homological algebra

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#### Homological algebra
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[[!include homological algebra - contents]]
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#Contents#
* table of contents
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## Idea

In an [[abelian category]] $\mathcal{A}$, _homological algebra_ is the [[homotopy theory]] of [[chain complexes]] in $\mathcal{A}$ up to [[quasi-isomorphism of chain complexes]]. Hence it is the study of the [[(infinity,1)-categorical localization]] of the [[category of chain complexes]] at the class of [[quasi-isomorphism of chain complexes|quasi-isomorphisms]], or in other words the [[derived (infinity,1)-category]] of $\mathcal{A}$.

When considering nonnegatively graded chain complexes, homological algebra may be viewed as a linearized version of the [[homotopy theory]] of [[homotopy types]] or [[infinity-groupoids]], by the [[Dold-Kan correspondence]]. When considering unbounded chain complexes, it may be viewed as a linearized and stabilized version, by the [[stable Dold-Kan correspondence]]. Conversely, we may view [[homotopical algebra]] as a nonabelian generalization of homological algebra.

Hence homological algebra is

* The study of a particularly simple sort of [[stable (∞,1)-categories]], namely those derived from categories of chain complexes.  See _[As a toolbox in stable homotopy theory](#ToolboxInStableHomotopyTheory)_ below and the discussion at [[cosmic cube]].

* The study of properties of [[modules]] over [[rings]] of various types, by the use of methods adapted from [[topology|topological]] [[homology theory]].

* A simple fragment of, and toolbox for, [[stable homotopy theory]] --- and hence, by extension, unstable [[homotopy theory]].  From this point of view,  an archetypical motivating example is the [[chain complex]] $C_\bullet(X)$ of [[singular chains]] in a [[topological space]] $X$, whose [[chain homology]] is the _[[singular homology]]_ $H_\bullet(X)$ of $X$, which is a linear approximation to the [[homotopy groups]] of $X$.  Accordingly, $C_\bullet(X)$ itself serves as a linearized approximation to the [[homotopy type]] of $X$.

### As a toolbox in stable homotopy theory
 {#ToolboxInStableHomotopyTheory}

With homological algebra being a topic in stabilized [[homotopy theory]], it is really the study of [[stable (∞,1)-categories]] [[(∞,1)-category of chain complexes|of chain complexes]] -- and thus, by the stable [Dold-Kan correspondence](module%20spectrum#StableDoldKanCorrespondence), of Eilenberg-MacLane [[module spectra]].

Historically this modern perspective has developed only in stages out of more "concrete" (more [[category theory|1-categorical]]) notions, which now form the body of homological algebra, in the form of a box of tools for computing linearized problems in homotopy theory. The following list indicates how these traditional notions serve to present constructions in stable homotopy theory.

1. The notion of _[[quasi-isomorphism]]_ between chain complexes -- [[chain maps]] which induce [[isomorphisms]] on [[homology groups]] -- is the stable version of [[weak homotopy equivalences]] of topological spaces. The _[[derived category]]_ of chain complexes $D(\mathcal{A})$ obtained by [[localization|localizing]] $Ch_\bullet(\mathcal{A})$ at these [[weak equivalences]] is the corresponding _[[homotopy category]]_, the context where all [[chain maps]] are identified up to [[chain homotopy]] between good representatives of these objects.  (On the other hand, in more general situations this correspondence is less immediate, and the notion of quasi-isomorphism may not be the best choice; see at _[[Whitehead theorem]]_.)

1. By the discussion at _[[localization]]_ the morphisms in $D(\mathcal{A})$ are [[zig-zags]] of [[chain maps]] that involve [[resolutions]] by non-isomorphic but [[quasi-isomorphism|quasi-isomorphic]] chain complexes. By the various [[model structures on chain complexes]] these resolutions can concretely be constructed as _[[injective resolutions]]_, _[[projective resolutions]]_ and/or more general sorts of resolutions (such as [[flat resolutions]], soft, flabby, etc.) of chain complexes, and much of the theory revolves around handling these.

1. Notably, [[functors]] between categories of chain complexes may extend to functors on these derived categories by evaluating them on suitable resolutions -- accordingly called _[[derived functors]]_.  (In homological algebra, the phrase "derived functor" is traditionally applied to the *homology groups* of what abstract homotopy theory calls the "derived functor", these being the invariants that one can compute.)  Much of the theory revolves around computing and characterizing derived functors, for instance in the definition of [[abelian sheaf cohomology]] and hence there are powerful tools for these computations, notably [[spectral sequences]].

1. However, the [[derived category]] $D(\mathcal{A})$ is still a rather coarse approximation to the full [[stable (∞,1)-category]] [[(∞,1)-category of chain complexes|of chain complexes]] in $\mathcal{A}$. There is a series of extra property and structures added to it which gives better approximations, and large parts of modern homological algebra study these:

   First of all the derived category is automatically a [[triangulated category]], which is a means of remembering the operation of _[[suspension]]_ and _de-suspension_ ([[looping]]) of chain complexes. Further structure added to these goes by names such as _[[enhanced triangulated category]]_.  A [[stable derivator]] is a strong enhancement which encodes basically all the requisite structure for internal computations.  Finally, the further promotion of these to _[[stable model categories]]_ or _[[pretriangulated dg-categories]]/linear [[A-∞ categories]]_ of chain complexes makes them capture the full information present in the [[stable (∞,1)-category]].

1. [[algebra|Algebra]] in [[stable homotopy theory]] is _[[higher algebra]]_ over [[E-∞ rings]], and _homological algebra_ provides approximations to that: by the [[stable Dold-Kan correspondence]] [[chain complexes]] of $R$-[[modules]] are a presentation for [[Eilenberg-Mac Lane spectrum|HR]]-[[module spectra]]. Moreover, [[A-infinity algebras]] in $HR$-module spectra [are presented by](http://www.ncatlab.org/nlab/show/model+structure+on+dg-algebras#RelationToAInfinityAlgebras) [[dg-algebras]], hence by ordinary [[associative algebras]] in [[chain complexes]]. Similarly [[E-infinity algebras]] [are presented by](http://www.ncatlab.org/nlab/show/model+structure+on+dg-algebras#RelationToEInfinityAlgebras) [[commutative dg-algebras]], hence by [[commutative monoid|commutative algebras]] internal to chain complexes. By variation of this theme a multitude of notions in [[higher algebra]] finds their representation in homological algebra, for instance [[L-∞ algebras]] in terms of [[dg-Lie algebras]]: [[Lie algebras]] internal to chain complexes.

### Non-abelian variants

There are variants of the tools of homological algebra that can also be applied to more non-linear phenomena, see for instance at [[Dold-Kan correspondence]] the section _[non-abelian case](Dold-Kan%20correspondence#StatementGeneral)_.  These include non-Abelian (co)homology and crossed and quadratic versions that use a small degree of non-linearity in the models.  These latter theories make extensive use of techniques from [[homotopical algebra]] in the wide sense of that term and [[simplicial homotopy theory]] to avoid the crushing of homotopical information that can occur when passing to chain complexes.

## Entries in homological algebra ##

[[!include homological algebra - contents]]

## References
 {#References}

The following lists references on homological algebra:

* [General](#ReferencesGeneral)

* [Lecture notes and course notes](#LectureNotes)

* [History](#ReferencesHistory)

### General
 {#ReferencesGeneral}

Classical accounts:

* {#EilenbergMacLane42} [[Samuel Eilenberg]], [[Saunders MacLane]], *Group Extensions and Homology*, Annals of Mathematics **43** 4 (1942) 757-831 [[doi:10.2307/1968966](https://doi.org/10.2307/1968966), [jstor:1968966](https://www.jstor.org/stable/1968966)]

* D. A. Buchsbaum, _Exact categories and duality_, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 ([JSTOR](http://www.jstor.org/stable/1993003))

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], *[[Homological Algebra]]*, Princeton Univ. Press (1956), Princeton Mathematical Series **19** (1999) [[ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19), [doi:10.1515/9781400883844](https://doi.org/10.1515/9781400883844), [pdf](http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf)]

* [[A. Grothendieck]], _[[Tohoku|Sur quelques points d'algèbre homologique]]_ (1957)  ([part1](http://projecteuclid.org/euclid.tmj/1178244774), [part2](http://projecteuclid.org/euclid.tmj/1178244774))

* {#Godement58} [[Roger Godement]], *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]

* [[Saunders MacLane]], *Homology*, Grundlehren der Mathematischen Wissenschaften **114**, Springer (1963, 1974), reprinted as: Classics in Mathematics, Springer (1995)  [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/maclane_homology.pdf), [doi:10.1007/978-3-642-62029-4](https://doi.org/10.1007/978-3-642-62029-4)]

* {#HiltonStammbach71} [[Peter Hilton]],  [[Urs Stammbach]], _A course in homological algebra_, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 ([doi:10.1007/978-1-4419-8566-8](https://link.springer.com/book/10.1007/978-1-4419-8566-8), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/hilton-stammbach.pdf))

* [[Saunders Mac Lane]], _Homology_ (1975) reprinted as Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8 ([doi:10.1007/978-3-642-62029-4](https://link.springer.com/book/10.1007/978-3-642-62029-4))

A standard modern textbook introduction is

* {#Weibel94} [[Charles Weibel]], *[[An introduction to homological algebra]]*,  Cambridge Studies in Adv. Math. **38**, Cambridge University Press (1994) [[doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136), [pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf)]

and a more systematic modern development of the theory is in sections 8 and 12-18 of

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_,  Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006)

Non-abelian variants of homological algebra are disussed for instance in

* [[Francis Borceux]], [[Dominique Bourn]], _[[Borceux-Bourn|Mal'cev, protomodular, homological and semi-abelian categories]]_, Mathematics and Its Applications __566__, Kluwer (2004)

The foundations of the formulation in the broader context of [[stable (∞,1)-category]] theory is in

* [[Jacob Lurie]], _[[Stable Infinity-Categories]]_, ([math.CT/0608228](http://www.arXiv.org/abs/math.CT/0608228))

Other textbooks include

* I. Bucur, A. Deleanu, _Introduction to the theory of categories and functors_, 1968

* [[Sergei Gelfand]], [[Yuri Manin]], *[[Methods of homological algebra]]*,  transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [[doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5)]

See also

* Wikipedia, _[Homological algebra](http://en.wikipedia.org/wiki/Homological_algebra)_

* Springer Online Encyclopeadia of Mathematics: [homological algebra](http://eom.springer.de/H/h047710.htm)

### Lecture notes and course notes
 {#LectureNotes}

* [[Alexander Beilinson]], _Introduction to homological algebra_ (handwritten notes, summer 2007, pdf) [lec1](http://www.math.uchicago.edu/~mitya/beilinson/hom-alg-1.pdf), [lec2](http://www.math.uchicago.edu/~mitya/beilinson/hom-alg-2.pdf), [lec3](http://www.math.uchicago.edu/~mitya/beilinson/hom-alg-3.pdf), [lec4](http://www.math.uchicago.edu/~mitya/beilinson/hom-alg-4.pdf)

* {#Collins06} Julia Collins, _Homological algebra_ (2006) [[pdf](http://www.maths.ed.ac.uk/~s0681349/HomologicalAlgebra.pdf)]

* [[Rick Jardine]], _Homological algebra_, course notes, 2009  ([index](http://www.math.uwo.ca/~jardine/papers/HomAlg/index.shtml))

* [[Peter May]], _Notes on Tor and Ext_ ([pdf](http://www.math.uchicago.edu/~may/MISC/TorExt.pdf))
 {#May}

* [[Pierre Schapira]], _Categories and homological algebra_, lecture notes (2011) ([pdf](http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf))

* [[Urs Schreiber]], _[[schreiber:Introduction to Homological Algebra]]_

### In constructive mathematics

Discussion of homological algebra in [[constructive mathematics]]:

* {#CoquandSpiwack} [[Thierry Coquand]], [[Arnaud Spiwack]], *Towards constructive homological algebra in type theory*, in: *Towards Mechanized Mathematical Assistants. MKM Calculemus 2007*,  Lecture Notes in Computer Science **4573** Springer (2007) [[doi:10.1007/978-3-540-73086-6_4](https://doi.org/10.1007/978-3-540-73086-6_4), [pdf](https://hal.inria.fr/inria-00432525/document)]

  > (via [[type theory]])

* [[Julio Rubio]], [[Francis Sergeraert]], _Constructive homological algebra and applications_ [[arXiv:1208.3816](http://arxiv.org/abs/1208.3816)]

### History
 {#ReferencesHistory}

* [[Charles Weibel]], _A history of homological algebra_, [dvi](http://www.math.rutgers.edu/~weibel/HA-history.dvi)

From the introduction of [Collins 2006](#Collins06):

> The word "[[homology]]" was first used in a [[topology|topological]] context by [[Poincaré]] in 1895, who used it to think about [[manifolds]] which were the [[boundaries]] of higher-dimensional manifolds. It was [[Emmy Noether]] in the 1920s who began thinking of homology in terms of [[groups]], and who developed algebraic techniques such as the idea of [[modules]] over a [[ring]]. These are both absolutely crucial ingredients in the modern theory of homological algebra, yet for the next twenty years homology theory was to remain confined to the realm of topology.

