stable homotopy category

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## Idea

The _stable homotopy category_ $Ho(Spectra)$ is the [[category]] of [[spectra]] and [[homotopy classes]] of morphisms between them, the object of study in classical [[stable homotopy theory]]. Equivalently this is the [[homotopy category of an (∞,1)-category]] of the [[stable (∞,1)-category of spectra]] and the latter is the proper context for [[stable homotopy theory]]. But with due care exercised, the stable homotopy category itself is useful.

The stable homotopy category may be thought of as the [[stabilization]] of the [[classical homotopy category]] $Ho(Top)$ under the operation of forming [[loop space objects]] $\Omega$ and [[reduced suspensions]] $\Sigma$: via forming [[suspension spectra]] $\Sigma^\infty$ every [[pointed object]] in the [[classical homotopy category]] maps to the stable homotopy category, and under this map the [[loop space]]- and [[reduced suspension]]-[[functors]] become inverse [[equivalence of categories|equivalences]] on the stable homotopy category.

$$
  \array{
     Ho(Top)^{\ast/}
      &
      \underoverset{\underset{\Omega}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{}
      &
     Ho(Top)^{\ast/}
     \\
     {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}}
     &&
     {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}}
     \\
     Ho(Spectra)
     &
     \underoverset{\underset{\Omega}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{\simeq}
     &
     Ho(Spectra)
  }
  \,.
$$

In contrast to the [[classical homotopy category]], the stable homotopy category is a [[triangulated category]] (a shadow of the fact that the [[(∞,1)-category of spectra]] is a [[stable (∞,1)-category]]). As such it may be thought of as a refinement of the [[derived category]] [[category of chain complexes|of chain complexes]] (of [[abelian groups]]): every [[chain complex]] gives rise to a [[spectrum]] and every [[chain map]] to a map between these spectra (the [[stable Dold-Kan correspondence]]), but there are many more spectra and maps between them than arise from chain complexes and chain maps.

Equipped with the [[smash product of spectra]] "$\wedge$" and with [[function spectra]] $[-,-]$, the stable homotopy category becomes a [[symmetric monoidal category|symmetric]] [[closed monoidal category]].

A ([[commutative monoid|commutative]]) [[monoid object]] with respect to $\wedge$ is ([[commutative ring spectrum|commutative]]) [[ring spectrum]].

For $E \in Ho(Spectra)$ any spectrum, then the [[functor]]

$$
  E_\bullet \coloneqq \pi_0((\Sigma^\bullet E) \wedge(-)) \;\colon\; Ho(Spectra) \longrightarrow Ab
$$

is a [[generalized homology theory]], while the functor

$$
  E^\bullet \coloneqq \pi_0[-,\Sigma^\bullet E] \;\colon\; Ho(Spectra)^\op \longrightarrow Ab
$$

is a [[generalized cohomology theory]].

## Definition

There are various different-looking ways to define the stable homotopy category.

### Via eventually defined maps

One of the first constructions of the stable homotopy category is due to ([Adams 74, part III, sections 2 and 3](#Adams74)), following ([Boardman 65](#Boardman65)). This _[[Adams category]]_ is defined to be the category of [[CW-spectra]] with [[homotopy classes]] (with respect to [[cylinder spectra]]) of "eventually defined" functions between them.

Historically this was advertised as being a construction free of tools of [[category theory]]. See ([Lewis-May-Steinberger 86, pages 1-3](#LewismaySteinberger86)) for review and critical assessment

### Via left homotopy
 {#ViaSpectrificationofSequentialPreSpectra}

([Lewis-May-Steinberger 86, "preamble" pages 1-7](#LewisMaySteinberger86), [Elmendorf-Kriz-May 95, p. 8](#ElmendorfKrizMay95), [Malkiewich 14](#Malkiewich14))

For $E$ a [[sequential spectrum|sequential]] [[pre-spectrum]] and $X$ a [[pointed topological space]], write

$$
  (E \wedge X)_n \coloneqq E_n \wedge X
$$

for the degreewise [[smash product]] of [[pointed topological spaces]]. Write $I_+$ for the unit [[interval]] with a base point adjoined, such that for any spectrum $E$, the [[smash product]] $E \wedge I_+$ is its _[[cylinder spectrum]]_.

A _[[left homotopy]]_ between morphism of pre-spectra $f,g \colon E_1 \longrightarrow E_2$ is a morphism of spectra

$$
  \phi \;\colon\; E_1\wedge I_+ \longrightarrow E_2
$$

such that $\phi|_0 = f$ and $\phi|_1 = g$.

In order for this to be the homotopy-correct notion, we need to apply it with [[domain]] a [[CW-spectrum]] and [[codomain]] an [[Omega-spectrum]].

Let $PreSpectra \stackrel{\overset{L}{\longrightarrow}}{\underset{\ell}{\longleftarrow}} Spectra$ be the 1-categorical [[adjunction]] between [[Omega-spectra]] and [[prespectra]], in the sense defined at [[coordinate-free spectrum]], where $\ell$ is the [[forgetful functor]] and $L$ is [[spectrification]].

There is also a [[CW-spectrum]]-replacement functor $\Gamma$.

Write then

$$
  [E_1,E_2]
  \coloneqq
  Hom(\Gamma E_1, L E_2)_{\sim_{left\;homotopy}}
$$

for the corresponding [[homotopy classes]] of maps.

### Via model structures

There are several [[model categories]] which exhibit [[model structures for spectra]], hence whose [[homotopy category of a model category]] is equivalent to the stable homotopy category.

The most lightweight of these is the [[Bousfield-Friedlander model structure]] of [[sequential spectrum|sequential]] [[pre-spectra]] in [[simplicial sets]] ([Bousfield-Friedlander  78](#BousfieldFriedlander78))

Model structures on more [[highly structured spectra]] include the [[model structure on symmetric spectra]], the [[model structure on orthogonal spectra]], and ultimately the [[model structure for excisive functors]]. It is only with these model structures that the [[smash product of spectra]] is represented by a [[symmetric monoidal smash product of spectra]] even before passing to the stable homotopy category.

A unified account of all thes model structures is at _[[Model categories of diagram spectra]]_.

## Properties

### Symmetric monoidal structure

The [[smash product of spectra]] makes the stable homotopy category into a [[symmetric monoidal category]].

An ([[commutative monoid|commutative]]) [[monoid object]] with respect to this is a ([[commutative ring spectrum|commutative]]) [[ring spectrum]]. A [[module object]] over such is a [[module spectrum]].

### Triangulated category structure

The [[homotopy fiber sequences]] of spectra gives the stable homotopy category the structure of a [[triangulated category]].

For **proof** see at _[[Introduction to Stable homotopy theory]]_ [this Prop](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated).

[[!include finite homotopy limits of spectra -- section]]

## Related concepts

* [[Ho(Top)]]

* [[Whitehead theorem]]

* [[Spectra]]

## References

The original direct definitions of the stable homotopy category (for precursors see at _[[Spanier-Whitehead category]]_) is due to

* {#Boardman65} [[Michael Boardman]], _Stable homotopy theory_, mimeographed notes, University of Warwick, 1965 onward

Early accounts include

* {#Vogt69} [[Rainer Vogt]], _Boardman's stable homotopy category_, lectures, spring 1969

* {#Cohen70} J. M. Cohen, _Stable Homotopy_, Springer Lecture Notes in Math., No. 165,  Springer-Verlag, Berlin, 1970.

* [[Dieter Puppe]], _On the stable homotopy category_, Topology and its application (1973)  ([[PuppeStableHomotopyCategory.pdf:file]])

* {#Adams74} [[Frank Adams]], Part III, section 2 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Switzer75} [[Robert Switzer]], _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

Expository introduction:

* {#Malkiewich14} [[Cary Malkiewich]], _The stable homotopy category_, 2014 ([pdf](https://people.math.binghamton.edu/malkiewich/stable.pdf), [[MalkiewichStableHomotopyCategory.pdf:file]])

A fun scan of the (pre-)history of the stable homotopy category:

* [[Peter May]], _The hare and the tortoise_, speech at [[Michael Boardman]]'s birthday meeting ([txt](http://hopf.math.purdue.edu/May/boardman.txt))

See also the references at _[[stable homotopy theory]]_.

Original articles realizing the stable homotopy category as the [[homotopy category]] of a [[model category]] include

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf))

Original articles in the context of [[highly structured spectra]] include

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics, 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))

* {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_, Amsterdam: North-Holland, 1995 pp. 213–253,  ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], _Equivariant stable homotopy theory_, in I.M. James (ed.), _[[Handbook of Algebraic Topology]]_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

A textbook account in the context of [[symmetric spectra]] is

* {#Schwede12} [[Stefan Schwede]], chapter II, section 1_[[Symmetric spectra]]_ (2012)

[[!redirects HoSpectra]]

[[!redirects Ho(Spectra)]]


stable homotopy homology theory

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## Idea

The [[generalized homology theory]] which is [[Brown representability theorem|represented]] by the [[sphere spectrum]] $\mathbb{S}$ is usually called _stable homotopy_, since the [[homology groups]] that it assigns to a suitable [[topological space]] $X$ are just the [[stable homotopy group]] of $X$:

$$
  H_\bullet(X,\mathbb{S})
  \;=\;
  \mathbb{S}_\bullet(X)
  \coloneqq
  \pi_\bullet\left( \mathbb{S} \wedge X_+  \right)
  \;=\;
  \simeq \pi_\bullet^{st}(X)
  \,.
$$

Hence the [[coefficient]] [[cohomology ring]] of stable homotopy homology theory (its value on the [[point]]) is the [[stable homotopy groups of spheres]]. This highlights that stable homotopy homology of any space $X$ is extremely hard, or impossible, to completely analyze, since this is true already for the coefficient ring over the point.

