \in U$, else $\chi_U(a) = 0$) are poset maps that preserve [[directed colimits]]. The [[specialization order]] of $i(L)$ is $L$ again.

Every $T_0$-space $X$ occurs as a [[subspace]] of some space $i(L)$ associated with an algebraic lattice. Explicitly, let $L(X)$ be the [[power set]] of the underlying set of the [[topology]], $P{|\mathcal{O}(X)|}$, and define

$$X \to (i\circ L)(X)$$

to take $x$ to $N(x) \coloneqq \{U \in \mathcal{O}(X): x \in U\}$. This gives a topological embedding of $X$ in $i(L(X))$.

+-- {: .un_remark}
###### Remark
On similar grounds, if $U \colon AlgLat \to Set$ is the forgetful functor, then the [2-image](http://ncatlab.org/nlab/show/stuff%2C+structure%2C+property#a_factorisation_system_14) of the projection functor $\pi \colon Set\downarrow U \to Set$ is the category of topological spaces $Top$. In more nuts-and-bolts terms, an object $(S, L, f \colon S \to U(L))$ gives a space with underlying set $S$ and open sets those of the form $f^{-1}(O)$, where $O$ ranges over the Scott topology on $L$. Notice that if $(f \colon S \to S', g \colon L \to L')$ is a morphism in $Set \downarrow U$, then $f$ is continuous with respect to these topologies. Therefore the projection $\pi \colon Set \downarrow U \to Set$ factors through the faithful forgetful functor $Top \to Set$. Thus, working in the factorization system (eso+full, faithful) on $Cat$, we have a faithful functor $2$-$im(\pi) \to Top$ filling in as the diagonal

$$\array{
Set \downarrow U & \to & Top \\
\downarrow & \nearrow & \downarrow \\
2\text{-}im(\pi) & \to & Set.
}$$

But notice also that $Set \downarrow U \to Top$ is [eso and full](http://ncatlab.org/nlab/show/ternary+factorization+system#examples_9). It is eso because any topology $\mathcal{O}(S)$ on $S$ can be reconstituted from the triple $(S, P{|\mathcal{O}(S)|}, x \mapsto N(x) \colon S \to P{|\mathcal{O}(S)|})$. We claim it is full as well. For, every continuous map $X \to X'$ between topological spaces induces a continuous map between their $T_0$ reflections $X_0 \to X_{0}'$, and since algebraic lattices like $P{|\mathcal{O}(X)|}$ (being continuous lattices) are [[injective objects]] in the category of $T_0$ spaces, we are able to complete to a diagram

$$\array{
X & \to & X_0 & \to & P{|\mathcal{O}(X)|} \\
\downarrow & & \downarrow & & \downarrow \\
X' & \to & X_{0}' & \to & P{|\mathcal{O}(X')|}
}$$

where the rightmost vertical arrow is Scott-continuous (and the horizontal composites are of the form $x \mapsto N(x)$). Finally, since $Set \downarrow U \to Top$ is eso and full, it follows that $2$-$im(\pi) \to Top$ is eso, full, and faithful, and therefore an equivalence of categories.

This connection is explored in more depth with the category of [[equilogical spaces]], which can be seen either as a category of (set-theoretic) [[equivalence relation|partial equivalence relations]] over $AlgLat$, or equivalently of (set-theoretic) total [[equivalence relations]] on $T_0$ topological spaces.
=--

### Relation to locally finitely presentable categories
 {#RelationToLocallyFinitelyPresentableCategories}

One of our definitions of algebraic lattice is: a poset $L$ which is locally finitely presentable when viewed as a category. The completeness of $L$ means that right adjoints $L \to Set$ are representable, given by $L(p, -) \colon L \to Set$, and we are particularly interested in those representable functors that preserve [[filtered colimits]]. These correspond precisely to finitely presentable objects $p$, which in lattice theory are usually called compact elements. These compact elements are closed under finite joins.

By [[Gabriel-Ulmer duality]], $L$ is determined from the join-semilattice of compact elements $K$ by $L \cong Lex(K^{op}, Set)$. Since the elements of $K^{op}$ are subterminal, we can also write $L \cong Lex(K^{op}, 2)$ where $2 = Sub(1)$.

+-- {: .num_theorem}
###### Theorem
**(Porst)**

If $C$ is a [[locally finitely presentable category]] and $X$ is an object of $C$, then

* The lattice of subobjects $Sub(X)$,

* the lattice of quotient objects (equivalence classes of epis sourced at $X$) $Quot(X)$,

* the lattice of congruences (internal equivalence relations) on $X$

are all algebraic lattices.
=--

This is due to [Porst](#Porst). Of course if $C$ is the category of algebras of an Lawvere theory, then the lattice of quotient objects of an algebra is isomorphic to its congruence lattice, as such $C$ is an [[exact category]].

### Congruence lattices

The following result is due to Grätzer and Schmidt:

+-- {: .num_theorem #GS}
###### Theorem
Every algebraic lattice is isomorphic to the congruence lattice of some [[model]] $X$ of some finitary algebraic theory.
=--

In particular, since every finite lattice is algebraic, every finite lattice arises this way. Remarkably, it is not known at this time whether every finite lattice arises as the congruence lattice of a *finite* algebra $X$. It has been conjectured that this is in fact **false**: see this [MO discussion](http://mathoverflow.net/a/196074/2926).

Another problem which had long remained open is the congruence lattice problem: is every *distributive* algebraic lattice the congruence lattice (or lattice of quotient objects) of some lattice $L$? The answer is negative, as shown by Wehrung in 2007: see this [Wikipedia article](http://en.m.wikipedia.org/wiki/Congruence_lattice_problem).