> In 1942 came the first move forward towards homological algebra as we know it today, with the arrival of a [paper by Samuel Eilenberg and Saunders MacLane](#EilenbergMacLane42). In it we find [[Hom]] and [[Ext]] defined for the very first time, and along with it the notions of a [[functor]] and [[natural isomorphism]]. These were needed to provide a precise language for talking about the properties of $Hom(A,B)$; in particular the fact that it varies naturally, contravariantly in $A$ and covariantly in $B$.

> Only three years later this language [was expanded](category#EilenbergMacLane45) to include [[category]] and [[natural equivalence]]. However, this terminology was not widely accepted by the mathematical community until the appearance of [Cartan and Eilenberg's book](#CartanEilenberg56). Cartan and Eilenberg's book was truly a revolution in the subject, and in fact it was here that the term "Homological Algebra" was first coined. The book used [[derived functors]] in a systematic way which united all the previous [[homology theories]], which in the past ten years had arisen in [[group theory]], [[Lie algebras]] and [[algebraic geometry]]. The sheer list of terms that were first defined in this book may give the reader an idea of how much of this project is due to the existence of that one book! They defined what it means for an object to be [[projective object|projective]] or [[injective object|injective]], and defined the notions of [[projective resolution|projective]] and [[injective resolutions]]. It is here that we find the first mention of $Hom$ being [[left exact functor|left exact]] and the first occurrence of $Ext^n$ as the [[right derived functors]] of $Hom$.


homological algebra - contents

**[[homological algebra]]**

(also [[nonabelian homological algebra]])

_[[schreiber:Introduction to Homological Algebra|Introduction]]_

**Context**

* [[additive and abelian categories]]

  * [[Ab-enriched category]]

  * [[pre-additive category]]

  * [[additive category]]

  * [[pre-abelian category]]

  * [[abelian category]]

  * [[Grothendieck category]]

* [[abelian sheaves]]

* [[semi-abelian category]]

**Basic definitions**

* [[kernel]], [[cokernel]]

* [[complex]]

  * [[differential]]

  * [[homology]]

* [[category of chain complexes]]

  * [[chain complex]]

  * [[chain map]]

  * [[chain homotopy]]

* [[chain homology and cohomology]]

  * [[quasi-isomorphism]]

  * [[homological resolution]]

  * [[cochain on a simplicial set|simplicial homology]]

  * [[generalized homology]]

* [[exact sequence]],

   * [[short exact sequence]], [[long exact sequence]], [[split exact sequence]]

* [[injective object]], [[projective object]]

  * [[injective resolution]], [[projective resolution]]

  * [[flat resolution]]

**Stable homotopy theory notions**

* [[derived category]]

* [[triangulated category]], [[enhanced triangulated category]]

* [[stable (∞,1)-category]]

  * [[stable model category]]

  * [[pretriangulated dg-category]]

  * [[A-∞-category]]

* [[(∞,1)-category of chain complexes]]

* [[derived functor]], [[derived functor in homological algebra]]

  * [[Tor]], [[Ext]]

  * [[homotopy limit]], [[homotopy colimit]]

* [[abelian sheaf cohomology]]

**Constructions**

* [[double complex]]

* [[Koszul-Tate resolution]], [[BRST-BV complex]]

* [[spectral sequence]]

  * [[spectral sequence of a filtered complex]]

  * [[spectral sequence of a double complex]]

  * [[Grothendieck spectral sequence]]

    * [[Leray spectral sequence]]

    * [[Serre spectral sequence]]

    * [[Hochschild-Serre spectral sequence]]

**Lemmas**

[[diagram chasing]]

* [[3x3 lemma]]

* [[four lemma]], [[five lemma]]

* [[snake lemma]], [[connecting homomorphism]]

* [[horseshoe lemma]]

* [[Baer's criterion]]

[[Schanuel's lemma]]

**Homology theories**

* [[singular homology]]

* [[cyclic homology]]

**Theorems**

* [[Dold-Kan correspondence]] / [[monoidal Dold-Kan correspondence|monoidal]], [[operadic Dold-Kan correspondence|operadic]]

  * [[Moore complex]], [[Alexander-Whitney map]], [[Eilenberg-Zilber map]]

* [[Eilenberg-Zilber theorem]]

  * [[cochain on a simplicial set]]

* [[universal coefficient theorem]]

* [[Künneth theorem]]


Recently, [[homological algebra]] started appearing in the study of numerical stability for [[finite element method]]s,

* Douglas N. Arnold, Differential Complexes and Numerical Stability, ICM 2002, Vol. I, 137-157, [scan](http://www.mathunion.org/ICM/ICM2002.1/Main/icm2002.1.0137.0158.ocr.pdf), [arXiv:math/0212391](http://arxiv.org/abs/math/0212391)
* Douglas N. Arnold, Richard S. Falk, Ragnar Winther, _Finite element exterior calculus: from Hodge theory to numerical stability_, Bull. Amer.
Math. Soc. (N.S.) __47__, 281-354, 2010, [arxiv/0906.4325](http://arxiv.org/abs/0906.4325), [MR2594630](http://www.ams.org/mathscinet-getitem?mr=2594630), [doi](http://dx.doi.org/10.1090/S0273-0979-10-01278-4)
* Douglas N. Arnold, Richard S. Falk, Ragnar Winther, _Finite element exterior calculus, homological techniques, and applications_, Acta Numer. __15__, 1-155, 2006, [MR2007j:58002](http://www.ams.org/mathscinet-getitem?mr=2269741), [doi](http://dx.doi.org/10.1017/S0962492906210018)
* D. N. Arnold, M. E. Rognes, _Stability of Lagrange elements for the mixed Laplacian_, Calcolo __46__ (2009), no. 4, 245–260,  [doi](http://dx.doi.org/10.1007/s10092-009-0009-6), [MR2563784](http://www.ams.org/mathscinet-getitem?mr=2563784)

The following vision may be relevant in future development of homological methods and applications of higher geometry in numerical analysis for [[hydrodynamic]] systems:

* [[Dennis Sullivan]], _Algebra, topology and algebraic topology of 3D ideal fluids_, [arxiv/1010.2721](http://arxiv.org/abs/1010.2721) (has some ideas on need of a version of [[higher geometry]], hence of homological methods in particular, is at the very end of the article)

[[!redirects homological algebra in finite element method]]
[[!redirects homological algebra in the finite element method]]
[[!redirects homological algebra in finite element methods]]
[[!redirects homological methods in finite element method]]
[[!redirects homological methods in the finite element method]]
[[!redirects homological methods in finite element methods]]
[[!redirects homological methods in FEM]]
[[!redirects homological methods in the FEM]]
[[!redirects homological methods in FEMs]]
[[!redirects homology and finite element method]]
[[!redirects homology and the finite element method]]
[[!redirects homology and finite element methods]]


homological and higher algebra -- table

| [[algebra]] | [[homological algebra]] |  [[higher algebra]] |
|-------------|---------|--------------------|
| [[abelian group]] | [[chain complex]] |  [[spectrum]] |
| [[ring]]    | [[dg-ring]] |  [[ring spectrum]] |
| [[module]]  | [[dg-module]] | [[module spectrum]]|


homological category

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### Context
#### Homological algebra
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[[!include homological algebra - contents]]
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#### Regular and exact categories
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[[!include regular and exact categories - contents]]
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# Homological category
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{: toc}

## Definition

A category $C$ is called __homological__ if it is

1. [[pointed category|pointed]]

1. [[regular category|regular]]

1. [[protomodular category|protomodular]].

## Properties

Many of the standard results of classical [[homological algebra]] in abelian categories extend to homological categories:

* the [[five lemma]]

* the [[nine lemma]]

* the [[snake lemma]]

* the [[long exact sequence in homology]]

* the Noether isomorphism theorem

* a version of the [[Jordan-Hölder theorem]].

A homological category which is [[Barr-exact]] and has finite [[coproducts]] is [[semiabelian category|semiabelian]].

## Examples

\begin{example}\label{CategoryOfGroupsIsHomological}
 The category [[Grp]] of all [[groups]] (including [[non-abelian groups]]) is homological. Namely it is

1. regular, by [this example](regular+category#CategoryOfGroupsIsRegular),

1. pointed protomodular by [this example](protomodular+category#GroupsFormAProtomodularCategory).

\end{example}

## Related entries

* [[abelian category]], [[semiabelian category]]

* [[Barr-exact category]]

* [[homological algebra]]

* [[pointed category]]

* [[protomodular category]]

* [[regular category]]

## References

* [[Francis Borceux]], [[Dominique Bourn]], Chapter IV in: *[[Mal'cev, protomodular, homological and semi-abelian categories]]*, Mathematics and Its Applications __566__, Kluwer, 2004 ([doi:10.1007/978-1-4020-1962-3](https://link.springer.com/book/10.1007/978-1-4020-1962-3))

* [[Francis Borceux]], [[Marco Grandis]], _Jordan-Hölder, modularity and distributivity in non-commutative algebra_,  J. Pure Appl. Algebra __208__ (2007),  no. 2, 665--689. ([doi](http://dx.doi.org/10.1016/j.jpaa.2006.03.004), [MR2007k:18021](http://www.ams.org/mathscinet-getitem?mr=2007k:18021))

* {#Janelidze09} [[Tamar Janelidze]], *Foundations of relative non-abelian homological algebra*, 2009 ([pdf](https://open.uct.ac.za/bitstream/item/4777/thesis_sci_2009_janelidze_t.pdf?sequence=1), [[Janelidze_NonabelianHomologicalAlgebra.pdf:file]], [hdl:11427/4891](http://hdl.handle.net/11427/4891))

[[!redirects homological categories]]


homological functor

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#### Homotopy theory
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#Contents#
* table of contents
{:toc}

## Definition

Let throughout $\mathcal{C}$ be a [[stable (∞,1)-category]], $\mathcal{A}$ an [[abelian category]],

A _homological functor_

$$
   \pi \;\colon\; \mathcal{C}\longrightarrow \mathcal{A}
$$

is an [[(∞,1)-functor]] that transforms every [[homotopy cofiber sequence]]

$$ X\to Y\to Z\to \Sigma X $$

in $\mathcal{C}$ into a [[long exact sequence]]

$$ \dots \to \pi(X)\to \pi(Y)\to \pi(Z)\to \pi(\Sigma X) \to \dots $$

in $\mathcal{A}$. One writes

$$
  \pi_n \coloneqq \pi\circ \Sigma^{-n}
$$

for $n \in \mathbb{N}$ and for $\Sigma$ the [[suspension]] functor.

## Examples

+-- {: .num_example}
###### Example

* $\mathcal{C}$ is arbitrary, $\mathcal{A}$ is the category of [[abelian groups]] and $\pi$ is $\pi_0 \mathcal{C}(S,-)$ for some object $S\in\mathcal{C}$

* $\mathcal{C}$ is equipped with a [[t-structure]], $\mathcal{A}$ is the [[heart of a stable (∞,1)-category|heart]] of the t-structure, and $\pi$ is the canonical functor.

* $\mathcal{C} = D(\mathcal{A})$ is the [[derived category]] of the abelian category $\mathcal{A}$ and $\pi=H_0$.

* Any of the above with $\mathcal{C}$ and $\mathcal{A}$ replaced by their [[opposite categories]].

* Any reduced [[generalized homology theory]].

=--

## Related concepts

* [[homotopical functor]]

* [[Adams spectral sequence]]

* [[Lurie spectral sequence]]

[[!redirects homological functors]]
[[!redirects homological (∞,1)-functor]]
[[!redirects homological (∞,1)-functors]]


homological perturbation theory

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#Contents#
* table of contents
{:toc}

## Idea

The aim of _Homological Perturbation Theory_ is to construct small [[chain complex]]es from large ones. It was originally developed for the calculation of chain models of the total spaces of [[fiber bundles]], but has since developed into a useful general computational tool.