Beware that the term _[[stable homotopy theory]]_, which would seem to be the canonical name for this [[generalized homology theory]], traditionally refers instead to the general [[homotopy theory]] of [[spectra]]. The full term _stable homotopy homology theory_ is used for emphasis, but clunky in practice.

The [[duality|dual]] [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theory]], co-represented by the [[sphere spectrum]], is called _[[stable cohomotopy theory]]_.

## Properties

### Relation to other homology theories

Along the canonical morphism of [[spectra]] $\mathbb{S} \to H R$ from the [[sphere spectrum]] to any [[Eilenberg-MacLane spectrum]] of a [[ring]] $R$ (which is the [[unit]] map of $H R$ in the [[(infinity,1)-category]] of [[E-infinity ring spectra]]) stable homotopy homology maps to [[ordinary homology]] with [[coefficients]] in $R$ (given notably by [[singular homology]]).

This is known as the _[[Hurewicz homomorphism]]_

More generally, for $E$ an [[E-infinity ring spectrum]], the unit $\mathbb{S} \to E$ induces a [[natural transformation]] from stable homotopy homology theory to the [[generalized homology theory]] co-represented by $E$.

This is known as the _[[Boardman homomorphism]]_.

## Related concepts

* [[ordinary homology]]

* [[bordism homology theory]]

* [[stable cohomotopy theory]]

## References

* [[Akhil Mathew]], _Torsion exponents in stable homotopy and the Hurewicz homomorphism_, Algebr. Geom. Topol. 16 (2016) 1025-1041 ([arXiv:1501.07561](https://arxiv.org/abs/1501.07561))


stable homotopy hypothesis

# Stable homotopy hypothesis

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## Idea

While the ordinary [[homotopy hypothesis]] relates [[∞-groupoids]] to [[topological spaces]], the **stable homotopy hypothesis** relates symmetric monoidal $\infty$-groupoids to [[spectra]].

## References

* {#JO12} [[Niles Johnson]] and [[Angelica Osorno]], *Modeling stable one-types*, [tac](http://tac.mta.ca/tac/volumes/26/20/26-20abs.html)

* {#GJO17} [[Nick Gurski]], [[Niles Johnson]], and [[Angelica Osorno]], *The 2-dimensional stable homotopy hypothesis*, [arxiv](https://arxiv.org/abs/1712.07218)


stable homotopy theory

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## Idea

_Stable homotopy theory_ is [[homotopy theory]] in the case that the operations of [[looping and delooping]] are _[[equivalence of (infinity,1)-categories|equivalences]]_.

As [[homotopy theory]] is the study of [[homotopy types]], so _stable homotopy theory_ is the study of [[stable homotopy types]]. As [[homotopy theory]] in generality is [[(∞,1)-category theory]] (or maybe [[(∞,1)-topos theory]]), so stable homotopy theory in generality is the theory of [[stable (∞,1)-categories]].

More specifically, if one thinks of classical [[homotopy theory]] as the study of (just) the [[(∞,1)-category]] $L_{whe}$[[Top]] $\simeq$ [[∞Grpd]] of [[topological spaces]] modulo [[weak homotopy equivalence]] ([[∞-groupoids]]), or rather of its [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho(Top)$, then stable homotopy theory is the study of the corresponding [[stabilization]]: To every suitable [[(∞,1)-category]] is associated its corresponding [[stable (∞,1)-category]] of [[spectrum objects]]. For $L_{whe}$[[Top]] this is the [[stable (∞,1)-category of spectra]], $Sp(L_{whe}Top)$. **Stable homotopy theory** is the study of $Sp(Top)$, or rather of its [[homotopy category of an (infinity,1)-category|homotopy category]], the [[stable homotopy category]] $Ho(Sp(L_{whe}Top))$.

For detailed introduction to the stable homotopy theory of spectra see at _[[Introduction to Stable homotopy theory]]_.

By definition a [[stable homotopy type]] is one on which [[suspension]] and hence [[looping and delooping]] act as an [[equivalence]]. Historically people considered in plain [[homotopy theory]] statements that became true after sufficiently many [[suspensions]], hence once the process of taking [[suspensions]] "stabilizes". Whence the name.

## Applications

### Higher algebra

The study of [[monoid object in a monoidal (infinity,1)-category]] in a [[stable (∞,1)-category]] is the [[homotopy theory|homotopy-theoretic]] version of [[commutative algebra]], hence _[[higher algebra]]_ and [[higher linear algebra]].

A tool of central importance in stable homotopy theory and its application to [[higher algebra]] is the [[symmetric monoidal smash product of spectra]] which allows us to describe [[A-∞ rings]] and [[E-∞ rings]] as ordinary [[monoid]] objects in a [[model category]] that presents $Sp(Top)$. ("[[brave new algebra]]").

## Related concepts

* [[homotopy theory]], [[homotopy type theory]]

* [[stable homotopy category]], [[(infinity,1)-category of spectra]]

* [[Spanier-Whitehead duality]], [[Anderson duality]]

  * [[S-theory]]

* [[stable (infinity,1)-category]]

* [[parameterized homotopy theory]], [[parameterized stable homotopy theory]]

* [[rational stable homotopy theory]]

* [[cohesive stable homotopy theory]]

* [[noncommutative stable homotopy theory]]

* [[chromatic homotopy theory]]

* [[K(n)-local stable homotopy theory]]

* [[generalized homology theory]], [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theory]]

* [[stable homotopy homology theory]]

When the spaces and spectra in question carry an [[infinity-action]] of a [[group]] $G$ the theory refines to

* [[equivariant stable homotopy theory]], [[global equivariant stable homotopy theory]].

## History
 {#History}

> Stable homotopy theory began around 1937 with the [[Freudenthal suspension theorem]]. $[$...$]$ Stable phenomena had of course appeared earlier, at least implicitly: [[reduced cohomology|reduced]] [[generalized homology|homology]] and [[generalized (Eilenberg-Steenrod) cohomology|cohomology]] are examples of functors that are invariant under suspension without limitation on dimension.

> Stable homotopy theory emerged as a distinct branch of [[algebraic topology]] with Adams' introduction of [[Adams spectral sequence|his eponymous spectral sequence]] and his spectacular conceptual use of the notion of stable phenomena in his solution to the [[Hopf invariant one]] problem.

> Its centrality was reinforced by two related developments that occurred at very nearly the same time, in the late 1950's. One was the introduction of [[generalized (Eilenberg-Steenrod) cohomology|generalized homology and cohomology theories]] and especially [[K-theory]], by Atiyah and Hirzebruch. The other was the work of Thom which showed how to reduce the problem of classifying manifolds up to cobordism to a problem, more importantly, a solvable problem, in stable homotopy theory $[$ [[Thom spectrum]] $]$.

> The reduction of geometric phenomena to solvable problems in stable homotopy theory
has remained an important mathematical theme, the most recent major success being Stolz's use of [[spin bordism|Spin cobordism]] to study the classication of manifolds with positive scalar curvature.

> In an entirely different direction, the early 1970's saw Quillen's introduction of higher [[algebraic K-theory]] and the recognition by Segal and others that it could be viewed as a construction in stable homotopy theory. With algebraic K-theory as an intermediary, there has been a growing volume of work that relates algebraic geometry to stable homotopy theory. With Waldhausen's introduction of the algebraic K-theory of spaces in the late 1970's, stable homotopy became a bridge between algebraic K-theory and the study of diffeomorphisms of manifolds.

> Within algebraic topology, the study of stable homotopy theory has been and remains the focus of much of the best work in the subject.

> ([Elmendorf-Kriz-May 95, p. 2](#ElmendorfKrizMay95))

## References

The original direct definitions of the [[stable homotopy category]] (for precursors see at _[[Spanier-Whitehead category]]_) is due to

* {#Boardman65} [[Michael Boardman]], _Stable homotopy theory_, mimeographed notes, University of Warwick, 1965 onward

Early accounts:

* {#Vogt69} [[Rainer Vogt]], _Boardman's stable homotopy category_, lectures, spring 1969

* {#Cohen70} J. M. Cohen, _Stable Homotopy_, Springer Lecture Notes in Math., No. 165,  Springer-Verlag, Berlin, 1970.

* [[Dieter Puppe]], _On the stable homotopy category_, Topology and its application (1973)  ([[PuppeStableHomotopyCategory.pdf:file]])

* {#Adams74} [[Frank Adams]], Part III, section 2 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Switzer75} [[Robert Switzer]], _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

A fun scan of the (pre-)history of the [[stable homotopy category]]:

* [[Peter May]], _The hare and the hedgehog_, speech at [[Michael Boardman]]'s birthday meeting ([txt](http://hopf.math.purdue.edu/May/boardman.txt))

Quick surveys:

* {#Malkiewich14} [[Cary Malkiewich]], _The stable homotopy category_, 2014 ([pdf](https://people.math.binghamton.edu/malkiewich/stable.pdf), [[MalkiewichStableHomotopyCategory.pdf:file]])

* [[Dylan Wilson]], _Introduction to stable homotopy theory_ ([pdf](https://etale.site/livetex/wcatss/sht1.pdf), [[WilsonStableHomotopyTheory.pdf:file]])

* [[Aaron Mazel-Gee]], _An introduction to spectra_ ([pdf](https://math.berkeley.edu/~aaron/writing/an-introduction-to-spectra.pdf))

Lecture notes:

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

* [[Urs Schreiber]], _[[Introduction to Stable homotopy theory]]_, Bonn 2016

* [[David Barnes]], [[Constanze Roitzheim]], _Foundations of Stable Homotopy Theory_, Cambridge University Press 2020 ([doi:10.1017/9781108636575](https://doi.org/10.1017/9781108636575))

* [[Denis Nardin]], _Introduction to stable homotopy theory_, Regensburg 2021 ([webpage](https://homepages.uni-regensburg.de/~nad22969/stable-homotopy-theory2020.php), [pdf](https://homepages.uni-regensburg.de/~nad22969/stable-homotopy-2020/stable-homotopy.pdf), [video](https://www.youtube.com/watch?v=v5Rr_Z8QsOs&list=PLQuS12Ap6va-VgI5cdIjdHquIH3sLCLdJ))

See also:

* Glossary for stable and chromatic honotopy theory ([[StableChromaticGlossary.pdf:file]])

Models with [[symmetric monoidal smash product of spectra]]:

for [[S-modules]]:

* {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]], _[[Handbook of Algebraic Topology]]_, Amsterdam: North-Holland, (1995) pp. 213–253,  ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

for [[symmetric spectra]]

* [[Stefan Schwede]], _Symmetric spectra_ ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

for [[orthogonal spectra]]:

* [[Stefan Schwede]], _[[Global homotopy theory]]_ (take $\mathcal{F} = \{1\}$, on p. 4, to be the trivial collection of groups, in order to specialize from [[global equivariant stable homotopy theory]] to plain stable homotopy theory).