### Completely distributive lattices

+-- {: .num_prop}
###### Proposition

The category of [[Alexandroff locales]] is equivalent to that of [[completely distributive lattice|completely distributive]] algebraic lattices.
=--

This appears as ([Caramello, remark 4.3](#Caramello)).

The [[completely distributive lattice|completely distributive]] algebraic lattices form a [[reflective subcategory]] of that of all distributive lattices. The reflector is called _[[canonical extension]]_.

## Related concepts

See also [[compact element]], [[compact element in a locale]].

[[!include locally presentable categories - table]]

## References

* [[Andrej Bauer]], [[Lars Birkedal]], [[Dana Scott]], _Equilogical Spaces_, Theoretical Computer Science, 315(1):35-59, 2004. ([web](http://math.andrej.com/2002/07/05/equilogical-spaces/))

* Olivia Caramello, _A topos-theoretic approach to Stone-type dualities_ ([arXiv:1103.3493](http://arxiv.org/abs/1103.3493))
 {#Caramello}

The relation to [[locally finitely presentable categories]] is discussed in

* [[Hans Porst]], _Algebraic lattices and locally finitely presentable categories_, Algebra Univers. **65**, 285–298 (2011). <https://doi.org/10.1007/s00012-011-0129-0> ([pdf](http://www.math.uni-bremen.de/~porst/dvis/PORST_AlgebraicLattices_revfinAU.pdf))
 {#Porst}

That every algebraic lattice is a congruence lattice is proved in

* G. Grätzer and E. T. Schmidt, _Characterizations of congruence lattices of abstract algebras_, Acta Sci. Math. (Szeged) 24 (1963), 34–59.

[[!redirects algebraic lattice]]
[[!redirects algebraic lattices]]


algebraic Lefschetz formula

Let $(C,d)$ be a nonnegative [[cochain complex]] of [[vector space]]s over a [[field]] of (total) finite dimension $dim C = \sum_{p=0}^\infty dim C^p \lt \infty$ and $f = (f^p)_{p\geq 0} :(C,d)\to (C,d)$ an endomorphism of cochain complexes.

The **algebraic Lefschetz formula** is the statement

$$
\sum_{p\geq 0} (-1)^p tr (f^p :C^p\to C^p) = \sum_{p\geq 0} (-1)^p tr (H^p(f):H^p(C)\to H^p(C)).
$$

Its special case for $f = id$ is the **Euler-Poincaré formula**

$$
\sum_{p\geq 0} (-1)^p dim C^p = \sum_{p\geq 0} (-1)^p dim H^p(C).
$$


algebraic limit field

[[!redirects algebraic limit fields]]
[[!redirects algebraic limit theorem]]
[[!redirects algebraic limit theorems]]

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## Idea ##

A [[field]] with a notion of a [[limit of a sequence|limit]] of a [[function]] that satisfy the [[algebraic limit theorems]].

## Definition ##

Let $F$ be a [[Heyting field]] and a [[Hausdorff function limit space]], where $x^{-1}$ is another notation for the [[reciprocal function]] $\frac{1}{x}$. $F$ is a __algebraic limit field__ if the __algebraic limit theorems__ are satisfied, i.e. if the limit preserves the field operations:

* for all elements $c \in S$,
$$\lim_{x \to c} 0(x) = 0$$

* for all elements $c \in S$ and functions $f:S \to C$ and $g:S \to C$ such that
$$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$
$$\lim_{x \to c} f(x) + g(x) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$$

* for all elements $c \in S$ and functions $f:S \to C$ such that
$$\lim_{x \to c} f(x) = c$$
$$\lim_{x \to c} -f(x) = -\lim_{x \to c} f(x)$$

* for all elements $c \in S$ and functions $f:S \to C$ and $g:S \to C$ such that
$$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$
$$\lim_{x \to c} f(x) - g(x) = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$$

* for all elements $c \in S$, integers $a \in \mathbb{Z}$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
$$\lim_{x \to c} a f(x) = a \lim_{x \to c} f(x)$$

* for all elements $c \in S$ and $a \in S$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
$$\lim_{x \to c} a f(x) = a \lim_{x \to c} f(x)$$

* for all elements $c \in S$,
$$\lim_{x \to c} 1(x) = 1$$

* for all elements $c \in S$ and functions $f:S \to C$ and $g:S \to C$ such that
$$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$
$$\lim_{x \to c} f(x) \cdot g(x) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$$

* for all elements $c \in S$, natural numbers $n \in \mathbb{N}$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
$$\lim_{x \to c} {f(x)}^n = {\left(\lim_{x \to c} f(x)\right)}^n$$

* for all elements $c \in S$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
if
$$\lim_{x \to c} f(x) \# 0$$
then
$$\lim_{x \to c} {f(x)}^{-1} = {\left(\lim_{x \to c} f(x)\right)}^{-1}$$

* for all elements $c \in S$, and functions $f:S \to C$, such that
$$\lim_{x \to c} f(x) = c$$
if
$$\lim_{x \to c} f(x) \# 0$$
then
$$\lim_{x \to c} f(x) \cdot {f(x)}^{-1} = 1$$

## See also ##

* [[function limit space]]
* [[difference quotient]]
* [[algebraic limit vector space]]
* [[Newton-Leibniz operator]]
* [[differentiable space]]


algebraic limit vector space

[[!redirects algebraic limit vector spaces]]

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## Definition ##

Given an [[algebraic limit field]] $F$, an __algebraic limit vector space__ is a $F$-[[vector space]].

## See also ##

* [[algebraic limit field]]

* [[difference quotient]]

* [[Newton-Leibniz operator]]


algebraic line bundle

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## Idea

The generalization of the concept of _[[holomorphic line bundle]]_ from [[complex analytic geometry]] to more general [[algebraic geometry]].

Algebraic line bundles on some [[variety]]/[[scheme]] form its [[Picard group]]/[[Picard scheme]].