## Homological perturbation lemma

Let $(X,d), (Y,d)$ be [[chain complexes]] over a [[commutative ring]] $R$ and let $f: X \to Y, \nabla: Y \to X$ be [[chain maps]], and $\Phi: X \to X$ be a [[chain homotopy]] such that

$$ f \nabla=1, \quad \nabla f= 1 + d\Phi + \phi d, $$

 $$ f\Phi = 0, \Phi \nabla=0, \Phi^2=0, \Phi d \Phi= - \Phi. $$

Let $X,Y$ have [[filtrations]] $F^*$ bounded below by $0$ and preserved by $\nabla,f, \Phi$ and the [[differentials]] on $X,Y$. Suppose $X$ has another differential $d^\tau$ with the property that

$$ (d^\tau -d)F^p X \subseteq F^{p-1} X $$

for all $p \geq 0$. The **Homological Perturbation Lemma** states that $Y$ can be given a new differential $d^\tau$ such that there is a [[quasi-isomorphism]] $(Y, d^\tau) \to (X, d^\tau)$.

The main point is that there is an explicit formula for the new chain homotopy as

$$\Phi^\tau= \sum _{r=0}^\infty \Phi (1+ d^\tau \Phi)^r.$$

There is considerable interest in describing the new differential in terms of a [[twisting cochain]].

This result derived from earlier work of G. Hirsch, E.H. Brown, Weishu Shih, and has been widely developed into a useful theoretical and computational tool by Guggenheim, Lambe, Stasheff and others.

## Applications

### BV-complexes and Wick's lemma
 {#BVComplexesAndWickLemma}

Homological perturbation theory is a key tool in the construction of [[BRST-BV complexes]], where the [[quantum BV complex]] is a perturbation of a [[classical BV-complex]]. See ([Gwilliam, section 2.5](#Gwilliam)). In this context [[Wick's lemma]] in [[quantum field theory]] is a direct consequence of the homological perturbation lemma ([Gwilliam, section 2.5.2](#Gwilliam)).

## References

Review includes

* [[Marius Crainic]], _On the perturbation lemma, and deformations_ ([arXiv:math/0403266](http://arxiv.org/abs/math/0403266))

* [[Johannes Huebschmann]], _A survey on homological perturbation theory_ ([pdf](http://math.univ-lille1.fr/~huebschm/data/talks/courant.pdf))

Other references include

* [[Ronnie Brown]], _The twisted Eilenberg-Zilber Theorem_,  Simposio di Topologia (Messina, 1964)  pp. 33--37 Edizioni Oderisi, Gubbio. ([pdf] (http://pages.bangor.ac.uk/~mas010/pdffiles/twistedez.pdf))

* {#BarnesLambe91} Donald W. Barnes, [[Larry Lambe|Larry A. Lambe]], _A fixed point approach to homological perturbation theory_, Proc. Amer. Math. Soc. 112 (1991), no. 3, 881--892.

* V Gugenheim, LA Lambe, JD Stasheff, _Perturbation theory in differential homological algebra II_, Illinois Journal of Mathematics 35:3 (1991) 357--373 [euclid](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-35/issue-3/Perturbation-theory-in-differential-homological-algebra-II/10.1215/ijm/1255987784.full)

For applications to  "make computable" a bicategory of isolated hypersurface singularities and matrix factorisations comp that has been studied in the context of [[topological field theory]], using the formulation in ([Barnes-Lambe 91](#BarnesLambe91))  is in

* [[Daniel Murfet]], _Computing with cut systems_, ([arXiv:1402.4541](http://arxiv.org/abs/1402.4541))

See also _[[linear logic]]_.

Discussion with an eye towards [[Hochschild cohomology]] and [[cyclic cohomology]] is in

* [[Larry Lambe|Larry A. Lambe]], _Homological perturbation theory, Hochschild homology and formal groups_ Contemp. Math. __189__, AMS, 1992 ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.64.8783&rep=rep1&type=pdf))

* V. Álvarez, J.A. Armario, P. Real, B. Silva, _Homological perturbation theory and computability Of Hochschild and cyclic homologies of cdgas_ ([pdf](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.6955))

Discussion in the context of [[BV-quantization]] is in section 2.5 of

* [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([pdf](https://people.math.umass.edu/~gwilliam/thesis.pdf))
 {#Gwilliam}

[[!redirects homological perturbation lemma]]


homological quantum field theory

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#### Quantum field theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _homological_ [[TQFT]] is a representation of the [[cobordism]] category that depends only on the [[homology]] of the [[hom-space]]s.

In $d = 2$ this is also called a _homological_ [[conformal field theory]]. The passage to homology forgets the conformal structure.
The refinement of this concept from [[homology group]]s to [[chain complex]]es is called [[TCFT]].

## Definition

Write $Bord$ for any given [[cobordism category]], regarded as a [[Top]]-[[enriched category]]. A **homological TQFT** is a [[symmetric monoidal functor|symmetric monoidal]] [[Ab]]-[[enriched functor]]

$$
  Z : H_\bullet(Bord) \to Ab
  \,.
$$

## Applications

* The [[string topology]] operations of a manifold are part of an HTQFT. See there for details.

## References

For 2-dimensional cobordisms with closed boundary HCFT has been considered in

* [[Veronique Godin]], _Higher string topology operations_ (2007)([arXiv:0711.4859](http://arxiv.org/abs/0711.4859))
 {#Godin}

A detailed treatment in $d = 2$ involving arbitrary sets of [[brane]]s is in section 2 of

* [[Sander Kupers]], _String topology operations_ MSc thesis (2011)

[[!redirects HCFT]]
[[!redirects HCFTs]]

[[!redirects HQFT]]
[[!redirects HQFTs]]


homological resolution

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#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[homological algebra]] a _homological resolution_ of a [[vector space]] or more generally of a [[module]] $N$ is a [[chain complex]] $V_\bullet$ whose [[chain homology]] $H_\bullet(V)$ reproduces $N$, in that

$$
  H_n(V)
  \;\simeq\;
  \left\{
    \array{
       N &\vert& n = 0
       \\
       0 &\vert& \text{otherwise}
    }
  \right.
$$

Of course if one regards $N$ itself as a chain complex $N_\bullet$ which is concentrated in degree 0, then $N_\bullet$ has this property, trivially. Therefore, typically when asking for a homological resolution $V_\bullet$ of $N$, it is understood that the entries of $V_\bullet$ have certain nice properties that $N$ is lacking.

For instance if $V_\bullet$ consists of [[projective modules]], then it is called a _[[projective resolution]]_ of $N$, or if it consists of [[injective modules]] then it is called an _[[injective resolution]]_, or if it consists of [[free modules]], then it is called a _[[free resolution]]_, etc. The module $N$ itself may be far from being projective or injective or free, etc. and so the corresponding resolution, if it exists, allows to nevertheless regard $N$ with tools applicable to these particularly nice classes of modules, up to chain homology

Typically in these constructions one demands not just that the [[chain homology]] of the resolution reproduces $N$ via _any_ isomorphism $H_0(V) \simeq N$, but that there exists a [[chain map]] $f_\bullet$ from $V$ to $N$ or from $N$ to $V$, such that it is this chain map which under passage to [[chain homology]] induces this isomorphism. A chain map which induces isomorphisms on chain homology groups is called a _[[quasi-isomorphism]]_, and so typically a _homological resolution_ means a choice of quasi-isomorphism from a module to a (particularly nice) chain complex.

Famous classes of examples of such resolutions are the injective and projective resolutions that are used to construct _[[derived functors in homological algebra]]_, see there for more.

Phrased this way, there is nothing special about starting with a single module, and more generally one may speak of [[resolutions]] of one chain complex by another chain complex with some better properties. The theory of [[homotopical categories]] such as [[model categories]] or [[fibration categories]]/[[cofibration categories]] is used to handle homological resolutions in this generality, see at _[[model structure on chain complexes]]_ for more on this.

Viewed from this general perspective of [[homotopical categories]], homological resolutions are a special case of the general concept of [[resolutions]] in [[homotopy theory]]. (In fact [[homological algebra]] may be understood as a fragment of [[stable homotopy theory]], see at _[[Dold-Kan correspondence]]_ and _[[stable Dold-Kan correspondence]]_ for more on this.)

## Examples

The ur examples of homological resolutions are the [[Koszul complexes]] or more generally [[Koszul-Tate resolutions]] of an $A$-[[module]] by [[free modules]] ove or of an $A$-[[associative algebra|algebra]] by free $A$-algebras.

For example, consider $A$ is a commutative [[augmented algebra]] over a [[field]] $k$ and $I$ an ideal in $A$. Resolve the [[quotient]] $A/I$ by a free $A$-algebra $R$ with a [[derivation]] [[differential]] so that $H(R) = A/I$ is concentrated in degree $0$ and all other homology vanishes. Iff $I$ is a [[regular ideal]], the [[Koszul complex]] will do; if $I$ is not regular, continue the process forming the [[Koszul-Tate resolution]], the algebraic analog of a [[Moore-Postnikov system]], which was indeed Tate's inspiration.

If the original object is itself [[graded object|graded]] or differential graded, the resolution will be bigraded by resolution degree and _internal degree_.

By **homological resolution of a quotient**, one means a _weak quotient_ or _homotopy quotient_ in some $\infty$-categorical [[homotopy theory]] context which is equivalent to a category of (co)chain complexes. This means: a homological resolution of a quotient is a (co)chain complex $C^\bullet$ of abelian groups whose _(co)homology_ $H^\bullet(C^\bullet)$ in some degree, usually in degree 0, is the desired quotient, $H^0(C) = desired quotient$.
Depending on the situation one may want to demand that the (co)homology in all other degrees vanish, in which case $C^\bullet$ would be [[homotopy theory|weakly equivalent]] to the desired quotient.

## Related concepts

* [[simplicial resolution]]

## References

Discussion for an audience of physicists in the context of [[BV-BRST formalism]] is in

* [[Marc Henneaux]], [[Claudio Teitelboim]], section 8.3 of _[[Quantization of Gauge Systems]]_, Princeton University Press, 1992

[[!redirects homological resolutions]]

[[!redirects Homological resolution]]
[[!redirects Homological resolutions]]


Homological resolution > history

< [[Homological resolution]]
[[!redirects Homological resolution -- history]]


homologically smooth dga

A [[differential graded algebra]] $A$ is __homologically smooth__ if as a dg-bimodule $_A A_A$ over itself it has a bounded resolution by finitely generated projective dg-bimodules.

The definition is due [[Kontsevich]].

[[!redirects homologically smooth differential graded algebra]]


homology

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#Contents#
* table of contents
{:toc}

## Of chain complexes

### Idea

Under the [[Dold-Kan correspondence]], [[∞-groupoids]] with strict abelian group structure (modeled by [[Kan complexes]] that are [[simplicial object|simplicial]] abelian groups) are identified with non-negatively graded [[chain complexes]] of abelian groups

$$
  N_\bullet : SimpAb \stackrel{\simeq}{\to} Ch_+
  \,.
$$

The **homology groups** of a [[chain complex]] of abelian groups are the image under this identification of the [[homotopy groups]] of the corresponding [[∞-groupoids]]. More details on this are at [[chain homology and cohomology]].

So at least for the case of chain complexes of abelian groups we have the slogan

+-- {: .standout}

**homology** = [[homotopy]] under [[Dold-Kan correspondence]]

=--

Of course historically the development of concepts was precisely the opposite: chain homology is an old fundamental concept in [[homological algebra]] that is simpler to deal with than [[simplicial homotopy groups]]. The computational simplification for chain complexes is what makes the [[Dold-Kan correspondence]] useful after all.

Conceptually, however, it can be useful to understand homology as a special kind of [[homotopy]]. This is maybe most vivid in the [[duality|dual]] picture: [[cohomology]] derives its name from that fact that [[chain homology and cohomology]] are dual concepts. But later generalizations of [[cohomology]] to [[generalized (Eilenberg-Steenrod) cohomology]] and further to [[nonabelian cohomology]] showed that the restricted notion of homology is an insufficient dual model for cohomology: what cohomology is really dual to is the more general concept of [[homotopy]].  More on this is at [[cohomotopy]] and [[Eckmann-Hilton duality]].