A survey of formalisms used in stable homotopy theory -- tools to present the [[triangulated category|triangulated]] [[homotopy category]] of a  [[stable (infinity,1)-category]] -- is in

* [[Neil Strickland]], _Axiomatic stable homotopy - a survey_ ([arXiv:math.AT/0307143](http://front.math.ucdavis.edu/0307.5143))

* [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]],  *Axiomatic stable homotopy theory*, Memoirs Amer. Math. Soc. **610** (1997) [[ISBN:978-1-4704-0195-5](https://bookstore.ams.org/memo-128-610), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf)]

An account in terms of [[(∞,1)-category theory]] is in section 1 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

Brief indications of open questions and future directions (as of 2013) of [[algebraic topology]] and [[stable homotopy theory]] are in

* [[Tyler Lawson]], _The future_, Talbot lectures 2013 ([pdf](http://math.mit.edu/conferences/talbot/2013/19-Lawson-thefuture.pdf))

[[!redirects stable homotopy theories]]


stable homotopy theory - contents

**[[stable homotopy theory]]**

* [[homological algebra]], [[higher algebra]]

_[[Introduction to Stable Homotopy Theory|Introduction]]_

#Ingredients

* [[homotopy theory]]

#Contents

* [[loop space object]]

* [[suspension object]]

* [[looping and delooping]]

* [[stable (∞,1)-category]]

  * [[stabilization]]

    * [[spectrum object]]

  * [[stable derivator]]

  * [[triangulated category]]

* [[stable (∞,1)-category of spectra]]

  * [[spectrum]]

  * [[stable homotopy category]]

* [[smash product of spectra]]

  * [[symmetric smash product of spectra]]

  * [[Spanier-Whitehead duality]]

  * [[A-∞ ring]]

  * [[E-∞ ring]]

[Edit this sidebar](/nlab/edit/stable+homotopy+theory+-+contents)


stable homotopy type

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#### Type theory
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[[!include type theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

A [[pointed object|pointed]] [[homotopy type]] $X_\ast$ could be called _linear_ (following the terminology of _[[linear model category]]_) if the canonical [[morphism]]

$$
  \Sigma \Omega X \longrightarrow X
$$

from the [[suspension object]] of its [[loop space object]] is an [[equivalence in an (infinity,1)-category|equivalence]].

By [[Goodwillie calculus]] the [[tangent (∞,1)-topos]] $T\mathbf{H}$ of some [[(∞,1)-topos]] $\mathbf{H}$ is the [[localization of an (∞,1)-category|localization]] of the [[classifying (∞,1)-topos]] $\mathbf{H}[X_\ast]$ (of pointed objects), that regard each pointed finite [[powering]] $X_\ast^\bullet$ of the generic pointed object $X_\ast$ as a linear homotopy type.

See at [excisive functor -- Characterization via the generic pointed object](https://ncatlab.org/nlab/show/excisive+infinity-functor#CharacterizationViaAGenericStableObject).

Therefore a pointed homotopy type $X_\ast$ such that all its finite pointed powers are linear could be called a _stable homotopy type_.

For $\mathbf{H}$ an [[(∞,1)-topos]] and $T \mathbf{H}$ its [[tangent (∞,1)-topos]], then all homotopy types in $T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$ are stable in this sense, these are equivalently the [[spectrum objects]] in $\mathbf{H}$.

## Related concepts

* [[sequential spectrum type]]

* [[dependent linear type theory]]

* [[equivariant homotopy type]]

* [[modal homotopy type]]

* [[geometric homotopy type]], [[cohesive homotopy type]]

## References

An account using [[modal homotopy type theory]] is in

* [[Mitchell Riley]], [[Eric Finster]], [[Daniel Licata]], _Synthetic Spectra via a Monadic and Comonadic Modality_, ([arXiv:2102.04099](https://arxiv.org/abs/2102.04099))

[[!redirects stable homotopy types]]


Stable Infinity-Categories

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#### Stable Homotopy theory
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=--

This entry provides a hyperlinked index for

* [[Jacob Lurie]], _Stable $\infty$-Categories_, ([arXiv:math.CT/0608228](http://www.arXiv.org/abs/math.CT/0608228)), now section 1 in _[[Higher Algebra]]_

on [[stable homotopy theory]] in terms of  _[[stable ∞-categories]]_.

Based on the theory of [[(∞,1)-category|(∞,1)-categories]] as developed in his book _[[Higher Topos Theory]]_, the author studies here $(\infty,1)$-categories of "stable objects", i.e. of objects that behave like [[spectrum|spectra]] in that for each object $X$ there not only its [[loop space object]] $\Omega X$ but also conversely, $X$ is the [[loop space object]] of another object $\Sigma X$.

The definition is very simple.
The homotopy category of a stable $(\infty,1)$-category is shown to be a [[triangulated category]]: the comparatively complicated axioms of triangulated categories follow from the simple $(\infty,1)$-categorical axioms. Large chunks of [[homological algebra]] is then re-examined from the more natural point of view of stable $(\infty,1)$-categories.

#Content#
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{:toc}

## Introduction##

## Stable $\infty$-Categories##

* [[stable (∞,1)-category]]

  * [[zero object]]

  * [[kernel]]

  * [[cokernel]]

  * [[homotopy fiber]]

  * [[fibration sequence]]

  * [[loop space object]]

  * [[homotopy limit]]

## The Homotopy Category of a Stable $\infty$-Category##

* [[homotopy category of an (∞,1)-category]]

* [[triangulated category]]

## Properties of Stable $\infty$-Categories##

## Exact Functors##

* [[exact functor]]

## t-Structures and Localization##

## Boundedness and Completeness##

## Stabilization##

* [[stabilization]]

* [[spectrum object]]

## The $\infty$-Category of Spectra##

* [[spectrum]]

* [[stable (∞,1)-category of spectra]]

## Excisive Functors##

## Filtered Objects and Spectral Sequences##

## The $\infty$-Categorical Dold-Kan Correspondence##

* [[Dold-Kan correspondence]]

## Homological Algebra##

* [[homological algebra]]

* [[derived category]]

* [[derived functor on a derived category]]

## The Universal Property of $D^-(A)$

## Presentable Stable $\infty$-Categories

* [[accessible (∞,1)-category]]

* [[presentable (∞,1)-category]]

## Accessible t-Structures##

category: reference

[[!redirects Stable ∞-Categories]]
[[!redirects Stable infinity-Categories]]


stable infinity-category > history

< [[stable infinity-category]]


stable map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A special kind of morphism between [[orbifolds]]. Counted by [[Gromov-Witten invariants]].

## Related concepts

* [[quantum cohomology]]

## References

According to p 43 in

* [[Dan Abramovich]], _Lectures on Gromov-Witten invariants of orbifolds_ ([arXiv:math/0512372](http://arxiv.org/abs/math/0512372))
 {#Abramovich05}

the notion originates in an email by [[Maxim Kontsevich]], reproduced there.

The definition is stated for instance as def. 2.3.3 in

* Weimin Chen, [[Yongbin Ruan]], _Orbifold Gromov-Witten Theory_, in _Orbifolds in mathematics and physics_ (Madison, WI, 2001), 25–85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 ([arXiv:math/0103156](http://arxiv.org/abs/math/0103156))
 {#ChenRuan01}

Lecture notes include

* [[William Fulton]], R. Pandharipande, _Notes on stable maps and quantum cohomology_ ([arXiv:alg-geom/9608011](http://arxiv.org/abs/alg-geom/9608011))

[[!redirects stable maps]]


stable model category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _stable model category_ is a 1-[[category]] structure used to present a [[stable (∞,1)-category]] in analogy to how a general [[model category]] encodes a (generally non-stable) [[(∞,1)-category]].

## Defintion

A **stable model category** $\mathcal{C}$ is

* a [[pointed model category]]

* such that the [[loop space object]] functor $\Omega$ and the [[suspension object]] functor $\Sigma$, are inverse [[equivalence of categories|equivalences]] on the [[homotopy category of a model category|homotopy category]] $Ho(C)$:

  $$
    \Omega
    \colon
    Ho(\mathcal{C})
     \stackrel{\overset{\simeq}{\longleftarrow}}{\underset{\simeq}{\longrightarrow}}
   Ho(\mathcal{C})
   \colon
  \Sigma
   \,.
  $$

## Properties

### Triangulated homotopy category

The [[homotopy category of a model category|homotopy category]] $Ho(\mathcal{C})$ of a stable model category, equipped with the [[reduced suspension]] functor $\Sigma \colon Ho(\mathcal{C})\overset{\simeq}{\to} Ho(\mathcal{C})$ is a [[triangulated category]] ([Hovey 99, section 7](#Hovey99)).

### Relation to stable $\infty$-categories

Stabilization of model categories is a model for the abstractly defined [[stabilization]] in [[(infinity,1)-category theory]] ([Robalo 12, prop. 4.15](#Robalo12)).