## Related concepts

* [[divisor (algebraic geometry)]]

* [[algebraic line n-bundle]]

[[!redirects algebraic line bundles]]


algebraic line n-bundle

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## Idea

The concept of [[line n-bundle]] in [[algebraic geometry]], classified by maps into the $n$-fold [[delooping]] $\mathbf{B}^n \mathbb{G}_m$ of the [[multiplicative group]].

* for $n = 0$, classified by the [[group of units]];

* for $n= 1$ these are [[algebraic line bundles]], classified by [[Picard group]], modulated by [[Picard stack]];

* for $n =2$ these are [[algebraic line 2-bundles]], classified by [[Brauer group]], modulated by [[Brauer stack]]

## Properties
 {#Properties}

According to ([Grothendieck 64, prop. 1.4](#Grothendieck64)) for $X$ a [[Noetherian scheme]] whose [[local rings]] have strict [[Henselian ring|Henselisations]] that are factorial (...explain...) then the [[cohomology groups]]

$$
  H^n(X,\mathbb{G}_m) = \pi_0 \mathbf{H}(X,\mathbf{B}^n \mathbb{G}_m)
$$

are all [[torsion groups]] for $n \geq 2$. (For $n = 2$ this is the [[Brauer group]].) See also [this MO discussion](http://mathoverflow.net/q/171638/381).

See also at _[[Friedlander-Milnor isomorphism conjecture]]_.

## Related concepts

* [[holomorphic line n-bundle]]

## References

* {#Grothendieck58} [[Alexander Grothendieck]], _Torsion homologique et sections rationnelles_, Séminaire Claude Chevalley, 3 (1958), Exp. No. 5, 29 p. ([Numdam](http://www.numdam.org/item?id=SCC_1958__3__A5_0))

* {#Grothendieck64} [[Alexander Grothendieck]], _Le groupe de Brauer : II. Théories cohomologiques_. Séminaire Bourbaki, 9 (1964-1966), Exp. No. 297, 21 p. ([Numdam](http://www.numdam.org/item?id=SB_1964-1966__9__287_0))

[[!redirects algebraic line n-bundles]]

[[!redirects algebraic line 2-bundle]]
[[!redirects algebraic line 2-bundles]]


algebraic microlocalization

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# Idea

As an analogue of the [[microlocalization]] in operator theory, T. Springer has introduced an algebraic microlocalization in the theory of filtered noncommutative rings.

[[microlocal analysis|Microlocal analysis]] using [[hyperfunctions]] instead of [[Schwartz distributions]] is also called algebraic microlocal analysis.

#References

An alternative way to algebraic microlocalization is given in

* Maria J. Asensio, [[Michel Van den Bergh]], [[Freddy Van Oystaeyen]], _A new algebraic approach to microlocalization of filtered rings_, Trans. Amer. Math. Soc. __316__, 2 (Dec. 1989) 537--553 [jstor](http://www.jstor.org/pss/2001360)

This is used in comparison to Kapranov's noncommutative geometry based on commutator expansion in

* [[Lieven Le Bruyn]], _Formal structures and representation spaces_, J. Algebra __247__, 616--635 (2002) [doi](https://doi.org/10.1006/jabr.2001.9019)

An introduction to the [[microlocal analysis]] of [[hyperfunctions]] is this:

* Goro Kato, Daniele C. Struppa: _Fundamentals of algebraic microlocal analysis_ ([ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0924.35001&format=complete))

[[!redirects algebraic microlocal analysis]]


algebraic model category

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## Idea

The structure of an _algebraic model category_ is a refinement of that of a [[model category]].

Where a bare [[model category]] structure is a [[category with weak equivalences]] refined by two [[weak factorization systems]] ((cofibrations, acyclic fibrations) and (acyclic cofibrations, fibrations)) in an algebraic model structure these are refined further to [[algebraic weak factorization systems]] plus a bit more.

This extra structure supplies more control over constructions in the model category. For instance its choice induces a [[weak factorization system]] also in every [[diagram]] [[category]] of the given model category.

## Definition

An _algebraic model structure_ on a [[homotopical category]] $(M,W)$ consists of a pair of algebraic weak factorization systems $(C_t, F)$, $(C,F_t)$ together with a _morphism of algebraic weak factorization systems_

$$(C_t,F) \to (C,F_t)$$

such that the underlying weak factorization systems form a model structure on $M$ with weak equivalences $W$.

A _morphism of algebraic weak factorization systems_ consists of a natural transformation
  $$\array{  &  \text{dom} f &  \\  {}^{C_{t}f}\swarrow & & \searrow {}^{{C}{f}}  \\ Rf & \stackrel{\xi_f}{\to}  & Qf \\  {}_{{F}{f}}\searrow & & \swarrow {}_{F_{t}f}  \\  & \text{cod} f &  }$$
comparing the two functorial factorizations of a map $f$ that defines a [[monad|colax comonad morphism]] $C_t \to C$ and a lax monad morphism $F_t \to F$.

## Properties

Every [[cofibrantly generated model category]] structure can be lifted to that of an algebraic model category.  It is not clear whether or not this is true for any [[accessible model category]].

Any algebraic model category has a fibrant replacement monad $R$ and a cofibrant replacement comonad $Q$. There is also a canonical [[distributive law]] $RQ \to QR$ comparing the two canonical bifibrant replacement functors.