### Definition

The category of abelian groups is in particular an [[abelian category]].
We can define [[chain complexes]] and their homology in any [[abelian category]] $C$.

Let $C$ be an [[abelian category]] and let
$$
  V_\bullet = ( \cdots  \to V_{n+1} \stackrel{\delta_n}{\to}
  V_n \stackrel{\delta_{n-1}}{\to}
  V_{n-1} \to \cdots
  )
$$
be a [[chain complex]] in $C$. For each integer $n \in \mathbb{N}$ this induces the following diagram of [[kernel]]s, [[cokernel]]s and [[image]]s

$$
  \array{
    && im \delta_n &&\to&& ker \delta_{n-1}
    \\
    & \nearrow && \searrow && \swarrow
    \\
    V_{n+1}
    &&\stackrel{\delta_n}{\to}&&
    V_n
    &&\stackrel{\delta_{n-1}}{\to}&&
    V_{n-1}
    \\
    & &&
    \swarrow && \searrow && \nearrow
    \\
    && coker \delta_n &&\stackrel{}{\to}&&
    im \delta_{n-1}
  }
$$

the **homology** $H_n(V)$ of $V$ in degree $n$ is the object

$$
  \begin{aligned}
    im(ker \delta_{n-1} \to  V_n \to coker \delta_{n})
    & \simeq
    coker(im \delta_n \to ker \delta_{n-1})
    \\
    & \simeq
    coker(V_{n+1} \to ker \delta_{n-1})
    \\
    & \simeq
    ker(coker \delta_n \to im \delta_{n-1})
    \\
    & \simeq
    ker(coker \delta_n \to V_{n-1})
  \end{aligned}
$$

* If $H_n(V) \simeq 0$ then one says that the complex $V$ is [[exact sequence|exact]] in degree $n$.

### Examples

In the special case that $C$ is the category of abelian groups, or of vector spaces, this definition reduces to the more familiar simpler statement:

the $n$-th homology group of the [[chain complex]] $V_\bullet$ is the quotient group

$$
  H_n(V) = ker(\partial_n) / im(\partial_{n+1})
  \,.
$$

## Generalized homology

By the [[Brown representability theorem]] every [[spectrum]] $A$ induces a [[generalized (Eilenberg-Steenrod) cohomology]] theory, and dually a **generalized homology theory**.

For $X$ a [[topological space]] and $A$ a [[spectrum]], the generalized homology of spectrum of $X$ with coefficients in $A$ is

$$
  X \wedge A := \Sigma^\infty(X)\wedge A
  \,,
$$

where on the right we have the [[smash product of spectra]] with the [[suspension spectrum]] of $X$ and on the left we abbreviate this to the [[(∞,1)-colimit|(∞,1)-tensoring]] of [[Spec]] over [[Top]].

The corresponding [[homology group]]s are the [[homotopy group]]s of this spectrum:

$$
  E_n(X,A) := \pi_n(X \wedge A) := [\Sigma^n \mathbb{S}, X \wedge A ]
  \,.
$$

where $\mathbb{S}$ is the [[sphere spectrum]]. For more see [[generalized homology]].

## Examples

* [[chain homology]]

  * [[relative homology]]

* [[simplicial homology]]

* [[singular homology]]

  * [[cellular homology]]

* [[generalized homology]]

* [[factorization homology]]

* [[Kan-Thurston Theorem]]

## Related concepts

* [[homology of loop spaces]]

* [[effective homology]]

* [[persistent homology]]

The relation between homology, cohomology and homotopy:

[[!include homotopy-homology-cohomology]]

The ingredients of homology and cohomology:

[[!include chains and cochains - table]]

## References

* {#Hilton69} [[Peter Hilton]] (ed.) _Category Theory, Homology Theory and Their Applications_,

  vol 1: Lecture Notes in Mathematics **86**, Springer (1969) [[doi:10.1007/BFb0079380](https://doi.org/10.1007/BFb0079380)]

  vol 2: Lecture Notes in Mathematics **92**, Springer (1969) [[doi:10.1007/BFb0080761](https://doi.org/10.1007/BFb0080761)]

  vol 3: Lecture Notes in Mathematics **99**, Springer (1969) [[doi:10.1007/BFb0081959](https://doi.org/10.1007/BFb0081959)]

[[!redirects homologies]]
[[!redirects homology group]]
[[!redirects homology groups]]


homology localization

#Contents#
* table of contents
{:toc}

## Idea

The notion of _homology localization_ is the formal dual of that of _[[cohomology localization]]_. See at _[[Bousfield localization of spectra]]_.

## Related concepts

* [[quasi-isomorphism]]

[[!redirects homology localizations]]

[[!redirects homology equivalence]]
[[!redirects homology equivalences]]


homology of loop spaces

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#Contents#
* table of contents
{:toc}

## Idea

The [[homology]] of ([[iterated based loop space|iterated]]) [[based loop spaces]] ([[ordinary homology]] or [[generalized homology]]) carries special [[mathematical structure|structure]], reflecting the [[∞-group]]-[[mathematical structure|structure]] of [[based loop spaces]].

In particular, under mild technical conditions (see [Milnor-Moore 65, p. 262 ](#MilnorMoore65), [Halperin 92](#Halperin92)) the [[Pontrjagin ring]]-structure induced by [[concatenation]] of loops enhances the homology [[coalgebra]] induced by the [[diagonal maps]] to that of a [[Hopf algebra]].

## Examples

* [[weight systems are cohomology of loop space of configuration space]]

## Related concepts

* [[Pontrjagin ring]]

## References

The "[[Pontryagin-multiplication]]" on loop/path spaces is due to

* [[Raoul Bott]], [[Hans Samelson]], *On the Pontryagin product in spaces of paths*, Commentarii Mathematici Helvetici **27** (1953) 320–337 [[doi:10.1007/BF02564566](https://doi.org/10.1007/BF02564566)]

named in honor of the analogous construction over [[compact Lie groups]] in:

* [[Lev Pontrjagin]], _Homologies in compact Lie groups_, Rec. Math. [Mat. Sbornik] N.S., 1939 Volume 6(48), Number 3, Pages 389–422 ([mathnet:5835](http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5835&option_lang=eng))

Further discussion of homology of ([[iterated loop space|iterated]]) [[loop spaces]]:

* [[Eldon Dyer]], [[Richard Lashof]], _Homology of Iterated Loop Spaces_,  American Journal of Mathematics Vol. 84, No. 1 (Jan., 1962), pp. 35-88 ([jstor:2372804 ](https://www.jstor.org/stable/2372804))

* {#MilnorMoore65} [[John Milnor]], [[John Moore]], p. 262 & Appendix of: _On the structure of Hopf algebras_, Annals of Math. __81__ (1965), 211-264 ([doi:10.2307/1970615](https://doi.org/10.2307/1970615), [pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf))

* [[Samuel Eilenberg]], [[John Moore]], _Homology and fibrations_, Comment. Math. Helv., 40 (1966), pp. 199-236 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/eilemoor.pdf), [doi:10.1007/BF02564371](https://doi.org/10.1007/BF02564371))

* [[William Browder]], _Homology Rings of Groups_, American Journal of Mathematics, Vol. 90, No. 1 (Jan., 1968) ([jstor:2373440](https://www.jstor.org/stable/2373440))

* {#Halperin92} [[Stephen Halperin]], *Universal enveloping algebras and loop space homology*, Journal of Pure and Applied Algebra **83** 3 (1992) 237-282 \[doi:10.1016/0022-4049(92)90046-I\]

* [[Jonathan A. Scott]], _Algebraic Structure in the Loop Space Homology Bockstein Spectral Sequence_,  Transactions of the American Mathematical Society **354** 8 (2002) 3075-3084 [[jstor:3073034](https://www.jstor.org/stable/3073034)]

* [[Victor Buchstaber]], [[Jelena Grbić]], *Hopf algebras and homology of loop suspension spaces* ([pdf](http://www.personal.soton.ac.uk/jg1u11/Papers/Buchstaber.Grbic.pdf), [[BuchstaberGrbicHopfAlgebrasAndHomology.pdf:file]]) in: V. Buchstaber et al. (eds.), *Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov’s Seminar 2012–2014*, American Mathematical Society Translations - Series 2
Advances in the Mathematical Sciences, 2014 ([ISBN:978-1-4704-1871-7](https://bookstore.ams.org/trans2-234))

See also:

* Wikipedia, _[Pontryagin product](https://en.wikipedia.org/wiki/Pontryagin_product)_

[[!redirects homology of a loop space]]

[[!redirects homology of iterated loop spaces]]

[[!redirects loop space homology]]
[[!redirects loop space homologies]]

[[!redirects homology of H-spaces]]


homology of MG

[[!redirects homology of MU]]

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#### Cohomology
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[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The ([[generalized homology|generalised]]) [[homology]] of [[Thom spectra]] $M G$, such as [[MO]] and [[MU]].

In the presence of a [[Thom isomorphism]] (e.g. for [[complex oriented cohomology]] of [[MU]]) this identifies with the homology of the [[classifying spaces]] $B G$, such as $B O$ and $B U$.

Feeding the homology of $M G$ into an [[Adams spectral sequence]] gives a way to compute its [[homotopy groups]].

This is for instance a key ingredient in [[Quillen's theorem on MU]] giving the [[homotopy groups of a spectrum|homotopy groups]] of $MU$.

## Statement

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

For $E$ a [[complex oriented cohomology theory]], there is an [[isomorphism]] of [[graded rings]]

$$
  E_\bullet(M U) \simeq (\pi_\bullet E)[b_1, b_2, \cdots]
  \,,
$$

where the $\{b_i \in E_\bullet(M U(1))\}$ are a [[dual basis]] to the basis $\{t^{i+1} \in E^\bullet(M U) \simeq t (\pi_\bullet E)[ [ t ] ]\}$ that is induced by the [[complex oriented cohomology theory|complex orientation]].

=--

([Lurie 10, lecture 7, prop. 2](#Lurie10))

## Related concepts

* [[Boardman homomorphism]]

* [[homology of MO]]

## References

* [[Frank Adams]], part II.6 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Kochmann96} [[Stanley Kochmann]], section 2.6 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Lurie10} [[Jacob Lurie]], lecture 7 of _[[Chromatic Homotopy Theory]]_, 2010, ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf))


homology of MO

## Idea

[[ordinary homology]] and [[generalized homology]] of [[MO]]

## related concepts

* [[homology of MU]]

## References

Lecture notes include

* {#Malkiewich11} [[Cary Malkiewich]], section 4-7 of _Unoriented cobordism and $M O$_, 2011 ([pdf](http://math.uiuc.edu/~cmalkiew/cobordism.pdf))


homology sphere

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#### Homotopy theory
+--{: .hide}
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _homology sphere_ is a [[topological space]] which need not be [[homeomorphism|homeomorphic]] to an [[n-sphere]], but which has the same [[ordinary homology]] as an $n$-sphere.

## Related concepts

* [[homotopy sphere]]

## Redirects

* Wikipedia, _[Homology sphere](https://en.wikipedia.org/wiki/Homology_sphere)_

[[!redirects homology spheres]]

[[!redirects homology n-sphere]]
[[!redirects homology n-spheres]]


homomorphism

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=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

In a restrictive sense, a homomorphism is a [[function]] between (the [[underlying sets]]) of two [[algebras]] that preserves the [[algebraic structure]].

More generally, a homomorphism is a function between [[structured sets]] that preserves whatever [[structure]] there is around. Even more generally, 'homomorphism' is just a synonym for '[[morphism]]' in any [[category]], the structured sets being generalised to arbitrary [[object]]s.

*Note: The word "homomorphism" has also traditionally been used for what we call a (weak) [[2-functor]] between [[bicategories]]*.

## Definitions

### Traditional (magmas, semigroups, groups, rngs)

Traditionally, a __homomorphism__ between two [[magmas]] $A$ and $B$ is a [[function]]

$$
  \phi\colon A \to B
$$

of the underlying [[set]]s that respects the binary operation in that for all $a_1, a_2$ in $A$ we have

$$
  \phi(a_1 \cdot a_2) = \phi(a_1) \cdot \phi(a_2)
  \,.
$$

This definition gives us the correct notion of __magma homomorphism__, __[[semigroup]] homomorphism__, and __[[group]] homomorphism__, but it is actually a bit of a coincidence that it works for groups.  It does *not* give the correct definition of [[monoid]] homomorphism, since it doesn\'t properly treat the [[identity elements]].  (However, the correct notion of monoid [[isomorphism]] can still be constructed from this inadequate definition of homomorphism.)