### As $A_\infty$-algebroid module categories
 {#AsCategoriesOfModules}

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

Let $\mathcal{C}$ be a _stable model category_ that is in addition

* a [[simplicial model category]];

* a [[cofibrantly generated model category]];
* a [[proper model category]];

* with a [[set]] $S$ of [[compact object|compact]] generators;

then there is a chain of [[simplicial Quillen adjunction|sSet-enriched]] [[Quillen equivalences]]  linking $\mathcal{C}$ to the the [[spectrum]]-[[enriched functor category]]

$$
  \mathcal{C} \simeq_Q A_S Mod \coloneqq Sp Cat(A_S, Sp)
$$

equipped with the [[global model structure on functors]],
where $A_S$ is the $Sp$-[[enriched category]] whose set of objects is $S$

=--

This is  in ([Schwede-Shipley, theorem 3.3.3](#SchwedeShipley))

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

An $Sp$-enriched category is a homotopy-theoretic analog of an [[Ab-enriched category]], which may be thought of as a many-object version of a [[ring]], a "[[ringoid]]". Accordingly, an $Sp$-enriched category is an $A_\infty$-ringoid. It is has a single object then (as a pointed category) it is an [[A-infinity algebra]].

=--

Hence:

+-- {: .num_cor}
###### Corollary

If if in prop. \ref{ModuleProp} there is just one compact generator $P \in \mathcal{C}$, then there is a one-object $Sp$-enriched category, hence an [[A-infinity algebra]] $A$, which is the [[endomorphisms]] $A \simeq End_{\mathcal{C}}(P)$, and the stable model category is its [[category of modules]]:

$$
  \mathcal{C} \simeq_Q A Mod
  \,.
$$

=--

This is  in ([Schwede-Shipley, theorem 3.1.1](#SchwedeShipley))

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

This may be thought of as a homotopy-theoretic analog of the [[Freyd-Mitchell embedding theorem]] for [[abelian categories]].

=--

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

One way to read this is that [[formal duals]] of presentable [[stable infinity-categories]] are a model for [[spaces]] in ("[[derived noncommutative geometry|derived]]") [[noncommutative geometry]].

=--

If $A$ is an [[Eilenberg-MacLane spectrum]], then this identifies the corresponding stable model categories with the [[model structure on unbounded chain complexes]].

This is ([Schwede-Shipley 03, theorem 5.1.6](#SchwedeShipley03)).

## Related concepts

* [[linear model category]]

## References

The concept originates with

* {#Hovey99} [[Mark Hovey]], section 7 of _Model Categories_ Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))

The classification theorems are due to

* {#SchwedeShipley03} [[Stefan Schwede]], [[Brooke Shipley]], _Classification of stable model categories_ or _Stable model categories are categories of modules_, Topology 42 (2003) 103-153 ([arXiv:0108143](http://arxiv.org/abs/math/0108143))

Discussion of the notion of stable model categories with the abstract notion of [[stabilization]] in [[(infinity,1)-category theory]] is in section 4.2 (prop. 4.15) of

* {#Robalo12} [[Marco Robalo]], _Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes_, June 2012 ([arxiv:1206.3645](http://arxiv.org/abs/1206.3645))

On ([[monoidal localization|monoidal]]) [[Bousfield localization of model categories|Bousfield localization]] of stable model categories:

* [[Luca Pol]], [[Jordan Williamson]], *The Left Localization Principle, completions, and cofree $G$-spectra*, J. Pure Appl. Algebra **224** 11 (2020) 106408 [[arXiv:1910.01410](https://arxiv.org/abs/1910.01410), [doi:10.1016/j.jpaa.2020.106408](https://doi.org/10.1016/j.jpaa.2020.106408)]

[[!redirects stable model categories]]


stable orthogonal group

#Contents#
* table of contents
{:toc}

## Definition

For each $n \in \mathbb{N}$ there is an inclusion

$$
  O(n) \hookrightarrow O(n+1)
$$

of the [[orthogonal group]] in dimension $n$ into that in dimension $n+1$. The _stable orthogonal group_ is the [[direct limit]] over this sequence of inclusions.

$$
  O \coloneqq {\underset{\to}{\lim}}_n O(n)
  \,.
$$

## Properties

### Homotopy groups

By the discussion at _[orthogonal group -- homotopy groups](orthogonal%20group#HomotopyGroups)_ we have that the [[homotopy groups]] of the stable orthogonal group are

| $n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|
| $\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ |0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |

or if we instead write down the [[order of a group|order]]:

| $n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|
| ${\vert\pi_n(O)\vert}$ | 2 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |

Via the [[J-homomorphism]] this is related to the [[stable homotopy groups of spheres]]:

[[!include image of J -- table]]

## Related concepts

* [[stable unitary group]]

* [[J-homomorphism]]

* [[topological K-theory]]

[[!redirects stable orthogonal groups]]

[[!redirects stabe orthogonal group]]


stable pro-object

Given any category $C$, we can form the corresponding category of [[pro-object]]s in $C$, which is denoted by $pro$-$C$. Since the category $\mathbb{0}$ with one morphism is a [[filtered category|coflitered]] category, within $pro$-$C$, we have all pro-objects of the form $X: \mathbb{0}\to C$.  Clearly such a functor is completely determined by the single object, $X(*)$, of $C$ to which it corresponds.  This gives a functor:

$$c: C\to pro-C$$

which embeds the category $C$ in $pro$-$C$. (This is really the [[Yoneda embedding]] in disguise.)

###Definition###
Any pro-object isomorphic in $pro$-$C$ to one of the form, $c(X)$, for $X$ an object of $C$, is called **stable** or **essentially constant**.

###Stability problem###

In any given categorical context, the so-called __stability problem__ is the problem of deciding what internal criteria can be applied to check if a given pro-object in that context, is or is not stable,

If $C$ is an [[abelian category]], it is relatively simple to give necessary and sufficient 'internal' conditions for a given pro-object to be *essentially constant*. It must be both [[essentially epimorphic pro-object|essentially epimorphic]] and [[essentially monomorphic pro-object|essentially monomorphic]].


stable proposition

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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In ([[modal logic|modal]]) [[logic]], a [[proposition]] $p$ is __stable__ under a [[modality]] $\diamond$ if $p \equiv \diamond{p}$.  If $\diamond$ is [[monadic modality|monadic]], then $p$ is stable iff $p \equiv \diamond{q}$ for some $q$.

(In terms of [[modal type theory]] and [[propositions as types]] this means that $p$ is a _[[modal type]]_.)

## Examples

In [[intuitionistic logic]], the default is the [[double negation]] modality $\neg\neg$.  Since $p \Rightarrow \neg\neg{p}$ regardless, $p$ is stable iff $\neg\neg{p} \Rightarrow p$.  Being stable is weaker than being [[decidable proposition|decidable]]; however, if every proposition is stable, then every proposition is decidable and the logic becomes [[classical logic|classical]].  (This is because $p$ is decidable iff $p \vee \neg{p}$ is stable.)  Double negation is monadic, so by the previous paragraph, $p$ is stable iff $p \equiv \neg\neg{q}$ for some $q$; in fact, $p$ is stable iff $p \equiv \neg{q}$ for some $q$.  (I guess that this has to do with [[negation]] forming a [[monadic adjunction]] with itself, or something like that.)  In the [[topological semantics]] of intuitionistic logic (where propositions correspond to [[open sets]]), the stable propositions correspond to the [[regular open sets]].

In the [[internal language]] of a [[sheaf topos|category of sheaves]] $Sh(X)$, a proposition $p$ is $\neg\neg$-stable iff it is enough for $p$ to hold on a dense open subset of $X$ to be able to conclude that $p$ holds on the whole of $X$. For instance, the proposition $f = 0$ about a section $f$ of the sheaf of real functions on $X$ is $\neg\neg$-stable.

[[!redirects stable proposition]]
[[!redirects stable propositions]]

[[!redirects stable property]]
[[!redirects stable properties]]

[[!redirects stable predicate]]
[[!redirects stable predicates]]


stable relation

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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

\tableofcontents

## Idea

A relation whose [[double negation]] implies the relation itself.

## Definition

A stable [[binary relation]] $R$ on a set $S$ is a relation such that for all elements $a \in S$ and $b \in S$, $\neg \neg R(a, b)$ implies that $R(a, b)$.

## Examples

* Every [[decidable relation]] is a stable relation.

* The [[denial inequality relation]] of a set is a stable relation.

* In any [[inequality space]], [[equality]] is a stable relation.

* [[Markov's principle]] states that the [[pseudo-order]] for the [[modulated Cauchy real numbers]] is a stable relation.

* The [[analytic Markov's principle]] states that the [[pseudo-order]] for the [[Dedekind real numbers]] is a stable relation.

* Given a relation $R$, the negation of the relation $\neg R$ is a stable relation.

* In the presence of the [[double negation law]], every relation is a stable relation.

## See also

* [[stable proposition]]

* [[stable equality]]

* [[denial inequality]]


stable space

A __stable space__, in the sense of [[shape theory]], is a space for which the corresponding Cech nerve prosimplicial set, $\check{C}(X,-) : Cov(X)\to Ho(sSets$), is a [[stable pro-object]].


stable splitting of mapping spaces

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Goodwillie calculus
+--{: .hide}
[[!include Goodwillie calculus - contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[stabilization]]/[[suspension spectrum]] $\Sigma^\infty  Maps(X,A)$ of  [[mapping spaces]] $Maps(X,A)$ between suitable [[CW-complexes]] $X, A$ happens to decompose as a [[direct sum]] of [[spectra]] (a [[wedge sum]]) in a useful way, related to the expression of the [[Goodwillie derivatives]] of the functor $Maps(X,-)$ and often expressible in terms of the [[configuration space (mathematics)|configuration spaces]] of $X$.

## Definition
 {#Definition}

The stable splitting of mapping spaces discussed [below](#StableSplittings) have summands given by [[configuration spaces of points]], or generalizations thereof. To be self-contained, we recall the relevant definitions here.