## Related pages

* [[algebraic weak factorization system]]
* [[accessible model category]]

[[!include algebraic model structures - table]]

## References

The notion was introduced in:

* [[Emily Riehl]], _Algebraic model structures_, New York J. Math. 17 (2011) 173-231 ([journa](http://nyjm.albany.edu/j/2011/17-10.html), [arXiv](http://arxiv.org/abs/0910.2733))

The algebraic analog of [[monoidal model categories]] is discussed in

* [[Emily Riehl]], _Monoidal algebraic model structures_ ([arXiv:1109.2883](http://arxiv.org/abs/1109.2883))

Review:

* [[Emily Riehl]], §12.9 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

See also:

* Patrick M. Schultz, *Algebraic Weak Factorization Systems in Double Categories*, PhD thesis, University of Orego (2014) [[hdl:1794/18429](http://hdl.handle.net/1794/18429), [pdf](https://scholarsbank.uoregon.edu/xmlui/bitstream/handle/1794/18429/Schultz_oregon_0171A_11048.pdf?sequence=1&isAllowed=y)]

* Gabriel Bainbridge, *Some Constructions of Algebraic Model Categories*, PhD thesis, Ohio State University (2021) [[pdf](https://etd.ohiolink.edu/apexprod/rws_etd/send_file/send?accession=osu1620719585729611&disposition=inline), [[Bainbridge-AlgebraicModelCategories.pdf:file]]]

[[!redirects algebraic model categories]]

[[!redirects algebraic model structure]]
[[!redirects algebraic model structures]]


algebraic model for modal logics

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# Algebras for modal logics
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## Idea

Classical [[propositional calculus]] has an algebraic [[model]], namely a [[Boolean algebra]]. With a bit of imagination, one can give it a combinatorial model in the line of [[Kripke semantics]]. As this ordinary propositional logic has no [[modal operators]], then the corresponding [[frame (modal logic)|frames]] have no relations, so are just sets.  If $W$ is such a set, (of worlds), a valuation $V: Prop \to 2^W$ just assigns to each $p \in Prop$ and $w\in W$ a truth value, $\top$ or $\bot$, (true or false).  We know however that there is a Boolean algebra structure around in that power set $2^W$ and the [[semantics]] extends the assignment given by $V$ to a map of algebras, from the term algebra based on the basic propositional language to the Boolean algebra of subsets of $W$. That is just to say that it builds up that extension of $V$ bit-by-bit on the terms. This gives an algebraic interpretation or representation of the terms of the logic in terms of the algebra of subsets of $W$, in other words an algebraic semantics.

[[modal logic|Modal logics]] also have an algebraic semantics based on a Boolean algebra, but with additional operators that model the [[modal operators]]. This is as well as the [[geometric models for modal logics|geometric semantics]] using [[frames (modal logic)|frames]].

## Boolean algebras with operators (BAOs)

### Boolean algebras

We will, here, consider a [[Boolean algebra]], $\mathbb{B }$, as an algebra, and in the notation,

$$\mathbb{B} = (B, +, \cdot, \overline{},0,1)$$

so, for example, for a set $S$, the [[power set]] Boolean algebra will be

$$\mathbb{P}(S) = (\mathcal{P}(S), \cup, \cap,-,\emptyset, S),$$

where $-A$ is shorthand for the complement, $S- A$, of $A$.

### Operators

The __operators__ that we need to add into the Boolean algebras do not always preserve all the structure:

+-- {: .num_defn}
###### Definition

A [[function]], $m : B\to B$ is called an __operator__ on the Boolean algebra, $\mathbb{B}$, if it is _additive_

$$m(x+y) = m x + m y.$$

The operator, $m$, is called __normal__ if $m(0)=0$.

=--

Any operator, $m$, in this sense has a dual $l : B\to B$ given by

$$l(x) = (m(x^-))^-.$$

As $m$ is additive, $l$ is __multiplicative__

$$l(x\cdot y) = l(x)\cdot l(y),$$

and has $l(1) = 1$ if $m$ is normal.

+-- {: .num_remark}
###### Remark

One of the myriad notations used for the generic modal operators $\lozenge$ and $\Box$, are $M$ and $L$, whence $M$ is 'possibility, and $L$ is 'necessity", and these gave the names to the operators above.

=--

+-- {: .num_defn}
###### Definition

A **Boolean algebra with operators,** or BAO, of type $n$ consists of a Boolean algebra $\mathbb{B}$, and a set, $m_i$, $i = 1,\ldots, n$ of operators on $B$.
=--

BAOs are sometimes called **modal algebras**, especially in the case that $n = 1$.  The term **polymodal algebra** is then used for the general case.

There is no need in the definition of BAOs to restrict to finitely many operators nor to have all the operators being unary. The general theory is discussed in the Survey by Goldblatt (see the references).

## Examples

+-- {: .num_example}
###### Example
**BAOs from frames.**

Let $\mathfrak{F} = (W ,R)$ be a [[frame (modal logic)|frame]]. We define on the power set Boolean algebra, $\mathbb{P}(W)$, the operator $m$ by, if $T\subseteq W$,

$$m(T) = \{w \in W : \exists t\in T, R w t \}$$

It perhaps pays to interpret this in the case where $R$ is a [[preorder]] and when it is an equivalence relation. In the first case, this will be the set of states less than or equal to something in $T$, in the second it is the union of all equivalence classes that contain an element of $T$.

+-- {: .num_lemma}
###### Lemma

The function $m$ is a normal operator.
=--

The proof is a simple manipulation of the definitions.

The dual operator $l$ is given by $l(T) = \{w\in W\mid \forall t\in T \neg R w t\}$. (Again look at this for the preorder and equivalence frame cases.)

It is easy to extend this example to $\mathfrak{F} = (W ,R_1,\ldots, R_n)$ with the result being a BAO of type $n$.
=--

+-- {: .num_example}
###### Example
**The Lindenbaum-Tarski algebra of a modal logic.**

Suppose $\Lambda \subseteq \mathcal{L}_\omega(n)$ is a [[normal modal logic]], then its [[Lindenbaum-Tarski algebra]] has a natural BAO structure, for which see the above page.
=--

##Varieties of modal and polymodal algebras

The following is a list of some of the main equationally defined classes of (poly)modal algebras. (For convenience each has been given a separate entry.)