A __rng homomorphism__ is a function between [[rngs]] that is a homomorphism for both the additive group and the multiplicative semigroup.  (For [[rings with identity]], this is again inadequate.)

### Identity-preserving (monoids, rings)

A __homomorphism__ between two [[monoids]] $A$ and $B$ is a semigroup homomorphism

$$
  \phi\colon A \to B
$$

of the underlying [[semigroup]]s that preserves [[identity elements]] in that we have

$$
  \phi(1_A) = 1_B
  \,.
$$

This definition of __monoid homomorphism__ is a special case (via [[delooping]]) of the definition of [[functor]].

It is a theorem that a semigroup homomorphism between [[groups]] must be a monoid homomorphism (and additionally must preserve [[inverse elements]], which is also necessary to be the *correct* definition of __group homomorphism__.)

A __[[ring]] homomorphism__ is a function between rings that is a homomorphism for both the additive group and the multiplicative monoid.  Traditional [[ring theory]] sometimes actually uses rng homomorphisms even when the rngs in question are assumed to have identity elements, so be careful when reading old books.

A [[rig]] **homomorphism** is a function between rigs that is a homomorphism for both the additive monoid and the multiplicative monoid.

A [[semi-ring]] **homomorphism** is a function between semi-rings that is a homomorphism for both the additive semigroup and the multiplicative monoid.

### General

More generally, a **homomorphism** between sets equipped with any algebraic structure is a map preserving this structure.  This can be made precise using [[Lawvere theories]], [[monads]], etc.

Generalizing further, we may simply treat 'homomorphism' as a synonym for '[[morphism]]' in any [[category]], although there is a strong tendency to use 'homomorphism' in the case of 'algebraic' categories: for example, nobody seems to speak of a homomorphism between [[topological space]]s ([[continuous map]]s), or between [[manifold]]s.  Here we put 'algebraic' in scare quotes to indicate the *field* of [[algebra]]; even morphisms in [[algebraic categories]] from other fields (such as the category of [[compacta]]) are not usually called homomorphisms.

## Examples

* an [[action]] homomorphism is an [[equivariant function]]

* a [[set]] homomorphism is an [[extensional function]]

* an [[inequality space]] homomorphism is a [[strongly extensional function]]

* [[homomorphism of L-infinity algebras|homomorphism of $L_\infty$-algebras]]

## Related concepts

* homomorphisms are to [[functions]] as [[logical relations]] are to [[relations]]

* [[crossed homomorphism]]

[[!redirects homomorphism]]
[[!redirects homomorphisms]]

[[!redirects magma homomorphism]]
[[!redirects magma homomorphisms]]
[[!redirects semigroup homomorphism]]
[[!redirects semigroup homomorphisms]]
[[!redirects monoid homomorphism]]
[[!redirects monoid homomorphisms]]

[[!redirects group homomorphism]]
[[!redirects group homomorphisms]]

[[!redirects homomorphism of groups]]
[[!redirects homomorphisms of groups]]

[[!redirects abelian group homomorphism]]
[[!redirects abelian group homomorphisms]]
[[!redirects rng homomorphism]]
[[!redirects rng homomorphisms]]
[[!redirects ring homomorphism]]
[[!redirects ring homomorphisms]]
[[!redirects semi-ring homomorphism]]
[[!redirects semi-ring homomorphisms]]
[[!redirects rig homomorphism]]
[[!redirects rig homomorphisms]]
[[!redirects algebra homomorphism]]
[[!redirects algebra homomorphisms]]

[[!redirects field homomorphism]]
[[!redirects field homomorphisms]]


homomorphism of L-∞ algebras

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###Context###
#### $\infty$-Lie theory
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#### Homotopy theory
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[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are two notions of [[homomorphism]] of [[L-∞ algebras|$L_\infty$-algebras]], one a special case of the other, depending on the [[homotopy theory|homotopy theoretic]] perspective:

If an [[L-∞ algebra|$L_\infty$]]-algebra $\mathfrak{g} = (V_\bullet, [\cdots])$ is regarded as a [[chain complex]]  $V_\bullet$ equipped with higher-[[arity|ary]]-bracket operations $[\cdots]$, then a homomorphism $\mathfrak{g} \xrightarrow{\;\;} \mathfrak{g}'$ is a [[chain map]] $V_\bullet \xrightarrow{\;\;} V'_\bullet$ which is compatible with the bracket operations.

However, even when so regarded, [[L-∞ algebras]] are [[objects]] of a [[homotopical category]], in fact of the [[category of fibrant objects]] of a [[model structure for L-∞ algebras|model structure for $L_\infty$-algebras]], so that the homotopy-correct morphisms out of $\mathfrak{g}$ are those out of a [[cofibrant resolution]] $\varnothing \xhookrightarrow{\;} \widehat {\mathfrak{g}} \xrightarrow{ \in \mathrm{W} } \mathfrak{g}$, hence are [[zig-zags]] of naive morphisms, as above, of the form $\mathfrak{g} \xleftarrow{\in \mathrm{W}} \widehat {\mathfrak{g}} \xrightarrow{\;\;\;} \mathfrak{g}'$.

But such poperly resolved morphisms are equivalently just the evident homomorphism of the associated [[Chevalley-Eilenberg algebra|Chevalley-Eilenberg]] [[dg-coalgebras]] $CE_\bullet(-)$ ([here](L-infinity-algebra#ReformulationInSemifreeDgCoalgebra)),
[[formal duality|dually]] of the [[Chevalley-Eilenberg algebra|CE]]-[[dg-algebras]] $CE^\bullet(-)$ ([here](L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra)):

$$
  CE_\bullet(\mathfrak{g}) \xrightarrow{\;\;\;\;} CE_\bullet(\mathfrak{g}')
  \,,
  \;\;\;\;\;\;\;\;
  CE^\bullet(\mathfrak{g}) \xleftarrow{\;\;\;\;} CE^\bullet(\mathfrak{g}')
  \,.
$$

These homotopy-correct homomorphisms of $L_\infty$-algebras are known under a variety of different names, including:

* "strong homotopy maps", abbreviated: "sh maps"

  (Stasheff following [Sugawara 60, Sec. 2](#Sugawara60), see also [Merkulov 02, 2.11](#Merkulov02)),

* "weak maps"

  ([Lada & Markl 1994, Rem. 5.3](#LadaMarkl94)),

* "$L_\infty$-morphisms"

  ([Kontsevich 1997, p. 12](#Kontsevich97))

## References

* {#LadaMarkl94} [[Tom Lada]], [[Martin Markl]], Rem. 5.3 of: _Strongly homotopy Lie algebras_, Communications in Algebra Volume 23, Issue 6, (1995) ([arXiv:hep-th/9406095](http://arxiv.org/abs/hep-th/9406095))

* {#Kontsevich97} [[Maxim Kontsevich]], *Deformation quantization of Poisson manifolds, I*,  Lett. Math. Phys. **66** (2003) 157-216 ([arXiv:q-alg/9709040](https://arxiv.org/abs/q-alg/9709040), [doi:10.1023/B:MATH.0000027508.00421.bf](https://doi.org/10.1023/B:MATH.0000027508.00421.bf))

* {#Merkulov02} [[Sergei A. Merkulov]], *Operads, deformation theory and $F$-manifolds* ([arXiv:math/0210478](https://arxiv.org/abs/math/0210478))

The terminology "strong homotopy" for the homotopy-coherent morphisms was originally borrowed from:

* {#Sugawara60} [[Masahiro Sugawara]], *On the homotopy-commutativity of groups and loop spaces*, Mem. College Sci. Univ. Kyoto Ser. A Math. 33(2): 257-269 (1960) ([DOI:10.1215/kjm/1250775911](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-33/issue-2/On-the-homotopy-commutativity-of-groups-and-loop-spaces/10.1215/kjm/1250775911.full))

[[!redirects homomorphisms of L-∞ algebras]]

[[!redirects homomorphism of L-infinity algebras]]
[[!redirects homomorphisms of L-infinity algebras]]

[[!redirects morphism of L-∞ algebras]]
[[!redirects morphisms of L-∞ algebras]]

[[!redirects morphism of L-infinity algebras]]
[[!redirects morphisms of L-infinity algebras]]

[[!redirects sh map]]
[[!redirects sh maps]]
[[!redirects sh-map]]
[[!redirects sh-maps]]

[[!redirects strong homotopy map]]
[[!redirects strong homotopy maps]]

[[!redirects weak map]]
[[!redirects weak maps]]


homothety

#Contents#
* table of contents
{:toc}

## Definition

### Traditional

In the original sense, a _homothety_ is a [[function]] from a [[Euclidean space]] to itself, which is a dilation centered at a given point of the Euclidean space. Hence a homothety of a [[Cartesian space]] $\mathbb{R}^n$ is a function $\mathbb{R}^n \to \mathbb{R}^n$ given by multiplication with some $\kappa \in \mathbb{R}-\{0\}$.

Such dilations are [[linear functions]] $\kappa \in GL(\mathbb{R}^n)$ which commute, in particular, with all [[orthogonal group|orthogonal transformations]] on $\mathbb{R}^n$. In view of this, in the context of [[Cartan geometry]] one says more generally that given a [[vector space]] $V$ and a [[group]] [[homomorphism]]
$\phi \colon G \longrightarrow  GL(V)$ to the [[general linear group]] of $V$, then a homothety of this data is an element $\kappa \in GL(V)$ such that for all $g \in G$ one has

$$
  \kappa \circ \phi(g) = \phi(g) \circ \kappa
  \,.
$$

Given then two [[G-structures]] on a [[smooth manifold]] modeled on $V$, then a homothety between these $G$-structures is a homothety of the local models which transforms the two $G$-structures into each other.

### General abstract

This has a succinct diagrammatic formulation in the formulation of $G$-structures on $V$-manifolds in [[differential cohesion]] (see at _[differential cohesion -- G-Structures](differential+cohesion#structures)_). Write

$$
  G \mathbf{Struc} \colon \mathbf{B}G\longrightarrow \mathbf{B}GL(V)
$$

for the [[delooping]] of the group homomorphism defining the kind of $G$-structure. Then a homothety is simply a [[homotopy]] from $G\mathbf{Struc}$ to itself

$$
  \array{
    & & \mathbf{B}G
    \\
    & \swarrow && \searrow^{\mathrlap{G\mathbf{Struc}}}
    \\
    & {}_{\mathllap{G\mathbf{Struc}}}\searrow &\swArrow_{\simeq}& \swarrow
    \\
    && \mathbf{B}GL(V)
  }
  \,.
$$

Accordingly, for $\mathbf{g}_1, \mathbf{g}_2 \colon \tau_X\longrightarrow G \mathbf{Struc}$ two $G$-structures on a $V$-manifold

$$
  \array{
     X && \longrightarrow && \mathbf{B}G
     \\
     & {}_{\mathllap{\tau_X}}\searrow &\swArrow_{\mathrlap{\mathbf{g}_i}}& \swarrow_{\mathrlap{G}\mathbf{Struc}}
     \\
     && \mathbf{B}GL(V)
  }
$$

then a homothety from $\mathbf{g}_1$ to $\mathbf{g}_2$ is a homotopy of homotopies of the form

$$
  \array{
     X && \longrightarrow && \mathbf{B}G
     \\
     & {}_{\mathllap{\tau_X}}
       \searrow
     &\swArrow_{\mathrlap{\mathbf{g}_1}}&
     \swarrow_{\mathrlap{G\mathbf{Struc}}}
     &\swArrow_{\simeq}&
      \searrow^{\mathrlap{G\mathbf{Struc}}}
     \\
     && \mathbf{B}GL(V) && =& \mathbf{B}GL(V)
  }
  \;\;\;\;\;\;
   \stackrel{\simeq}{\Rightarrow}
  \;\;\;\;\;\;
  \array{
     X && \longrightarrow && \mathbf{B}G
     \\
     & {}_{\mathllap{\tau_X}}\searrow &\swArrow_{\mathrlap{\mathbf{g}_2}}& \swarrow_{\mathrlap{G\mathbf{Struc}}}
     \\
     && \mathbf{B}GL(V)
  }
$$

## References

* Encyclopedia of Mathematics, _[Homothety](http://www.encyclopediaofmath.org/index.php/Homothety)_

[[!redirects homotheties]]


Homotopical Algebra

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#### Model category theory
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#### Homotopy theory
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[[!include homotopy - contents]]
=--
=--
=--

This page is about the book

* [[Daniel Quillen]]

  __Homotopical Algebra__

  Lecture Notes in Mathematics 43,

  Springer 1967

  [doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438)

  TeX-typeset manuscript: [PDF](https://aareyanmanzoor.github.io/assets/books/homotopical-algebra.pdf).

introducing the theory of [[model categories]] as a tool in [[homotopy theory]].