The following Def. \ref{ConfigurationSpacesOfnPoints} is not the most general definition of [[configuration spaces of points]] that one may consider in this context, instead it is streamlined to certain applications. See Remark \ref{ComparisonToNotationInLiterature} below for comparison of notation used here to notation used elsewhere.

+-- {: .num_defn #ConfigurationSpacesOfnPoints}
###### Definition
**([[configuration spaces of points]])**

Let $X$ be a [[manifold]], possibly with [[manifold with boundary|boundary]].

For $n \in \mathbb{N}$, the _**configuration space of $n$ distinguishable points** in $X$ disappearing at the boundary_ is the [[topological space]]

\[
    \label{DistinguishableConfigurationSpaceJustForX}
    \mathrm{Conf}^{ord}_{n}(X)
    \;\coloneqq\;
      \big(
        X^n \setminus \mathbf{\Delta}_X^n
      \big)
      / \partial(X^n)
\]

which is the [[complement]] of the [[fat diagonal]] $\mathbf{\Delta}_X^n \coloneqq \{(x^i) \in X^n | \underset{i,j}{\exists} (x^i = x^j) \}$ inside the $n$-fold [[product space]] of $X$ with itself, followed by [[quotient space|collapsing]] any configurations with elements on the [[boundary]] of $X$ to a common [[pointed topological space|base point]].

Then the _**configuration space of $n$ in-distinguishable points** in $X$ is the further [[quotient topological space]]

  \[
    \label{ConfigurationSpaceJustForX}
    \mathrm{Conf}_{n}(X)
    \;\coloneqq\;
    Conf_n^{ord}(X)/\Sigma_n
    \;=\;
    \Big(
      \big(
        X^n \setminus \mathbf{\Delta}_X^n
      \big)
      / \partial(X^n)
    \Big)
    /\Sigma(n)
    \,,
  \]

where $\Sigma(n)$ denotes the evident [[action]] of the [[symmetric group]] by [[permutation]] of factors of $X$ inside $X^n$.

  More generally, let $Y$ be another [[manifold]], possibly with [[manifold with boundary|boundary]]. For $n \in \mathbb{N}$, the _**configuration space of $n$ points** in $X \times Y$ vanishing at the boundary and distinct as points in $X$_ is the [[topological space]]

  \[
    \label{ConfigurationSpaceWithXAndY}
    \mathrm{Conf}_{n}(X,Y)
    \;\coloneqq\;
    \Big(
    \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^n
    \big)
    / \partial(X^n \times Y^n)
    \Big)
    /\Sigma(n)
  \]

  where now $\Sigma(n)$ denotes the evident [[action]] of the [[symmetric group]] by [[permutation]] of factors of $X \times Y$ inside
  $X^n \times Y^n \simeq (X \times Y)^n$.

  This more general definition reduces to the previous case for $Y = \ast \coloneqq \mathbb{R}^0$ being the point:
  $$
    \mathrm{Conf}_n(X)
    \;=\;
    \mathrm{Conf}_n(X,\ast)
    \,.
  $$

Finally the _**configuration space of an arbitrary number of points** in $X \times Y$ vanishing at the boundary and distinct already as points of $X$_ is the [[quotient topological space]] of the [[disjoint union space]]

$$
  Conf\left( X, Y\right)
  \;\coloneqq\;
  \left(
  \underset{n \in \mathbb{n}}{\sqcup}
      \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^k
    \big)
    /\Sigma(n)
  \right)/\sim
$$

by the [[equivalence relation]] $\sim$ given by

$$
  \big(
    (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n)
  \big)
  \;\sim\;
  \big(
    (x_1, y_1), \cdots, (x_{n-1}, y_{n-1})
  \big)
  \;\;\;\; \Leftrightarrow
  \;\;\;\; (x_n, y_n) \in \partial (X \times Y)
  \,.
$$

This is naturally a [[filtered topological space]] with filter stages

$$
  Conf_{\leq n}\left( X, Y\right)
  \;\coloneqq\;
  \left(
  \underset{k \in \{1, \cdots, n\}}{\sqcup}
      \big(
      (
        X^k \setminus \mathbf{\Delta}_X^k
      )
      \times
      Y^k
    \big)
    /\Sigma(k)
  \right)/\sim
  \,.
$$

The corresponding [[quotient topological spaces]] of the filter stages reproduces the above configuration spaces of a fixed number of points:

$$
  Conf_n(X,Y)
  \;\simeq\;
  Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y)
  \,.
$$

=--

+-- {: .num_remark #ComparisonToNotationInLiterature}
###### Remark
**(comparison to notation in the literature)**

The above Def. \ref{ConfigurationSpacesOfnPoints} is less general but possibly more suggestive than what is considered for instance in [Bödigheimer 87](#Boedigheimer87). Concretely, we have the following translations of notation:

  $$
    \array{
      \text{ here: }
        &&
      \array{ \text{ Segal 73,} \\ \text{ Snaith 74}: }
        &&
      \text{ Bödigheimer 87: }
      \\
      \\
      Conf(\mathbb{R}^d,Y)
        &=&
      C_d( Y/\partial Y )
        &=&
      C( \mathbb{R}^d, \emptyset; Y )
      \\
      \mathrm{Conf}_n\left( \mathbb{R}^d \right)
      & = &
      F_n C_d( S^0 ) / F_{n-1} C_d( S^0 )
      & = &
      D_n\left( \mathbb{R}^d, \emptyset; S^0  \right)
      \\
      \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right)
      & = &
      F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y )
      & = &
      D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y  \right)
      \\
      \mathrm{Conf}_n( X ) && &=& D_n\left( X, \partial X; S^0  \right)
      \\
      \mathrm{Conf}_n( X, Y  ) && &=& D_n\left( X, \partial X; Y/\partial Y  \right)
    }
  $$
  Notice here that when $Y$ happens to have [[empty space|empty]] [[boundary]], $\partial Y = \emptyset$, then the [[pushout]]

$$
  Y / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast
$$

is $Y$ with a [disjoint basepoint attached](pointed+topological+space#ForgettingAndAdjoiningBasepoints). Notably for $Y =\ast$ the [[point space]], we have that

$$
  \ast/\partial \ast = S^0
$$

is the [[0-sphere]].

=--

## Statements
 {#Statements}

### Prelude: Equivalence to the infinite configuration space

First recall the following equivalence already before [[stabilization]]:

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

For

1. $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],

1. $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that  $Y / \partial Y$ is [[connected topological space|connected]],

the [[scanning map]] constitutes a [[homotopy equivalence]]

$$
  Conf\left(
    \mathbb{R}^d, Y
  \right)
  \overset{scan}{\longrightarrow}
  \Omega^d \Sigma^d (Y/\partial Y)
$$

between

1. the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. \ref{ConfigurationSpacesOfnPoints})

1. the $d$-fold [[loop space]] of the $d$-fold [[reduced suspension]] of the [[quotient space]] $Y / \partial Y$ (regarded as a [[pointed topological space]] with basepoint $[\partial Y]$).

In particular when $Y = \mathbb{D}^k$ is the [[closed ball]] of [[dimension]] $k \geq 1$ this gives a [[homotopy equivalence]]

$$
  Conf\left(
    \mathbb{R}^d, \mathbb{D}^k
  \right)
  \overset{scan}{\longrightarrow}
  \Omega^d S^{ d + k }
$$

with the $d$-fold [[loop space]] of the [[n-sphere|(d+k)-sphere]].

=--

([May 72, Theorem 2.7](#May72),  [Segal 73, Theorem 3](#Segal73))

### Stable splitting of mapping spaces
 {#StableSplittings}

+-- {: .num_prop #StableSplittingOfMappingSpacesOutOfEuclideanSpace}
###### Proposition
**([[stable splitting of mapping spaces]] out of [[Euclidean space]]/[[n-spheres]])**

For

1. $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],

1. $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that  $Y / \partial Y$ is [[connected topological space|connected]],

there is a [[stable weak homotopy equivalence]]

$$
  \Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
$$

between

1. the [[suspension spectrum]] of the [[configuration space of points|configuration space]] of an arbitrary number of points in $\mathbb{R}^d \times Y$ vanishing at the boundary and distinct already as points of $\mathbb{R}^d$ (Def. \ref{ConfigurationSpacesOfnPoints})

1. the [[direct sum]] (hence: [[wedge sum]]) of [[suspension spectra]] of the [[configuration space of points|configuration spaces]] of a fixed number of points in $\mathbb{R}^d \times Y$, vanishing at the boundary and distinct already as points in $\mathbb{R}^d$ (also Def. \ref{ConfigurationSpacesOfnPoints}).

Combined with the [[stabilization]] of the [[scanning map]] [[homotopy equivalence]] from Prop. \ref{ScanningMapEquivalenceOverCartesianSpace} this yields a [[stable weak homotopy equivalence]]

\[
  \label{StableSplittingOfMappingSpacesOutOfSphere}
  \Sigma^\infty
  \,
  \Omega^d \Sigma^d (Y/\partial Y)
  \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow}
  \Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
\]

between the latter direct sum and the [[suspension spectrum]] of the [[mapping space]] of pointed [[continuous functions]] from the [[n-sphere|d-sphere]] to the $d$-fold [[reduced suspension]] of $Y / \partial Y$.

=--

([Snaith 74, theorem 1.1](#Snaith74), [Bödigheimer 87, Example 2](#Boedigheimer87))

In fact, by [Bödigheimer 87, Example 5](#Boedigheimer87) this equivalence still holds with $Y$ treated on the same footing as $\mathbb{R}^d$, hence with $Conf_n(\mathbb{R}^d, Y)$ on the right replaced by $Conf_n(\mathbb{R}^d \times Y)$ in the well-adjusted notation of Def. \ref{ConfigurationSpacesOfnPoints}:

$$
  \Sigma^\infty
  \,
  Maps_{cp}\big(\mathbb{R}^d, \Sigma^d (Y / \partial Y)\big)
  =
  \Sigma^\infty
  \,
  Maps^{\ast/}\big( S^d, \Sigma^d (Y / \partial Y)\big)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus}
  \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)
  \,.
$$

### In terms of Goodwillie-Taylor towers
 {#InTermsOfGoodwillieTowers}

We discuss the interpretation of the above stable splitting of mapping spaces from the point of view of [[Goodwillie calculus]], following [Arone 99, p. 1-2](#Arone99), [Goodwillie 03, p. 6](#Goodwillie03).