* [[temporal algebras]];

* [[closure algebras]];

* [[monadic algebras]].

## Related concepts

* [[geometric model for modal logic]]

* [[modal type theory]]

## References

General books on modal logics that include information on algebraic models include:

* [[Patrick Blackburn]], M. de Rijke and Y. [[Venema]], _Modal Logic_, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.

* [[Marcus Kracht]], _Tools and Techniques in Modal Logic,_ Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999.

There is an excellent short survey article (versions of which are available on the web):

*  [[Robert Goldblatt]], _Algebraic Polymodal Logic: A Survey_, the Logic Journal of the
IGPL, 8, (2000) pages 393–450, Special Issue on Algebraic Logic, edited by Istvan Nemeti and
Ildiko Sain.

Discussion of modal logic in terms of [[coalgebra]] and [[terminal coalgebra of an endofunctor]] is in

* Corina Cirstea, Alexander Kurz, Dirk Pattinson, Lutz Schröder and Yde Venema, _Modal logics are coalgebraic_ ([pdf](http://eprints.soton.ac.uk/267144/1/ModalCoalgRev.pdf))

[[!redirects modal algebra]]
[[!redirects modal algebras]]
[[!redirects polymodal algebra]]
[[!redirects polymodal algebras]]
[[!redirects algebraic model for modal logic]]
[[!redirects algebraic model for modal logics]]
[[!redirects algebraic models for modal logic]]
[[!redirects algebraic models for modal logics]]


algebraic model structures - table

**Algebraic model structures:** [[Quillen model structures]], mainly on [[locally presentable categories]], and their constituent [[categories with weak equivalences]] and [[weak factorization systems]], that can be equipped with further algebraic structure and "freely generated" by small data.

| structure | small-set-generated | small-category-generated | algebraicized |
| -- | -- | -- | -- |
| [[weak factorization system]] | [[combinatorial wfs]] | [[accessible wfs]] | [[algebraic wfs]] |
| [[model category]] | [[combinatorial model category]] | [[accessible model category]] | [[algebraic model category]] |
| [[transfinite construction of free algebras|construction method]] | [[small object argument]] | same as $\to$ | [[algebraic small object argument]] |


algebraic number

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## Definition

An **algebraic number** is a [[root]] of a non-zero [[polynomial]] with [[integer]] [[coefficients]] (or, equivalently, with [[rational number|rational]] coefficients). Equivalently, an element $\alpha$ of a [[field extension]] $K$ of the [[rational numbers]] $\mathbb{Q}$ is *algebraic* if the [[subfield]] $\mathbb{Q}(\alpha)$ is a finite degree [[field extension|extension]], i.e., is [[finite dimensional vector space|finite-dimensional]] as a vector space over $\mathbb{Q}$.

Since the [[rational numbers]] are a [[subfield]] of the [[complex numbers]], and since the [[complex numbers]] are an [[algebraically closed field]], algebraic numbers are naturally regarded as a sub-field of [[complex numbers]]

$$
  \mathbb{Q} \hookrightarrow \mathbb{C}
  \,.
$$

$,$

> **Visualisation of the (countable) field of algebraic numbers in the complex plane.** Colours indicate degree of the polynomial the number is a root of (red = linear, i.e. the rationals, green = quadratic, blue = cubic, yellow = quartic...). Points becomes smaller as the integer polynomial coefficients become larger. View shows integers 0,1 and 2 at bottom right, $+i$ near top.

> due to Stephen J. Brooks [here](https://en.wikipedia.org/wiki/File:Algebraicszoom.png)

But the collection of all algebraic numbers forms itself already an [[algebraically closed field|algebraically closed]] [[field]], typically denoted $\overline{\mathbb{Q}}$, as this is the [[algebraic closure]] of the field $\mathbb{Q}$ of [[rational numbers]]. This also follows easily from the equivalent definition of algebraic numbers in terms of finite degree extensions. The [[absolute Galois group]] $Gal(\overline{\mathbb{Q}}, \mathbb{Q})$ is peculiar, see [there](absolute+Galois+group#OfTheRationalNumbers).

Given a [[field]] $k$, an algebraic __[[number field]]__ $K$ over $k$ is a finite-degree extension of $k$. By default, the term "algebraic number field" means an algebraic number field over the rational numbers. If $\alpha$ is an algebraic number over $\mathbb{Q}$ then $\mathbb{Q}[\alpha]$ is a number field, however the field of all algebraic numbers is *not* a number field.

An [[algebraic integer]] is a root of a [[monic polynomial]] with integer coefficients. Equivalently, an element $\alpha$ of a field extension $K$ of $\mathbb{Q}$ is an algebraic integer if the [[ring]] $\mathbb{Z}[\alpha]$ is of finite [[rank]] as a $\mathbb{Z}$-module. It follows easily from this characterization that the collection of all algebraic integers forms a commutative ring.

## Related concepts

* [[algebraic number theory]]

  * **algebraic number**, [[algebraic integer]]

  * [[number field]]

* [[transcendental number]]

## References

See also

* Wikipedia, _[Algebraic numbers](https://en.wikipedia.org/wiki/Algebraic_number)_

[[!redirects algebraic number]]
[[!redirects algebraic numbers]]


algebraic number theory

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## Idea

Algebraic number theory studies [[algebraic numbers]], [[number fields]] and related [[algebra|algebraic]] [[structures]].