#Contents#
* table of contents
{:toc}

## Chapter I. Axiomatic homotopy theory

### 1. The axioms

* [[fibration]]

* [[cofibration]]

* [[weak equivalence]]

* [[lifting property]]

### 2. The loop and suspension functors

* [[loop space]]

* [[suspension]]

### 3. Fibration and cofibration sequences

* [[homotopy fiber]]

* [[fiber sequence]]

* [[cofiber sequence]]

### 4. Equivalence of homotopy theories

* [[Quillen equivalence]]

### 5. Closed model categories

* [[model category]]

## Chapter II. Examples of simplicial homotopy theories

### 1. Simplicial categories

* [[simplicially enriched category]]

### 2. Closed simplicial model categories

* [[simplicial model category]]

### 3. Topological spaces, sumplicial sets, and simplicial groups

* [[classical model structure on topological spaces]]

* [[classical model structure on simplicial sets|on simplicial sets]]

* [[model structure on simplicial groups|on simplicial groups]]

### 4. $s A$ as a model category

* [[simplicial object]]

* [[model structure on simplicial abelian groups]]

* [[model structure on chain complexes]]

* [[Dold-Kan correspondence]]

### 5. Homology and cohomology

* [[homological algebra]]

### 6. Modules over a simplicial ring

* [[simplicial ring]]

category: reference


homotopical algebra

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#Contents#
* table of contents
{:toc}

## Idea

See [[higher algebra]].

## History

At first, [[homotopy theory]] was restricted to [[topological spaces]], while [[homological algebra]] worked in a variety of (mainly algebraic) examples. Whitehead proposed around 1949 the subject of [[algebraic homotopy]] theory, to deal with classical homotopy theory of spaces via algebraic models. This idea did not extend to homotopy methods in general setups of course, but it had concrete modelling and calculations for topological spaces in mind. In the 1960s Grothendieck introduced fundamental groups and cohomology in the setup of [[topos|topoi]], which were a wider and more modern setup. In retrospective, he considered exactness axioms which he introduced in [[Tohoku]] in a context of homological algebra to be conceptually a kind of reasoning bringing understanding to general spaces, such as topoi. Quillen in the late 1960s introduced an axiomatics (the structure of a [[model category]]) on a category to be able to do a great deal of homotopy theory; this is the first instance of much abstract machinery which has as a common name 'homotopical algebra'; and [[rational homotopy theory]] was its first success. A large part (but maybe not all) of homological algebra can be subsumed as the [[derived functor]]s that make sense in model categories, and at least the categories of chain complexes can be treated via Quillen model structures.

Nowadays, there are lots of new formalisms ([[derivators]], [[homotopical categories]], ...) of homotopical algebra and very many examples ([[simplicial sets]], [[cubical sets]], [[crossed complexes]], strict $\infty$-categories, $\mathbb{A}^1$-homotopy theory, homotopy theory for operator algebras, dg-Lie algebras, dg-operads, ...). A strong impetus has been given by Grothendieck's program in [[Pursuing Stacks]] and the work of the Bangor school, that is, [[Ronnie Brown]] and [[Tim Porter]] with coauthors, of [[André Joyal]], of [[Carlos Simpson]] and his school, R. Jardine, [[Charles Rezk]], [[Georges Maltsiniotis]] and [[Denis-Charles Cisinski]], [[Jacob Lurie]], ...) which followed them; in that program [[homotopy types]] are related to higher [[stacks]]; thus the natural setup is realized to be that of $\infty$-categories. For simplicity, $(\infty,1)$-categories (usually modeled by [[quasicategories]], [[Segal categories]], simplicially enriched categories etc.) were understood first and by now systematically, cf. [[Higher Topos Theory]]. This modern language is, unlike more axiomatic presentations on $1$-categories with structure like Quillen model categories, more rarely referred to as homotopical algebra.

The $n$Lab entry [[homotopy theory]] takes homotopy in a rather general sense including not only classical homotopy, but also algebraic homotopy, homotopical algebra in the narrow sense (model categories and variants) and $(\infty,1)$-categories.

## Related entries

* [[higher structures]]

* [[homotopical algebraic quantum field theory]]

[[!redirects homotopical algebras]]


homotopical algebraic geometry

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#Contents#
* table of contents
{:toc}

## Idea

_Homotopical algebraic geometry_ is a [[homotopy theory|homotopical]] generalization of [[algebraic geometry]], where the [[affine schemes]] are not necessarily [[commutative algebras]] in the usual sense, but rather [[commutative algebra in an (∞,1)-category|commutative algebra objects]] in an arbitrary [[symmetric monoidal (infinity,1)-category]].
In other words, it is the specialization of [[higher geometry]] where the local models are [[commutative algebra in an (∞,1)-category|commutative algebras]] in a [[symmetric monoidal (infinity,1)-category]].

Interesting examples can be obtained by starting with

* the ordinary category of [[sets]] (yielding ordinary [[algebraic geometry]])
* the [[(infinity,1)-category]] of [[simplicial commutative rings]] (yielding [[derived algebraic geometry]]),
* the [[(infinity,1)-category of spectra]] (yielding [[spectral algebraic geometry]]).

As in ordinary [[algebraic geometry]], there are two equivalent definitions of [[scheme]]: the so-called [[functor of points]] approach, or the approach via [[locally ringed toposes]]; we consider the former approach below.

## Definition

Given a [[symmetric monoidal (infinity,1)-category]] $C$, let $CAlg(C)$ denote the [[(infinity,1)-category]] of [[commutative algebra in an (∞,1)-category|commutative algebra objects]] in $C$.

+-- {: .num_defn}
###### Definition
A **$C$-prestack** is an [[infinity-prestack]] on the [[opposite (infinity,1)-category]] of $CAlg(C)$.
In other words it is a functor $CAlg(C) \to Spc$ to the [[(infinity,1)-category of spaces]].
=--

Let $\tau$ be a subcanonical [[Grothendieck topology]] on $CAlg(C)^{op}$.

+-- {: .num_defn}
###### Definition
A **$(C,\tau)$-stack** is an [[infinity-stack]] on the [[opposite (infinity,1)-category]] of $CAlg(C)$, with respect to the topology $\tau$.
Let $Stk(C, \tau)$ denote the [[(infinity,1)-category]] of $(C,\tau)$-stacks.
The [[Yoneda embedding]] is denoted
  $$ Spec : CAlg(C) \hookrightarrow Stk(C,\tau) $$
and a stack is called **affine** if it is in the [[essential image]].
=--

Let $P$ be a class of morphisms in $Stk(C, \tau)$ which is stable by [[base change]].
The triple $(C, \tau, P)$ is a called a **homotopical algebraic geometry context** (*HAG context*).

+-- {: .num_defn}
###### Definition
A **$(C,\tau, P)$-scheme** is a $(C,\tau)$-stack $X$ such that there exists a family of morphisms $(X_\alpha \to X)_\alpha$, each in $P$, such that $X_\alpha$ are affine and the induced morphism
  $$ \coprod_\alpha X_\alpha \longrightarrow X $$
is an [[epimorphism]].
Let $Sch(C, \tau, P)$ denote the [[(infinity,1)-category]] of $(C,\tau,P)$-schemes.
=--

## Related concepts

* [[derived algebraic geometry]]
* [[spectral algebraic geometry]]

## References

* [[Jacob Lurie]], [[Derived Algebraic Geometry]], thesis, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG.pdf).

* [[Jacob Lurie]], [[Structured Spaces]].

An alternate exposition of the theory, using the presentations by [[model categories]] (hence the various [[model structures on simplicial presheaves]]), is given in

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry I: topos theory_, 2002, [arXiv:math/0207028](http://arxiv.org/abs/math/0207028).

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry II: geometric stacks and applications_, 2004, [arXiv:math/0404373](http://arxiv.org/abs/math/0404373).

An exposition in the style of Toën-Vezzosi but in the language of [[(infinity,1)-categories]] can be found in

* [[Mauro Porta]], Tony Yue Yu, _Higher analytic stacks and GAGA theorems_, [arXiv:1412.5166](http://arxiv.org/abs/1412.5166).

For an extension of homotopical algebraic geometry from $\infty$-stacks valued in $Spc$ to those valued in augmentation categories, a special class of [[generalized Reedy categories]], see.

* Scott Balchin, _Augmented Homotopical Algebraic Geometry_, ([arXiv:1711.02640](https://arxiv.org/abs/1711.02640))


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#### Higher algebra
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[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[axioms]] of traditional [[algebraic quantum field theory]] or of [[locally covariant perturbative quantum field theory]] -- where one assigns a plain [[algebra]] (a [[C*-algebra]] for [[non-perturbative QFT]] and a  [[formal power series algebra]] for [[perturbative QFT]]) of [[observables]] to a [[spacetime]] region (a "[[local net of observables]]") -- fail when one considers [[gauge theory]] on [[spacetimes]] more general than [[Minkowski spacetime]] ([[AQFT on curved spacetimes]]):

The nature of [[gauge fields]] implies that to re-construct their global configuration (including non-trivial topology as in [[instanton sectors]]) one needs to locally remember the [[groupoid]] of [[gauge transformations]] (see [Eggertsson 14](https://ncatlab.org/schreiber/show/bachelor+thesis+Eggertsson), [Schenkel 14](#Schenkel14), [Schreiber 14](https://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields) for exposition), and hence, after [[quantization]], some [[higher algebra]] of observables.

The program of _homotopical algebraic quantum field theory_ ([Benini-Schenkel 16](#BeniniSchenkel16), see [Schenkel 17](#Schenkel17) for survey) is to lift the axioms of [[AQFT on curved spacetimes]] to certain [[local nets]] of [[homotopical algebra|homotopical algebras]] such as to properly capture [[gauge theory]] with topologically non-trivial [[gauge field]] configurations.

The axioms as considered in [Schenkel 17](#Schenkel17) have some  similarity with that of [[factorization homology]], but also do take into account the [[causality]] condition for [[quantum fields]] on [[Lorentzian manifolds]].