Observe that the [[configuration space of points]] $Conf_n(X,Y)$ from Def. \ref{ConfigurationSpacesOfnPoints}, given by the formula (eq:ConfigurationSpaceWithXAndY)

$$
  Conf_n(X,Y)
  \;\coloneqq\;
  \Big(
    \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^n
    \big)
    / \partial(X^n \times Y^n)
  \Big)
  /\Sigma(n)
$$

is the [[quotient]] by the [[symmetric group]]-[[action]] of the _[[smash product]]_ $Conf_n(X) \wedge (Y/\partial Y)^n$ of the plain Configuration space $Conf_n(X)$  (eq:ConfigurationSpaceJustForX) (regarded as a [[pointed topological space]] with basepoint the class of the [[boundary]] $\left[\partial\left(X^n\right)\right]$) with the analogous [[pointed topological space]] given by $Y$, the latter in fact being (since here we do not form the [[complement]] by the [[fat diagonal]]) an $n$-fold [[smash product]] itself:

$$
  Y^{\times_n}/\partial (Y^{\times_n})
  \;\simeq\;
  ( Y/\partial Y )^{\wedge_n}
  \,.
$$

Hence in summary:

\[
  \label{ConfSplitsAsSmashProduct}
  Conf_n(X, Y)
  \;\simeq\;
  Conf^{ord}_n(X) \wedge_{\Sigma(n)} \left( Y/\partial Y \right)^{\wedge_n}
  \,,
\]

where

$$
  Conf_n^{ord}(X)
  \;\coloneqq\;
  \left(
    X^{\times_n} \setminus \mathbf{\Delta}_X^n
  \right)/ \partial(X^n)
$$

is the ordered configuration space (eq:DistinguishableConfigurationSpaceJustForX).

This construction, regarded as a [[functor]] from [[pointed topological spaces]] to [[spectra]]

$$
  \array{
    Top^{\ast/}
    &\longrightarrow&
    Spectra
    \\
    Z
      &\mapsto&
    \Sigma^\infty Conf^{ord}_n(X) \wedge_{\Sigma(n)} Z^{\wedge_n}
  }
$$

is an [[n-homogeneous (∞,1)-functor]] in the sense of [[Goodwillie calculus]], and hence the partial [[wedge sums]] as $n$ ranges

\[
  \label{IdentifyingTheGoodwillieTaylorStage}
  Z
    \;\mapsto\;
  \underset{k \in \{1, \cdot, n\}}{\bigoplus}
  \Sigma^\infty Conf^{ord}_k(X) \wedge_{\Sigma(k)} Z^{\wedge_k}
\]

are [[n-excisive (∞,1)-functors]]. Moreover, by the stable splitting of mapping spaces (eq:StableSplittingOfMappingSpacesOutOfSphere) of Prop. \ref{StableSplittingOfMappingSpacesOutOfEuclideanSpace}, there is a [[projection]] morphism onto the first $n$ [[wedge sum|wedge summands]]

\[
  \label{ProjectionMaps}
  \array{
    Maps_{cp}(\mathbb{R}^d, \Sigma^d Z)
    &=&
    Maps^{\ast/}( S^d, \Sigma^d Z)
    &\simeq&
    \underset{k \in \mathbb{N}}{\oplus}
      \Sigma^\infty Conf^{ord}_k(\mathbb{R}^d) \wedge_{\Sigma(k)} Z^{\wedge_k}
    \\
    &&
    &&
    \Big\downarrow {}^{\mathrlap{ p_n }}
    \\
    &&
    &&
     \underset{k \in \{1, \cdot, n\}}{\bigoplus}
     \Sigma^\infty Conf^{ord}_k( \mathbb{R}^d ) \wedge_{\Sigma(k)} Z^{\wedge_k}
  }
\]

and this is [[n-connected morphism|(n+1)k-connected]] when $Z$ is [[n-connected object|k-connected]].

By [[Goodwillie calculus]] this means that (eq:IdentifyingTheGoodwillieTaylorStage) are, up to [[equivalence in an (infinity,1)-category|equivalence]], the stages

\[
  \label{TheGoodwillieStagesOfTheMappingSpaceFunctor}
  P_n Maps^{\ast/}( S^d, \Sigma^d (-))
  \;\colon\;
  Z \mapsto
  \underset{k \in \{1, \cdot, n\}}{\bigoplus}
  \Sigma^\infty Conf^{ord}_k(S^d, Z)
\]

at $Z \in Top^{\ast/}$ of the [[Goodwillie-Taylor tower]] for the [[mapping space]]-functor

$$
  Maps_{cp}(\mathbb{R}^d, \Sigma^d (-))
  =
  Maps^{\ast/}( S^d, \Sigma^d (-))
  \;\colon\;
  Top^{\ast/} \longrightarrow Top^{\ast/}
  \,.
$$

Therefore the stable splitting theorem \ref{StableSplittingOfMappingSpacesOutOfEuclideanSpace} may equivalently be read as expressing the mapping space functor as the [[limit]] over its [[Goodwillie-Taylor tower]].

([Arone 99, p. 1-2](#Arone99), [Goodwillie 03, p. 6](#Goodwillie03))

$\,$

### Lax closed structure on $\Sigma^\infty$

Notice that the first stage in the [[Goodwillie-Taylor tower]] of $Maps(S^d, \Sigma^d(-))$ is

$$
  \begin{aligned}
    P_1 Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y) )
    & =
    \Sigma^\infty Conf^{ord}_1( \mathbb{R}^d , Y )
    \\
    & \simeq
    \Sigma^\infty
      \underset{\simeq S^0}{\underbrace{Conf^{ord}_1( \mathbb{R}^d )}}
      \wedge (Y/\partial Y)
    \\
    & \simeq
     \Sigma^\infty (Y/\partial Y)
    \\
    & \simeq
     \Omega^d \Sigma^d  \Sigma^\infty (Y/\partial Y)
    \\
    & \simeq
    Maps\left(  \Sigma^\infty S^d, \Sigma^d (Y/\partial Y) \right)
  \end{aligned}
$$

Here in the first step we used (eq:TheGoodwillieStagesOfTheMappingSpaceFunctor), in the second step we used (eq:ConfSplitsAsSmashProduct). Under the brace we observe that space of configurations of a single point in $\mathbb{R}^d$ is trivially $\mathbb{R}^d$ itself, which is [[contractible topological space|contractible]] $\mathbb{R}^d \simeq \ast$ and, due to [[empty set|empty]] [[boundary]] of $\mathbb{R}^d$, contributes a [[0-sphere]]-factor to the [[smash product]], which disappears. In the last last two steps we trivially rewrote the result to exhibit it as a [[mapping spectrum]].

Therefore the projection $p_1$ (eq:ProjectionMaps) to the first stage of the [[Goodwillie-Taylor tower]] is of the form

$$
  p_1
  \;\colon\;
  \Sigma^\infty Maps\left(  S^d , \Sigma^d (Y /\partial Y) \right)
  \longrightarrow
  Maps
  \left(
    \Sigma^\infty S^d, \Sigma^\infty \Sigma^d (Y / \partial Y)
  \right)
  \,.
$$

Since $\Sigma^\infty$ is a [[strong monoidal functor]] ([here](suspension+spectrum#StrongMonoidalness)), there is a canonical comparison morphism of this form, exhibiting the induce [[closed functor|lax closed]]-structure on $\Sigma^\infty$. Probably $p_1$ coincides with that canonical morphism, up to equivalence.

> Does it?

## Related concepts

* [[mapping spectrum]]

## References

The theorem is originally due to

* {#Snaith74} [[Victor Snaith]],  _A stable decomposition of $\Omega^n S^n X$_, Journal of the London Mathematical Society 7 (1974), 577 - 583 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/snaiths.pdf))

using the homotopy equivalence before stabilization due to

* {#May72} [[Peter May]], _The geometry of iterated loop spaces_, Springer 1972 ([pdf](https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf))

* {#Segal73} [[Graeme Segal]], _Configuration-spaces and iterated loop-spaces_, Invent. Math. __21__ (1973), 213–221. MR 0331377 ([pdf](http://dodo.pdmi.ras.ru/~topology/books/segal.pdf))

An alternative proof is due to

* {#Cohen80} [[Ralph Cohen]], _Stable proof of stable splittings_, Math. Proc. Camb. Phil. Soc., 1980, 88, 149 ([doi:10.1017/S030500410005742X](https://doi.org/10.1017/S030500410005742X), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/247D9F951F8AB99000E4FF6CB2DB9EA2/S030500410005742Xa.pdf/div-class-title-stable-proofs-of-stable-splittings-div.pdf))

Review and generalization:

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], *Stable splittings of mapping spaces*, in: *Algebraic topology*, Lecture Notes in Mathematics **1286**, Springer (1987) 174-187 [[pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf), [[BoedigheimerStableSplittings87.pdf:file]], [doi:10.1007/BFb0078741](https://doi.org/10.1007/BFb0078741)]

Interpretation in terms of the [[Goodwillie-Taylor tower]] of mapping spaces is due to

* {#Arone99} [[Greg Arone]], _A generalization of Snaith-type filtration_, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. ([pdf](https://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02405-8/S0002-9947-99-02405-8.pdf))

* {#Ching05} [[Michael Ching]], _Calculus of Functors and Configuration Spaces_, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 ([pdf](https://www3.amherst.edu/~mching/Work/skye.pdf))

* {#Goodwillie03} [[Thomas Goodwillie]], p. 6 of _Calculus. III. Taylor series_, Geom. Topol. 7 (2003), 645--711 ([journal](http://www.msp.warwick.ac.uk/gt/2003/07/p019.xhtml), [arXiv:math/0310481](http://arxiv.org/abs/math/0310481)))

A proof via [[nonabelian Poincaré duality]]:

* Lauren Bandklayder, _Stable splitting of mapping spaces via nonabelian Poincaré duality_ ([arxiv:1705.03090](https://arxiv.org/abs/1705.03090))

See also:

* [[Douglas Ravenel]], *What we still don't understand about loop spaces of spheres*, Contemporary Mathematics 1998 ([pdf](https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf), [[Ravenel_LoopSpacesOfSpheres.pdf:file]])

[[!redirects stable splittings of mapping spaces]]


stable tangent bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

By the _stable tangent bundle_ of a [[smooth manifold]] one means the [[Whitney sum]] of the actual [[tangent bundle]] with any [[trivial vector bundle]].