An [[algebraic number]] is a [[root]] of a [[polynomial]] [[equation]] with [[integer]] [[coefficients]] (or, equivalently with [[rational number|rational]] coeffients). An [[algebraic integer]] is a root of a [[monic polynomial]] with integer coefficients. Given a [[field]] $k$ a (algebraic) __[[number field]]__ $K = k[P]$ over $k$ is the minimal [[field]] containing all the roots of a given polynomial $P$ with coefficients in $k$. Usually one considers algebraic number fields over rational numbers.

The main direction in algebraic number theory is the [[class field theory]] which roughly studies finite abelian extensions of number fields. The one dimensional class field theory stems from the ideas of Kronecker and Weber, and results of Hilbert soon after them. Main results of the theory belong to the first half of the 20th century (Hilbert, Artin, Tate, Hasse...) and are quite different for the [[local field]] from the [[global field]] case. Generalizations for higher dimensional fields came later under now active higher class field theory, which is usually formulated in terms of algebraic K-theory and is closely related to deep questions of algebraic geometry (Tate, Kato, Saito etc.).

## Related entries

The circle of $n$Lab entries belonging or related closely to algebraic number theory is in its infancy, and the partial list of entries some of which are started and most of which are to be created should include (the entries grouped by similarity)

* [[field]], [[characteristic]], [[division ring]], [[ideal]]

* [[integer]], [[rational number]], [[irrational number]], [[period]]

* [[p-adic number]], [[profinite group]], [[idele]], [[adele]]

* [[separable extension]], [[normal extension of fields]], [[Galois group]], [[Galois extension]], [[abelian extension of fields]], [[cyclotomic field]]

* [[Brower group]], [[Galois cohomology]]

* [[Dedekind ring]], [[principal ideal ring]], [[unique factorization domain]], [[integral closure]]

* [[algebraic number]], [[algebraic closure]], [[number field]]

* [[discriminant]], [[resultant]], [[Euclid algorithm]]

* [[Diophantine equation]], [[Matiyasevich theorem]]

* [[function field]]

* [[discrete valuation]], [[valuation ring]], [[valuation ideal]], [[archimedean valuation]]

* [[local field]], [[global field]], [[complete field]]

* [[conductor]], [[ideal class group]], [[Picard group]], [[Milnor K-group]]

* [[class field theory]], [[global class field theory]], [[local class field theory]], [[higher class field theory]], [[Hilbert class field]]

* [[reciprocity law]], [[Artin reciprocity law]], [[Weil reciprocity law]]

* [[arithmetic scheme]], [[Arakelov geometry]], [[field with one element]]

* [[L-function]], [[motive]], [[Riemann conjecture]], [[algebraic K-theory]], [[Grothendieck Galois theory]]

## Literature

* {#FroehlichCassels67} [[Albrecht Fröhlich]], [[J. W. S. Cassels]] (eds.), _Algebraic number theory_, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, [pdf](https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf), [errata pdf](https://www.ma.imperial.ac.uk/~buzzard/errata.pdf) by [[Kevin Buzzard]])

* {#Cassels86} [[J. W. S. Cassels]], *Local Fields*, Cambridge University Press, 1986 (ISBN:9781139171885, [doi:10.1017/CBO9781139171885](https://doi.org/10.1017/CBO9781139171885))

* {#Neukirch92} [[Jürgen Neukirch]], _Algebraische Zahlentheorie_ (1992), English translation _Algebraic Number Theory_,  Grundlehren der Mathematischen Wissenschaften 322, 1999 ([pdf](http://www.plouffe.fr/simon/math/Algebraic%20Number%20Theory.pdf))

* [[Albrecht Fröhlich]], Martin J. Taylor, _Algebraic number theory_, Cambridge Studies in Advanced Mathematics 27, 1993

* [[James Milne]], *Algebraic number theory*, 2020 ([pdf](https://www.jmilne.org/math/CourseNotes/ANT.pdf))

The following survey of Connes-Marcolli work has an accessible quick introduction to algebraic number theory

* P. Almeida, _Noncommutative geometry and arithmetics_, Russian Journal of Mathematical Physics 16, No. 3, 2009, pp. 350–362, [doi](http://dx.doi.org/10.1134/S1061920809030030), see also nLab:arithmetic and noncommutative geometry

See also

* Alexander Schmidt, _Higher dimensional class field theory from a topological point of view_, [page](http://www.mathi.uni-heidelberg.de/~schmidt/papers/schmidt21-en.html)


algebraic quasi-category

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### Context
#### Higher category theory
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[[!include higher category theory - contents]]
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#### Quasi-category theory
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[[!include quasi-category theory contents]]
=--
=--
=--

An **algebraic quasi-category** is a [[quasi-category]] equipped with a _choice_ of (inner) [[horn]] fillers.

Algebraic quasi-categories give a [[algebraic definition of higher categories|algebraic definition of]] [[(∞,1)-categories]].

For more see  the section [Algebraic  fibrant models for higher categories](http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects#AlgebaicHigherCategories) at [[model structure on algebraic fibrant objects]].

[[!redirects algebraic quasi-category]]
[[!redirects algebraic quasi-categories]]
[[!redirects algebraic quasicategory]]
[[!redirects algebraic quasicategories]]


algebraic set theory

[[!redirects Algebraic set theory]]

#Contents#
* table of contents
{:toc}

## Idea

The new insight taken as a starting point in _algebraic set theory_ (AST) is that [[models]] of [[set theory]] are in fact [[algebra over an algebraic theory|algebras]] for a suitably presented [[algebraic theory]], and that many familiar [[set theory|set-theoretic]] conditions (such as well-foundedness) are thereby related to familiar algebraic ones (such as [[free functor|freeness]]). ([Awodey](#Awodey))

##Description of the project##

Also sometimes called _categorical_ (meaning category-theoretic) _set theory_, _algebraic set theory_ started to be developed in 1988 by [[André Joyal]] and [[Ieke Moerdijk]] and was first presented in detail as a book in 1995 by them. AST is a robust framework based on [[category theory]] to study and organize [[set theory|set theories]] and to construct [[model of a set theory|models of set theories]]. The aim of AST is to provide a uniform categorical semantics or description of set theories of different kinds (classical or [[constructive mathematics|constructive]]; bounded, [[predicative mathematics|predicative]] or impredicative; [[well-founded set|well founded]] or ill founded, ...), the various constructions of the [[von Neumann hierarchy|cumulative hierarchy]] of [[pure set]]s, [[forcing]] models, [[sheaf]] models and [[realizability|realisability]] models.