## Related concepts

* [[higher structures]]

* [[locally covariant perturbative algebraic quantum field theory]]

## References
 {#References}

For review and exposition see

* {#Schenkel17} [[Alexander Schenkel]], _Towards Homotopical Algebraic Quantum Field Theory_, talk at _[Foundational and Structural Aspects of Gauge Theories](https://indico.mitp.uni-mainz.de/event/76/overview)_, Mainz Institute for Theoretical Physics, 29 May – 2 June 2017. ([pdf](http://aschenkel.eu/Mainz17.pdf))

* {#Schenkel14} [[Alexander Schenkel]], _On the problem of gauge theories in locally covariant QFT_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([[SchenkelTrento2014.pdf:file]]) (with further emphasis on this point in the companion talk [Schreiber 14](field+bundle#Schreiber14))

* {#Schenkel17b} [[Alexander Schenkel]], _From Fredenhagen's universal algebra to homotopy theory and operads_, talk at _Quantum Physics meets Mathematics_, Hamburg, December 2017 ([pdf slides](http://aschenkel.eu/Hamburg17.pdf))

* {#BeniniSchenkel19} [[Marco Benini]], [[Alexander Schenkel]], _Higher Structures in Algebraic Quantum Field Theory_,  Proceedings of LMS/EPSRC Symposium _[[Higher Structures in M-Theory 2018]]_, Fortschritte der Physik 2019  ([arXiv:1903.02878](https://arxiv.org/abs/1903.02878), [doi:10.1002/prop.201910015]( https://doi.org/10.1002/prop.201910015))

* [[Simen Bruinsma]], _Coloring Operads for Algebraic Field Theory_, in: Proceedings of _[[Higher Structures in M-Theory 2018]]_ Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 ([arXiv:1903.02863](https://arxiv.org/abs/1903.02863) [doi:10.1002/prop.201910004](https://doi.org/10.1002/prop.201910004))

Construction and axiomatization of gauge field AQFT via [[homotopy theory]] and [[homotopical algebra]] is being developed in

* {#BDS} [[Marco Benini]], [[Claudio Dappiaggi]], [[Alexander Schenkel]], _Quantized Abelian principal connections on Lorentzian manifolds_, Communications in Mathematical Physics 2013 ([arXiv:1303.2515](http://arxiv.org/abs/1303.2515))

* {#BeniniSchenkelSzabo15} [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Homotopy colimits and global observables in Abelian gauge theory_ ([arXiv:1503.08839](http://arxiv.org/abs/1503.08839))

* {#BeniniSchenkel16} [[Marco Benini]], [[Alexander Schenkel]], _Quantum field theories on categories fibered in groupoids_, Communications in Mathematical Physics November 2017, Volume 356, Issue 1, pp 19–64 ([arXiv:1610.06071](https://arxiv.org/abs/1610.06071))

The [[stack]] of [[Yang-Mills theory|Yang-Mills]] [[gauge fields]] is discussed in

* {#BeniniSchenkelSchreiber17} [[Marco Benini]], [[Alexander Schenkel]], [[Urs Schreiber]], _The stack of Yang-Mills fields on Lorentzian manifolds_, Commun. Math. Phys. 359, 765–820 (2018) ([arXiv:1704.01378](https://arxiv.org/abs/1704.01378), [doi:10.1007/s00220-018-3120-1](https://doi.org/10.1007/s00220-018-3120-1))

An [[operad]] for [[local nets of observables]] in [[AQFT]] is considered in

* {#BeniniSchenkelWoike17} [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], _Operads for algebraic quantum field theory_ ([arXiv:1709.08657](https://arxiv.org/abs/1709.08657))

and its [[model structure on algebras over an operad]] (with respect to the [[model structure on chain complexes]]) is discussed in

* {#BeniniSchenkelWoike18} [[Marco Benini]], [[Alexander Schenkel]], [[Lukas Woike]], _Homotopy theory of algebraic quantum field theories_, Lett. Math. Phys. 109, 1487-1532 (2019) ([arXiv:1805.08795](https://arxiv.org/abs/1805.08795), [doi:10.1007/s11005-018-01151-x](https://doi.org/10.1007/s11005-018-01151-x))

The [[Boardman-Vogt resolution]] of the [[operad]] for [[local nets of observables]] ([Benini-Schenkel-Woike 17](#BeniniSchenkelWoike17)), lifting it to homotopy AQFT, is considered in

* {#Yau18} [[Donald Yau]], _Homotopical Quantum Field Theory_, World Scientific 2019 ([arXiv:1802.08101](https://arxiv.org/abs/1802.08101), [doi:10.1142/11626](https://doi.org/10.1142/11626))

Assignment of [[BV-BRST complexes]] as a homotopy AQFT is discussed in

* {#BeniniBruinsmaSchenkel19} [[Marco Benini]], [[Simen Bruinsma]], [[Alexander Schenkel]], _Linear Yang-Mills theory as a homotopy AQFT_, Commun. Math. Phys. 378, 185–218 (2020) ([arXiv:1906.00999](https://arxiv.org/abs/1906.00999), [doi:10.1007/s00220-019-03640-z](https://doi.org/10.1007/s00220-019-03640-z))

Discussion of [[orbifold|orbifolding]] via [[categorification]]:

* [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], [[Lukas Woike]], _Categorification of algebraic quantum field theories_, Lett. Math. Phys. 2021 ([h)arXiv:2003.13713](https://arxiv.org/abs/2003.13713))

Application to [[semi-topological 4d Chern-Simons theory]]:

* [[Marco Benini]], [[Alexander Schenkel]], Benoit Vicedo, _Homotopical analysis of 4d Chern-Simons theory and integrable field theories_ ([arXiv:2008.01829](https://arxiv.org/abs/2008.01829))

More on relevant [[model category]]-structures:

* Victor Carmona, *Algebraic Quantum Field Theories: a homotopical view* ([arXiv:2107.14176](https://arxiv.org/abs/2107.14176))

See also:

* [[Simen Bruinsma]], [[Christopher J. Fewster]], [[Alexander Schenkel]], *Relative Cauchy evolution for linear homotopy AQFTs* ([arXiv:2108.10592](https://arxiv.org/abs/2108.10592))

On the [[time slice axiom]] in homotopy AQFT:

* [[Marco Benini]], [[Victor Carmona]], [[Alexander Schenkel]], *Strictification theorems for the homotopy time-slice axiom* [[arxiv:2208.04344](https://arxiv.org/abs/2208.04344)]

Discussion of the [[model structure on modules in a monoidal model category|model structure on modules]] for representations of higher [[local nets of observables]]:

* Angelos Anastopoulos, [[Marco Benini]], *Homotopy theory of net representations*, Rev. Math. Phys. (2023) [[arXiv:2201.06464](https://arxiv.org/abs/2201.06464), [doi:10.1142/S0129055X23500083](https://doi.org/10.1142/S0129055X23500083)]

Relation to [[factorization algebras]]:

* [[Marco Benini]], Giorgio Musante, [[Alexander Schenkel]], *Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory* [[arXiv:2212.02546](https://arxiv.org/abs/2212.02546)]

Review:

* [[Alexander Schenkel]], *Quantum field theories on Lorentzian manifolds*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#SchenkelMar2023), slides:[[Schenkel-at-QFTAndCobordism2023.pdf:file]], video:[YT](https://www.youtube.com/watch?v=YiQNAgcwZYQ)]

Application to the [[CS/WZW correspondence]]:

* [[Marco Benini]], [[Alastair Grant-Stuart]], [[Alexander Schenkel]], *The linear CS/WZW bulk/boundary system in AQFT*, Annales Henri Poincaré (2023) [[arXiv:2302.06990](https://arxiv.org/abs/2302.06990)]

[[!redirects homotopical algebraic quantum field theories]]

[[!redirects homotopical AQFT]]

[[!redirects homotopy AQFT]]
[[!redirects homotopy algebraic quantum field theory]]


homotopical category

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\section{Idea}

By a *homotopical category* authors tend to mean something like a [[relative category]]/[[category with weak equivalences]], possibly satisfying further axioms (notably [[two-out-of-six]], as in [Dwyer, Hirschhorn, Kan & Smith 2004](#DwyerHirschhornKanSmith04)), but in any case a [[1-category]] equipped with a [[class]] of morphisms to be called *[[weak equivalences]]* and satisfying some [[extra properties]].

The terminology is that of a [[concept with an attitude]]: One is interested in the [[localization]]/[[homotopy category]] with respect to the weak equivalences, or rather in the [[simplicial localization]], hence in the [[(infinity,1)-category|$(\infty,1)$-category]] and hence "the [[homotopy theory]]" presented by this data.

(An earlier proposal by [Grandis 1992](#Grandis92), [1994](#Grandis94) to say "homotopical categories" for 1-categories equipped instead with *[[extra structure]]* of [[homotopies]]/[[2-morphisms]] subject (only) to [[horizontal composition]] does not seem to have caught on.)

Accordingly, a [[functor]] between the underlying categories of homotopical categories which preserves the weak equivalences is called a *[[homotopical functor]]*.

\section{Definition}

Most authors seem to agree that a *homotopical category* is at least a [[strict category|strict]] [[1-category]] equipped with a sub-[[class]] of [[morphisms]] to be called the *[[weak equivalences]]*, closed under [[composition]] and containing all [[identity morphisms]], hence forming a [[wide subcategory]] (a [[relative category]]).

The authors [Dwyer, Hirschhorn, Kan & Smith (2004)](#DwyerHirschhornKanSmith04) require that the weak equivalences satisfy the *[[2-out-of-6-property]]* (this includes all [[model categories]], see [here](two-out-of-six+property#ModelCategoriesSatisfyTwoOutOfSix)) and give a detailed discussion. (Notice that --- with the assumption that all [[identity morphisms]] are among the weak equivalences --- [[2-out-of-6]] *[implies](two-out-of-six+property#2outof3)* the [[2-out-of-3 property]] required for [[categories with weak equivalences]].)

This definition of "homotopical category" has found more followers, e.g. [Szumiło (2014)](#Szumiło14), [Hekking (2017)](#Hekking17).

But some authors [[Bergner (2014)](#Bergner14), [Riehl (2019)](#Riehl19)] use the term "homotopical category" more vaguely, apparently thinking at least of [[categories with weak equivalences]] but focusing on examples that do satisfy also the  [[two-out-of-six property]] (without mentioning this property).

On the other hand, [Arndt (2015)](#Arndt15) seems to use "homotopical categories" as a synonym for "[[relative categories]]" and [Szumiło (2019)](#Szumiło19) seems to use it as synonymous with "[[category with weak equivalences]]", see also [Bergner (2019)](#Bergner19).

\section{Related concepts}

* [[relative category]]

* [[category of fibrant objects]]

* [[cofibration category]]

* [[Waldhausen category]]

* [[model category]]

* [[calculus of fractions]]

* [[resolution]]

\section{References}

An early notion of "homotopical categories" as 1-categories equipped with [[homotopies]] subject to [[horizontal composition]] (hence [[extra structure]] mess than but in the direction of [[2-category]]-structure):

* {#Grandis92} [[Marco Grandis]], *On the categorical foundations of homological and homotopical algebra*, Cahiers de Topologie et Géométrie Différentielle Catégoriques **33** 2 (1992)  135-175 [[numdam:CTGDC_1992__33_2_135_0](http://www.numdam.org/item/?id=CTGDC_1992__33_2_135_0)]

* {#Grandis94} [[Marco Grandis]], *Homotopical algebra in homotopical categories*, Applied Categorical Structures **2** (1994) 351–406 [[doi:10.1007/BF00873039](https://doi.org/10.1007/BF00873039)]

The notion of "homotopical categories" as [[relative categories]] with the requirement that the weak equivalences satisfy the 2-out-of-6 property:

* {#DwyerHirschhornKanSmith04} [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], p. 23 & pp. 96 of: *[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]*, Mathematical Surveys and Monographs **113** AMS (2004) [[ISBN: 978-1-4704-1340-8](https://bookstore.ams.org/surv-113-s), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf)]

Authors following this terminology:

* {#Szumiło14} [[Karol Szumiło]], *Two Models for the Homotopy Theory of Cocomplete Homotopy Theories*, Bonn (2014) [[arXiv:1411.0303](https://arxiv.org/abs/1411.0303), [hdl:20.500.11811/6136](https://hdl.handle.net/20.500.11811/6136)]

* {#Hekking17} [[Jeroen Hekking]], *Segal Objects in Homotopical Categories & K-theory of Proto-exact Categories*, Leiden (2017) [[pdf](https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/hekking_master.pdf)]

* [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], *Homotopical inverse diagrams in categories with attributes*, Journal of Pure and Applied Algebra **225** 4 (2021) 106563 [[doi:10.1016/j.jpaa.2020.106563](https://doi.org/10.1016/j.jpaa.2020.106563)]

On the other hand, "homotopical categories" is used as synonymous with "[[category with weak equivalences]]" in:

* {#Szumiło19} [[Karol Szumiło]], [slide 10](https://hott.github.io/HoTT-2019/conf-slides/Szumilo.pdf#page=10) of: *Internal Languages of Higher Categories* (2019) [[pdf](https://hott.github.io/HoTT-2019/conf-slides/Szumilo.pdf)]

Usage of "homotopical categories" understood with more relaxed or unspecified axioms on the weak equivalences but focusing on examples which are homotopical in the [above](#DwyerHirschhornKanSmith04) sense:

* {#Bergner14} [[Julie Bergner]], *An introduction to homotopical categories*, lecture at MSRI (2014) [part 1:[YT](https://youtube.com/watch?v=OgbaZHpUeD0), 2:[YT](https://youtube.com/watch?v=SKZT7x_0i_Y)]

* {#Riehl19} [[Emily Riehl]], *Homotopical categories: from model categories to (∞,1)-categories* [[arXiv:1904.00886](https://arxiv.org/abs/1904.00886)] in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds,) *[[Stable categories and structured ring spectra]]*,  MSRI Book Series, Cambridge University Press (2022) [ISBN:9781009123297]

Usage of "homotopical categories" as, apparently, synonymous with *[[relative categories]]*:

* {#Arndt15} [[Peter Arndt]], *Homotopical categories of logics*, in: *The Road to Universal Logic*, Studies in Universal Logic Birkhäuser (2015) [[pdf](https://www.math.uni-duesseldorf.de/~arndt/research/HomotopicalCategoriesOfLogics.pdf), [doi:10.1007/978-3-319-10193-4_2](https://doi.org/10.1007/978-3-319-10193-4_2)]

see also:

* {#Bergner19} [[Julie Bergner]], MAA review (2019) [[web](https://www.maa.org/press/maa-reviews/higher-categories-and-homotopical-algebra)] of: [[Denis-Charles Cisinski]]'s *[[Higher Categories and Homotopical Algebra]]*

  > "the two subjects [[[homotopy theory]] and [[category theory]]] have come together in a deep way in the development of what one might call higher homotopical categories. The idea is to consider something like a category, but whose morphisms from one object to another form a topological space, rather than simply a set, and for which composition might only be defined up to homotopy. Such a structure turns out to have several other interpretations: a certain kind of higher category for which various higher morphisms are invertible (often called an (∞,1)-category or simply ∞-category), or even as a category with weak equivalences in the sense of abstract homotopy theory."