Specifically, for $\{G_k\}_{k \in \mathbb{N}}$ a sequence of [[Lie groups]], a "G-structure on the stable tangent bundle" is an actual [[G-structure|Gk-structure]] on any such direct sum $T X \oplus \underline{\mathbb{R}}^n$, for some $n \in \mathbb{N}$.

In particular, for $G = 1$ the [[trivial group]], a stable [[frame manifold|framing]] is a [[trivial bundle|trivialization]] of the stable tangent bundle.

## Examples

* [[stable complex structure]]

* ...

## Related concepts

* [[stable normal bundle]]

* [[cobordism theory]]

[[!redirects stable tangent bundles]]

[[!redirects stable framing]]
[[!redirects stable framings]]


stable unitary group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

For $n \in \mathbb{N}$ write $U(n)$ for the [[unitary group]] in dimension $n$ and $O(n)$ for the [[orthogonal group]] in dimension $n$, both regarded as [[topological group]]s in the standard way. Write $B U(n) , B O(n) \in$ [[Top]] for the corresponding [[classifying space]].

Write

$$
  [X, B O(n)] := \pi_0 Top(X, B O(n))
$$

and

$$
  [X, B U(n)] := \pi_0 Top(X, B U(n))
$$

for the set of [[homotopy]]-classes of [[continuous function]]s $X \to B U(n)$.

=--

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

This is equivalently the set of [[isomorphism]] classes of $O(n)$- or $U(n)$-[[principal bundles]]  on $X$ as well as of rank-$n$ real or complex [[vector bundles]] on $X$, respectively:

$$
  [X, B O(n)] \simeq O(n) Bund(X) \simeq Vect_{\mathbb{R}}(X,n)
  \,,
$$

$$
  [X, B U(n)] \simeq U(n) Bund(X) \simeq Vect_{\mathbb{C}}(X,n)
  \,.
$$

=--

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

For each $n$ let

$$
  U(n) \to U(n+1)
$$

be the inclusion of [[topological group]]s given by inclusion of $n \times n$ [[matrices]] into $(n+1) \times (n+1)$-matrices given by the block-diagonal form

$$
  \left[g\right] \mapsto
  \left[
    \array{
      1 & [0]
      \\
      [0] & [g]
    }
  \right]
  \,.
$$

This induces a corresponding sequence of morphisms of classifying spaces, def. \ref{BUn}, in [[Top]]

$$
  B U(0) \hookrightarrow B U(1) \hookrightarrow B U(2) \hookrightarrow \cdots
  \,.
$$

Write

$$
  B U
  \;\coloneqq\;
  {\lim_{\to}}_{n \in \mathbb{N}} B U(n)
$$

for the [[homotopy colimit]] (the "homotopy [[direct limit]]") over this diagram (see at _[[homotopy colimit]]_ the section _[Sequential homotopy colimits](homotopy+limit#SequentialHocolims)_).

=--

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

The [[topological space]] $B U$ is **not** equivalent to $B U(\mathcal{H}) $, where [[U(ℋ)|$U(\mathcal{H})$]] is the [[unitary group]] on an infinite-dimensional [[separable Hilbert space|separable]] [[Hilbert space]] $\mathcal{H}$. In fact the latter is [[contractible]], hence has a  [[weak homotopy equivalence]] to the point

$$
  B U(\mathcal{H}) \simeq *
$$

while $B U$ has nontrivial [[homotopy group]]s in arbitrary higher degree (by [[Kuiper's theorem]]).

But there is the [[subgroup]] $U(\mathcal{H})_{\mathcal{K}} \subset U(\mathcal{H})$ of unitary operators that differ from the [[identity]] by a [[compact operator]]. This is essentially $U = \Omega B U$. See [below](#Uk).

=--

## Properties

### Classifying space for topological K-theory

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

Write $\mathbb{Z}$ for the set of [[integers]] regarded as a [[discrete topological space]].

The product spaces

$$
  B O \times \mathbb{Z}\,,\;\;\;\;\;B U \times \mathbb{Z}
$$

are [[classifying spaces]] for real and complex [[topological K-theory]], respectively: for every compact Hausdorff topological space $X$, we have an isomorphism of groups

$$
  \tilde K(X)
  \simeq
  [X, B U ]
  \,.
$$

$$
  K(X)
  \simeq
  [X, B U \times \mathbb{Z}]
  \,.
$$

=--

See for instance ([Friedlander, prop. 3.2](#Friedlander)) or ([Karoubi, prop. 1.32, theorem 1.33](#Karoubi)).

+-- {: .proof}
###### Proof

First consider the statement for reduced cohomology $\tilde K(X)$:

Since a [[compact topological space]] is a [[compact object]] in [[Top]] (and using that the [[classifying space]]s $B U(n)$ are (see there) [[paracompact topological space]]s, hence normal, and since the inclusion morphisms are closed inclusions (...)) the [[hom-functor]] out of it commutes with the [[filtered colimit]]

$$
  \begin{aligned}
    Top(X, B U)
    &=
    Top(X, {\lim_\to}_n B U(n))
    \\
    & \simeq {\lim_\to}_n Top(X, B U (n))
  \end{aligned}
  \,.
$$

Since $[X, B U(n)] \simeq U(n) Bund(X)$, in the last line the colimit is over [[vector bundle]]s of all ranks and identifies two if they become isomorphic after adding a trivial bundle of some finite rank.

For the full statement use that by prop. \ref{missing} we have

$$
  K(X) \simeq H^0(X, \mathbb{Z}) \oplus \tilde K(X)
  \,.
$$

Because $H^0(X,\mathbb{Z}) \simeq [X, \mathbb{Z}]$ it follows that

$$
  H^0(X, \mathbb{Z}) \oplus \tilde K(X)
  \simeq
  [X, \mathbb{Z}] \times [X, B U]
  \simeq
  [X, B U \times \mathbb{Z}]
  \,.
$$

=--

There is another variant on the classifying space

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

Let

$$
  U_{\mathcal{K}}
  =
  \left\{
    g \in U(\mathcal{H}) | g - id \in \mathcal{K}
  \right\}
$$

be the group of unitary operators on a [[separable Hilbert space]] $\mathcal{H}$ which differ from the identity by a [[compact operator]].

=--

Palais showed that:

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

$U_\mathcal{K}$ is a [[homotopy equivalence|homotopy equivalent]] model for $B U$. It is in fact the [[norm closure]] of the evident model of $B U$ in [[U(ℋ)|$U(\mathcal{H})$]].

Moreover $U_{\mathcal{K}} \subset U(\mathcal{H})$ is a [[Banach Lie group|Banach Lie]] [[normal subgroup]].

=--

Since $U(\mathcal{H})$ is [[contractible]], it follows that

$$
  B U_{\mathcal{K}} \coloneqq U(\mathcal{H})/U_{\mathcal{K}}
$$

is a model for the [[classifying space]] of reduced complex [[topological K-theory]] [[KU]].

## Related concepts

* [[stable orthogonal group]]

* [[U(ℋ)]], [[PU(ℋ)]]

* [[topological K-theory]] [[KU]]

* [[unstable topological K-theory]]

## References

Discussion in the context of [[topological K-theory]]:

* {#Hatcher} [[Allen Hatcher]], _Vector bundles and K-theory_ ([web](http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))

* {#Friedlander} [[Eric Friedlander]], _An introduction to K-theory_ ([pdf](http://users.ictp.it/~pub_off/lectures/lns023/Friedlander/Friedlander.pdf))

* {#Karoubi} [[Max Karoubi]], _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], _[[Spin geometry]]_, Princeton University Press (1989)

On the [[H-space]] structure on $B U \times \mathbb{Z}$:

* [[Peter May]],  p. 205 (213 of 251) in: _[[A Concise Course in Algebraic Topology]]_.

[[!redirects stable unitary groups]]


stable vector bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[vector bundle]] (typically considered in [[complex-analytic geometry]] or [[algebraic geometry]]) is called _(semi-)stable_ if it is a _[[GIT-stable point|(semi-)stable point]]_ in the [[moduli space of bundles]] in the sense of [[geometric invariant theory]].

Under suitable conditions this is equivalent to a certain [[inequality]] on the [[slope of a coherent sheaf|slopes]] of the sub-bundles (see [below](#OverARiemannSurface)), and this inequality is what tends to be stated as the definition of stability of vector bundles.

For more discussion (informal and formal) of this concept of stability see at _[[Bridgeland stability condition]]_.