Instead of focusing on categories of sets AST focuses on categories of [[class]]es. The basic tool of AST is the notion of a [[category with class structure]] (a category of classes equipped with a class of small maps (the intuition being that their fibres are small in some sense), powerclasses and a [[universe|universal object]]) which provides an axiomatic framework in which models of set theory can be constructed. The notion of a class category permits both the definition of [[Zermelo-Fraenkel algebra|ZF-algebras]] and related structures expressing the idea that the hierarchy of sets is an algebraic structure on the one hand and the interpretation of the first order logic of elementary set theory on the other. The subcategory of sets in a class category is an [[topos|elementary topos]] and every elementary topos occurs as the topos of sets in some class category.

The class category itself always embeds into the [[ideal completion]] of a topos. The interpretation of the logic is that in every class category the universe is a model of basic intuitionistic set theory $\mathbf{BIST}$ that is logically [[complete model|complete]] with respect to class category models. Therefore class categories generalize both topos theory and intuitionistic set theory. AST founds and formalizes set theory on the ZF-algebra with operations union and successor (singleton) instead of on the membership relation. The [[ZFC|ZF axioms]] are nothing but a description of the free ZF-algebra just as the [[natural numbers object|Peano axioms]] are a description of the free [[monoid]] on one generator. In this perspective the models of set theory are algebras for a suitably presented [[algebraic theory]]. Using an auxiliary notion of small map it is possible to extend the axioms of a topos and provide a general theory for uniformly constructing models of set theory out of toposes.

##Naming##

There are two reasons for referring to this research as "algebraic set theory": The first reason is that the models of set theory that are produced by these methods are algebras for an abstractly presented "theory", in a precise, technical sense known to category theorists as a [[monad]]. The notion of an algebra for a monad subsumes and generalizes that of a model for a conventional algebraic theory, such as groups, rings, modules, etc. Indeed, the first significant work in this style on the applications of [[category theory]] to the study of [[set theory]] was the monograph ([Joyal-Moerdijk 1995](#JoyalMoerdijk)) _Algebraic set theory_.

The second reason is that we believe the locution "algebraic logic" should properly refer to categorical logic rather than just the logic of Boole and his modern proponents, since categorical logic subsumes such lattice theoretic methods and not the other way around. Hence the term "algebraic set theory" rather than "categorical set theory". This is in keeping with the use of "algebraic" to mean, essentially, "functorial" in modern algebraic topology, algebraic geometry, etc. (Awodey, [Why "algebraic set theory"?](http://www.phil.cmu.edu/projects/ast/whyast.html))

### Stack semantics
[[michaelshulman:stack semantics]] provides a structural and somewhat more uniform way of treating "classes" in [[topos]] theory.

## See also

* [[category of classes]]

* [[category with class structure]]

## References

Introductions and further pointers are at

* [Algebraic Set Theory Website](http://www.phil.cmu.edu/projects/ast/index.html)

Two introductions are:

* [[Steve Awodey]], _An Outline of Algebraic Set Theory_ ([archived pdf](https://web.archive.org/web/20070614063821/https://caae.phil.cmu.edu/projects/ast/Papers/awodey_outline.pdf))

* [[Benno van den Berg]], [[Ieke Moerdijk]], _A Unified Approach to Algebraic Set Theory_ [arXiv](http://arxiv.org/abs/0710.3066)

A standard textbook is

* [[André Joyal]], [[Ieke Moerdijk]], _Algebraic set theory_, Cambridge University Press (1995)
 {#JoyalMoerdijk}

A reference for BIST is

* [[Steve Awodey]], [[Carsten Butz]], [[Alex Simpson]], [[Thomas Streicher]], ["Relating first-order set theories, toposes and categories of classes"](https://doi.org/10.1016/j.apal.2013.06.004), _Annals of Pure and Applied Logic_ 165 (2014)

[[!redirects algebraic set theories]]
[[!redirects small maps]]
[[!redirects small map]]


algebraic small object argument

# The algebraic small object argument

* table of contents
{: toc}

## Idea

The ordinary [[small object argument]] is a way of constructing (often [[combinatorial wfs|combinatorial]]) [[weak factorization systems]] from a set of "generators".  While powerful and useful, it has several defects, such as:

* Its result is not uniquely determined.  In particular, it does not converge: we just go on until we've gone on long enough, but going on longer would produce a different result.

* Relatedly, it has no universal property.  This makes it hard to deal with category-theoretically.

* It does not suffice to construct every weak factorization system, not even every [[accessible wfs|accessible]] one.

The algebraic small object argument is a refinement of the small object argument, due to [Garner](#Garner), that remedies these defects.

## Definition

...

## Properties

* The result of the algebraic small object argument is an [[algebraic weak factorization system]], which is "freely generated" by the input data in an appropriate sense.

* If the input data is a set of arrows (rather than a category or double category), then the algebraic right class consists of the [[algebraically injective objects]] for the generating class of arrows.

* If the underlying category is [[locally presentable category|locally presentable]], then this awfs is in particular an [[accessible weak factorization system]].  Conversely, any accessible wfs can be generated by the algebraic small object argument; see [Rosicky](#Rosicky).