[[!redirects homotopical categories]]


homotopical cohomology theory

#Idea#

Under certain conditions on a given [[homotopy theory]] on a category $C$ (namely if $C$ satisfies the axioms of a [[category of fibrant objects]]), the morphisms in the homotopy category $Ho(C)$ are represented already by spans

$$
  \array{
    \hat X
    &\to&
    A
    \\
    \downarrow^\simeq
    \\
    X
  }
$$
([[anafunctor|ana-morphisms]]) instead of longer sequences of zig-zags.

In such a case it makes sense to address such a span as a [[nonabelian cocycle]] on $X$ with coefficients in $A$ and to regard

$$
  H(X,A) := Ho(C)(X,A)
$$
as the **cohomology** of $X$ with coefficients in $A$.
From the point of view of generalized [[sheaf cohomology]] this can be understood by recognizing the replacement object $\hat X$ appearing here as a [[descent|codescent object]].

#Literature#

The fundamental ideas and facts are given in

* Kenneth S. Brown, _[[BrownAbstractHomotopyTheory.pdf:file]]_, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 ([[BrownAHT]])

This develops the theory of localization in [[category of fibrant objects|categories of fibrant objects]] and then applies it to cohomology with coefficients in sheaves with values in combinatorial spectra.

Homotopical cohomology theory in the context of the [[homotopy theory]] of simplicial presheaves (presheaves with values in simplicial sets) has been much developed by Jardine. For instance

* J. Jardine, _Cocycle categories_ ([arXiv](http://arxiv.org/abs/math.AT/0605198))

[[!redirects Homotopical cohomology theory]]


Homotopical cohomology theory > history

< [[Homotopical cohomology theory]]
[[!redirects Homotopical cohomology theory -- history]]


homotopical enrichment

## Related concepts

* [[weak enrichment]]

* [[enriched (∞,1)-category]].


homotopical functor

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#Contents#
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## Definition

A **homotopical functor** $F : C \to D$ between two [[homotopical category|homotopical categories]] $C$ and $D$ is a [[functor]] which preserves the [[weak equivalences]] of $C$ (i.e., which sends all weak equivalences of $C$ to weak equivalences of $D$).

## Related concepts

* [[category with weak equivalences]], [[homotopical category]]

* [[derived functor]]

* [[simplicial localization]]

* [[homological functor]]

* [[Ken Brown's lemma]]

[[!redirects homotopical functors]]

[[!redirects homotopy functor]]
[[!redirects homotopy functors]]


homotopical structure on C*-algebras

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#Contents#
* table of contents
{:toc}

## Idea

By [[Gelfand duality]], [[commutative C*-algebras]] are  [[equivalence of categories|equivalent]], dually, to suitable [[topological spaces]]. In the spirit of [[noncommutative topology]] one may therefore regard general $C^\ast$-algebras as a [[Isbell duality|formal dual]] to generalized topological spaces and ask if the standard [[homotopy theory]] of topological spaces -- as notably expressed by the standard [[model structure on topological spaces]] -- generalizes to [[noncommutative topology]].

By the general discussion at [[simplicial localization]], for a [[category]] to present a [[homotopy theory]] it is sufficient to equip it with the structure of a _[[category with weak equivalences]]_ / _[[homotopical category]]_. Such structures we discuss here. The standard such "noncommutative homotopy" structures reproduce, after [[stabilization]], [[KK-theory]] and [[E-theory]] of $C^\ast$-algebras as their [[triangulated category|triangulated]] [[homotopy categories]].

For practical purposes it is, as usual, useful to enhance the data of the weak equivalences by further ([[cofibration|co]]-)[[fibration]]auxiliary data.
While there cannot be a suitable [[model category]] structure on [[C*Alg]] itself (([Uuye 10]{#Uuye10}); there is however one on [[l.m.c.-C*-algebras]] ([JoachimJohnson 07](#JoachimJohnson07))), there are suitable structures of a [[category of fibrant objects]] ([Uuye 10](#Uuye10)). These we discuss here. For actual [[model category]]-structures see at _[[model structure on operator algebras]]_.

## Definitions

### Category-theoretic preliminaries

Write [[C*Alg]] for the [[category]] of [[C*-algebras]] and [[*-algebra]]-[[homomorphisms]] between them. We regard this as a [[monoidal category]] $(C^\ast Alg, \otimes)$ with the _maximal_ [[tensor product of C*-algebras]].

Write $C^\ast Alg_{sep} \hookrightarrow C^\ast Alg$ for the [[full subcategory]] on the [[separable topological space|separable]] $C^\ast$-algebras.

Write [[Top]] for the category of [[compactly generated topological spaces|compactly generated]] [[weakly Hausdorff topological spaces]].

The [[forgetful functor]] which forgets the [[associative algebra]]-structure and the [[*-algebra]]-structure lands in compactly generated weakly Hausdorff spaces and hence is of the form

$$
  U \;\colon\; C^\ast Alg \to Top
  \,.
$$

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

For $A,B \in $ [[C*Alg]], let

$$
  C^\ast Alg(A,B) \hookrightarrow Top(U(A), U(B)) \in Top
$$

be the set of $C^\ast$-algebra homomorphisms equipped with the [[subspace topology]] from the [[mapping space]] of the underlying topological spaces.

This makes $C^\ast Alg$ a [[Top]]-[[enriched category]].

=--

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

For $A \in $ [[C*Alg]] and $X \in $ [[Top]] be [[compact topological space|compact]], the there is a [[natural isomorphism]]

$$
  Top(X,U(A)) \simeq U(C(X) \otimes A)
  \,.
$$

This defines a [[powering]] of $C^\ast Alg$ over $Top_{cpt}$, by

$$
  A^X \coloneqq C(X) \otimes A
  \,.
$$

=--

([Uuye, lemma 2.2](#Uuye))

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

For every $A \in C^\ast Alg$, the [[hom-functor]]

$$
  C^\ast Alg(A,-) \;\colon\; C^\ast Alg \to Top
$$

preserves [[pullbacks]].

=--

([Uuye, corollary 2.6](#Uuye))

### Plain homotopy theory of $C^\ast$-algebras

Write $I \coloneqq [0,1] \in $ [[Top]] for the standard topological [[interval]].

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

For $f,g \colon A \to B$ two morphisms in [[C*Alg]], a [[right homotopy]] $\eta \colon f \Rightarrow g$ is a morphism $\eta \colon A \to B^I$ such that

$$
  \array{
    && B
    \\
    & {}^{\mathllap{f}}\nearrow& \uparrow^{\mathrlap{B^{i_0}}}
    \\
    A &\stackrel{\eta}{\to}& B^I
    \\
    & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{B^{i_1}}}
    \\
    && B
  }
  \,.
$$

=--

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

Right homotopy is an [[equivalence relation]]. We write $[A,B]$ for the set of right homotopy-[[equivalence classes]] of morphisms $A \to B$. We have a [[natural isomorphism]]

$$
  [A,B] \simeq \pi_0 C^\ast Alg(A,B)
$$

These are the [[hom-sets]] of a [[category]], the [[homotopy category]] $Ho(C^\ast Alg)$

=--

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

Call a morphism $f \colon A \to B$ in [[C*Alg]] a **[[homotopy equivalence]]** of $C^\ast$-algebras if it is an [[isomorphism]] in $Ho(C^\ast Alg)$.

=--

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

Call a morphism $f \colon A \to B$ in [[C*Alg]] a **Schochet fibration** if for all $D \in C^\ast Alg$ the map

$$
  C^\ast(D,f) \;\colon\; C^\ast(D,A) \to C^\ast(D,B)
$$

is a [[Serre fibration]] in [[Top]].

=--

([Schochet 84](#Schochet84), [Uuye, def. 2.14, prop. 2.18](#Uuye))

+-- {: .num_theorem}
###### Theorem

The category [[C*Alg]] equipped with weak equivalences being the homotopy equivalences of def. \ref{HomotopyEquivalence} and with fibrations being the Schochet fibrations of def. \ref{SchochetFibration} is a [[category of fibrant objects]].

=--

([Schochet 84](#Schochet84), [Uuye, theorem 2.19](#Uuye)).

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

The canonical functorial [[path space object]] of a $C^\ast$-algebra $A$ in this structure is $A^I$. It follows that the corresponding [[loop space object]] is

$$
  \Omega A = C_0((0,1), A)
  \,.
$$

Dually, interpreted as a space in [[noncommutative topology]], this corresponds to the [[suspension]] of the space that corresponds to $A$.

=--

### Stabilization at the compact operators

Write $\mathcal{K} \in C^\ast Alg$ for the [[C*-algebra]] of [[bounded operators]] on an infinite-dimensional [[separable Hilbert space]].

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

Say a morphism $f \colon A \to B$ in [[C*Alg]] is a **stable homotopy equivalence** if $f \otimes id_{\mathcal{K}}$ is a homotopy equivalence, def. \ref{HomotopyEquivalence}.

Say $f$ is a **stable Schochet fibration** of $f \otimes id_{\mathcal{K}}$ is a Schochet fibration, def. \ref{SchochetFibration}.

=--

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

The category [[C*Alg]] becomes a [[category of fibrant objects]] with weak equivalences the stable homotopy equivalences and fibrations the stable Schochet fibrations.

=--

([Uuye, prop. 2.24](#Uuye))

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

(...) kk-groups (...)

=--

### KK-theory

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

Say a morphism $f \colon A \to B$ in $C^\ast Alg_{sep}$ is a **KK-equivalence** if for all $D \in C^\ast Alg_{sep}$ the morphism

$$
  KK(D,f) \;\colon\; KK(D,A) \to K(D,B)
$$

is an [[isomorphism]], where $KK$ is the [[KK-theory]]-category.

=--

+-- {: .num_theorem}
###### Theorem

The category $C^\ast Alg_{sep}$ carries the structure of a [[category of fibrant objects]] whose weak equivalences are the KK-equivalences of def. \ref{KKEquivalence}, and whose fibrations are the Schochet fibrations, def. \ref{SchochetFibration}.

=--

([Uuye, theore, 2.29](#Uuye))

## References

The various structures of a [[category of fibrant objects]] on [[C*Alg]] are discussed in

* [[Otgonbayar Uuye]], _Homotopy theory for $C^\ast$-algebras_,  Journal of Noncommutative Geometry,  ([arXiv:1011.2926](http://arxiv.org/abs/1011.2926))
 {#Uuye10}

using notions from

* [[Claude Schochet]], _Topological methods for C∗-algebras. III. Axiomatic homology_, Pacific J. Math. 114 (1984), no. 2, 399–445. MR 757510 (86g:46102)
 {#Schochet84}

A [[model category|model categorical]] approach is presented in

* [[Paul Arne Østvær]], _Homotopy theory of $C^*$-algebras_, Frontiers in Mathematics, Springer Basel, 2010, ([arxiv/0812.0154](http://arxiv.org/abs/0812.0154), [pdf](http://folk.uio.no/paularne/file0.pdf))

A [[model category]] structure presenting [[KK-theory]] not on $C^\ast$-algebras itselt but on [[l.m.c.-C*-algebras]] is discussed in

* [[Michael Joachim]], [[Mark Johnson]], _Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category_ ([arXiv:0705.1971](http://arxiv.org/abs/0705.1971))
 {#JoachimJohnson07}

[[!redirects homotopical stricture on C-star algebras]]