* Not every accessible *algebraic* wfs can be generated by the algebraic small object argument as above: every accessible wfs admits *some* algebraic realization that's generated by the algebraic SOA above, but it could admit other algebraic realizations that are not.  However, there is a further refinement of the algebraic SOA, due to [Bourke and Garner](#BourkeGarnerI), that takes as input a *double* category, and does suffice to generate all accessible algebraic wfs.

## Related concepts

* [[small object argument]]
* [[algebraically injective object]]
* [[accessible weak factorization system]]
* [[algebraic weak factorization system]]

[[!include algebraic model structures - table]]

## References

* {#Garner} [[Richard Garner]], _Understanding the small object argument_, [arXiv](http://arxiv.org/abs/0712.0724).

* [[Emily Riehl]], _Algebraic model structures_, ([arXiv:0910.2733](http://arxiv.org/abs/0910.2733)).

* [[Thomas Athorne]], _The coalgebraic structure of cell complexes_, [TAC](http://www.tac.mta.ca/tac/volumes/26/11/26-11abs.html)

* [[Tobias Barthel]] and [[Emily Riehl]], *On the construction of functorial factorizations for model categories*, Algebr. Geom. Topol. Volume 13, Number 2 (2013), 1089-1124, [projecteuclid](https://projecteuclid.org/euclid.agt/1513715550), [arxiv](https://arxiv.org/abs/1204.5427)

* {#BourkeGarnerI} [[John Bourke]] and [[Richard Garner]], _Algebraic weak factorisation systems I: accessible AWFS_, [arXiv](http://arxiv.org/abs/1412.6559).

* [[John Bourke]] and [[Richard Garner]], _Algebraic weak factorisation systems II: categories of weak maps_, [arXiv](http://arxiv.org/abs/1412.6560).

* {#Rosicky} J. Rosicky, _Accessible model categories_, [arxiv](https://arxiv.org/abs/1503.05010)

[[!redirects Garner's small object argument]]
[[!redirects algebraic SOA]]


algebraic space

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###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _algebraic space_ is an [[object]] in the [[sheaf topos]] over the [[fppf-site]], that has [[representable morphism of stacks|representable]] [[diagonal]] and an [[étale cover]] ([[atlas]]) by a [[scheme]].

In [[algebraic geometry]] one can glue [[affine schemes]] in various topologies; this way one obtains various kinds of locally affine [[ringed space]]s. For example, [[scheme]]s locally affine in [[Zariski topology]]. [[etale topology|Étale topology]] is finer than Zariski, hence the category of locally affine (ringed) spaces in étale topology is larger than the category of schemes. __Algebraic spaces__ make a category which includes the category of all schemes and is close to the category of locally affine spaces in [[étale topology]], namely it consists of those [[ringed space]]s which may be obtained as a [[quotient object|quotient]] of a scheme $S$ by an [[equivalence relation]] $R\subset S\times S$ which is a closed subscheme, and whose projections $p_1,p_2: R\to S$ are étale morphisms of schemes.

## Definition

Write $C_{fppf}$ for the [[fppf-site]] (over some [[scheme]], as desired).

+-- {: .num_defn}
###### Definition

An **algebraic space** is

* an object $X \in Sh(C_{fppf})$ in the [[sheaf topos]];

* whose [[diagonal]] morphism $X \to X \times X$ is [[representable morphism of stacks|representable]];

* and for which there exists $U \in C$ and a morphism $U \to X$ which is

  * surjective;

  * [[étale morphism|étale]].

=--

In this form this appears as [de Jong, def. 35.6.1](#deJong).

## Properties

### Characterization as presheaf on affine schemes

* [[Artin representability theorem]]

## Related concepts

* [[algebraic stack]]

## References

Algebraic spaces are the topic of part 4 (tag 0ELT) in

* [[Aise Johan de Jong]], _[[The Stacks Project]]_
{#deJong}

Monographs:

* [[Donald Knutson]], *Algebraic spaces*, Lecture Notes in Mathematics, **203**, Springer (1971) [[doi:10.1007/BFb0059750](https://link.springer.com/book/10.1007/BFb0059750)]

* [[Martin Olsson]], *Algebraic Spaces and Stacks*, Colloquium Publications **62** (2016) [[doi:10.1090/coll/062](https://doi.org/10.1090/coll/062), [ISBN:978-1-4704-2798-6](https://bookstore.ams.org/coll-62)]

Lecture notes:

* [[James Milne]], section 7 of _[[Lectures on Étale Cohomology]]_

* G. B. Winters, _An elementary lecture on algebraic spaces_, in: P. Salmon (eds) Categories and Commutative Algebra. C.I.M.E. Summer Schools __58__ (Varena 1971), C.I.M.E., Ed. Cremonese, Roma 1973; reprint Springer 2010, [doi 2010](https://doi.org/10.1007/978-3-642-10979-9_9)

Definition in [[E-∞ geometry]] is in

* [[Jacob Lurie]], section 1.3 of _[[Quasi-Coherent Sheaves and
Tannaka Duality Theorems]]_

Some related MO questions:

* [why-is-this-not-an-algebraic-space](http://mathoverflow.net/questions/9043/why-is-this-not-an-algebraic-space),

* [Can an algebraic space fail to have a universal map to a scheme?](http://mathoverflow.net/questions/4587/can-an-algebraic-space-fail-to-have-a-unviersal-map-to-a-scheme)

* [What are the Benefits of Using Algebraic Spaces over Schemes?](http://mathoverflow.net/q/3194/447)

* [Commutative rings to algebraic spaces in one jump?](http://mathoverflow.net/q/11226/447)

[[!redirects algebraic spaces]]


algebraic spaces > history

< [[algebraic spaces]]
[[!redirects algebraic spaces -- history]]