## Definition

### Over a Riemann surface / over an algebraic curve
 {#OverARiemannSurface}

For $\Sigma$ a [[Riemann surface]], a [[complex vector bundle]] $E \to \Sigma$ over $\Sigma$ is called **(slope-)stable** if for all non-trivial subbundles $K \hookrightarrow E$ the [[inequality]]

$$
  \mu(K) \lt \mu(E)
$$

between their [[slope of a coherent sheaf|slopes]] holds, i.e. if the inequality

$$
  \frac{deg(K)}{rank(K)}
  \lt
  \frac{deg(E)}{rank(E)}
$$

holds between the fractions of [[degree of a coherent sheaf|degree]] and [[rank]] of the vector bundles  holds.

e.g. ([Huybrechts-Lehn 96, bottom of p. 24](#HuybrechtsLehn96))

### Over a general (Noetherian) scheme

e.g. ([Huybrechts-Lehn 96, def. 1.2.4, def. 1.2.12](#HuybrechtsLehn96))

## Examples

Every [[line bundle]] is slope-stable. The [[extension]] of a degree-0 line bundle by a degree-1 line bundle is stable.

e.g. ([Huybrechts-Lehn 96, example 1.2.10](#HuybrechtsLehn96))

## Properties

### Relation to GIT-stability
 {#RelationToGITStability}

The slope-(semi-)stable vector bundles are essentially the [[stable point|(semi-)stable points]] in the sense of [[geometric invariant theory]] in the [[moduli space of bundles]]. The precise statement is discussed in ([King 94](#King94)) reviewed for instance in ([Saiz 09, section 2.3](#Saiz09).

### Relation to connections

The [[Narasimhan–Seshadri theorem]] identifies [[moduli spaces]] of stable vector bundles over [[complex curves]] with those of certain [[flat connections]].

The [[Donaldson-Uhlenbeck-Yau theorem]] relates semi-stable vector bundles over [[Kähler manifolds]] to [[Hermite-Einstein connections]].

Still more generally, the [[Kobayashi-Hitchin correspondence]] relates semi-stable vector bundles over [[complex manifolds]] to Hermite-Einstein connections.

### Relation to Bridgeland stability conditions

Slope-stability of vector bundles is a special case of a [[Bridgeland stability condition]], see [there](Bridgeland+stability+condition#SlopeStabilityOfVectorBundles) For review see e.g. ([Engenhorst 14, sections 3 and 4](#Engenhorst14)) and see [King 94](#King94).

## Related concepts

* [[Harder-Narasimhan filtration]]

* [[moduli space of bundles]]

* [[stable point]]

* [[positive line bundle]], [[ample line bundle]]

* [[Bridgeland stability condition]]

* [[geometric invariant theory]]

## References

The concept was introduced in

* [[David Mumford]], _Geometric invariant theory_, Ergebnisse Math. Vol 34 Springer (1965)

* F. Takemoto, _Stable vector bundles on algebraic surfaces_, Nagoya Math. J.  47 (1972) 29-48 ([euclid](http://projecteuclid.org/euclid.nmj/1118798682)); _II_, 52 (1973) ([euclid](http://projecteuclid.org/euclid.nmj/1118794885))

*  {#MumfordFogartyKirwan65} [[David Mumford]], John Fogarty, Frances Clare Kirwan, _Geometric invariant theory_, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) __34__, Springer-Verlag (1965)

Review is in

* {#AtiyahBott83} [[Michael Atiyah]], [[Raoul Bott]], section 7 of _The Yang-Mills equations over Riemann surfaces_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 ([jstor](http://www.jstor.org/stable/37156), [lighning summary](http://math.stackexchange.com/a/295505/58526))

A textbook account is in

* {#HuybrechtsLehn96} [[Daniel Huybrechts]], [[Manfred Lehn]], _The Geometry of the Moduli Spaces of Sheaves_, 1996 ([[HuybrechtsLehn.pdf:file]])

More discussion with regards to [[geometric invariant theory]] and [[Bridgeland stability conditions]] is in

* {#King94} [[Alastair King]], _Moduli of representations of finite dimensional algebras_, The Quarterly Journal of Mathematics 45.4 (1994): 515-530 ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.623.649&rep=rep1&type=pdf))

* {#Engenhorst14} Jan Engenhorst, _Bridgeland Stability Conditions in Algebra, Geometry and Physics_, 2014 ([pdf](https://www.freidok.uni-freiburg.de/fedora/objects/freidok:9595/datastreams/FILE1/content))

See also

* Paolo de Bartolomeis, [[Gang Tian]], _Stability of complex vector bundles_, J. Diff. Geometry __43__:2 (1996) ([pdf](https://www.intlpress.com/JDG/archive/1996/43-2-231.pdf))

* Wei-Ping Li, Zhenbo Qin, _Stable vector bundles on algebraic surfaces_ ([pdf](http://arxiv.org/pdf/math/9411233.pdf))

* {#Saiz09} Alfonso Zamora Saiz, _On the stability of vector bundles_, Master thesis 2009 ([[SaizStableBundles.pdf:file]])

Discussion for [[equivariant vector bundles]] is in

* [[C. S. Seshadri]], _Moduli of $\pi$-Vector Bundles over an Algebraic Curve_, In: Marchionna E. (eds) Questions on Algebraic Varieties. C.I.M.E. Summer Schools, vol 51. Springer, Berlin, Heidelberg ([doi:10.1007/978-3-642-11015-3_5](https://doi.org/10.1007/978-3-642-11015-3_5))

* Oscar García-Prada, _Invariant connections and vortices_,  Commun.Math. Phys. (1993) 156: 527 ([doi:10.1007/BF02096862](https://doi.org/10.1007/BF02096862))

[[!redirects stable vector bundles]]

[[!redirects stable coherent sheaf]]
[[!redirects stable coherent sheaves]]


stable weak homotopy equivalence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The analog in [[stable homotopy theory]]  of [[weak homotopy equivalences]] in classical [[homotopy theory]].

The stable weak equivalences of [[sequential spectra]] in [[simplicial sets]] form the weak equivalences in the [[Bousfield-Friedlander model structure]] for [[stable homotopy theory]].

Beware that for other types of spectra there may be subtle corrections to this statement. For instance for [[symmetric spectra]] the maps that are stable weak equivalences on the underlying sequential spectra are guaranteed to be weak equivalences in the [[model structure on symmetric spectra]] only on [[semistable symmetric spectra]].

## Definition

### For sequential spectra

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

The [[stable homotopy groups]] of a [[sequential spectrum]] $X$, is the $\mathbb{Z}$-[[graded abelian groups]] given by the [[colimit]] of [[homotopy groups]] of the component spaces (or of their [[geometric realization]] if they are given as [[simplicial sets]])

$$
  \pi_\bullet(X)
  \coloneqq
  \underset{\longrightarrow}{\lim}_k
  \pi_{\bullet+k}({ X_n })
  \,.
$$

This constitutes a [[functor]]

$$
  \pi_\bullet \;\colon\; SeqSpec(sSet) \longrightarrow Ab^{\mathbb{Z}}
  \,.
$$

=--

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

A morphism $f \colon X \longrightarrow Y$ of [[sequential spectra]], is called a [[stable weak homotopy equivalence]], if its image under the [[stable homotopy group]]-functor of def. \ref{StableHomotopyGroups} is an [[isomorphism]]

$$
  \pi_\bullet(f) \;\colon\; \pi_\bullet(X) \longrightarrow \pi_\bullet(Y)
  \,.
$$

=--

## Properties

### Closure properties

(e.g. [MMSS 00, theorem 7.4 ](#MMSS00))

## References

* {#MMSS00} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society, 82 (2001), 441-512 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf))

[[!redirects stable weak homotopy equivalences]]


stably compact space

[[!redirects stably compact]]
[[!redirects stably compact]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

Stably compact spaces are [[topological spaces]] which share many of the desirable properties of [[compact Hausdorff spaces]], such as [[compactness]] and [[local compactness]], without being [[Hausdorff]] or even [[T1]].

They are also a convenient setting for [[convergence]] in an [[order theory|ordered]] setting, being deeply linked to [[compact ordered spaces]].

## Definition

A [[topological space]] $X$ is called **stably compact** if the following conditions are met:

* $X$ is [[T0]];
* $X$ is [[compact]];
* $X$ is [[locally compact]];
* $X$ is [[sober]];
* $X$ is **coherent**, meaning that the intersection of two [[compact]] [[saturated set|saturated]] subsets is compact.

Note that the latter notion of coherence is slightly different than the one given at [[coherent space]].

## Connection with compact ordered spaces

(...)

## Connection with compact regular bitopological spaces

(...) For details see ([Jung-Moshier 2006](#jungmoshier06)).

## Related concepts

* [[compact ordered space]]
* [[compact Hausdorff space]], [[compact space]], [[locally compact space]]

## References

* {#jung2} Achim Jung, _Stably compact spaces and the probabilistic powerspace construction_, ENTCS 87, 2004 ([doi:10.1016/j.entcs.2004.10.001](https://doi.org/10.1016/j.entcs.2004.10.001)).

* {#jungmoshier06} Achim Jung and M. Andrew Moshier. _On the bitopological nature of Stone duality._  Technical Report CSR-06-13. School of Computer Science, University of Birmingham, December 2006, 110 pages. $[$[pdf](https://www.cs.bham.ac.uk/~axj/pub/papers/Jung-Moshier-2006-On-the-bitopological-nature-of-Stone-duality.pdf)$]$

[[!redirects stably compact spaces]]
[[!redirects stably compact]]
[[!redirects stably compact topological space]]
[[!redirects stably compact topological spaces]]
[[!redirects stable compactness]]


stably locally connected topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[sheaf topos]] $\mathcal{E}$ is called **stably locally connected** if it is a [[locally connected topos]]

$$
  (\Pi_0 \dashv \Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\longrightarrow}}{\stackrel{\overset{\Delta}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}} Set
$$

such that the extra [[left adjoint]] $\Pi_0$ in addition preserves finite [[product]]s (the [[terminal object]] and binary products).

This means it is in particular also a [[connected topos]].

If $\Pi_0$ preserves even all [[finite limit]]s then $\mathcal{E}$ is called a [[totally connected topos]].

If a stably locally connected topos is also a [[local topos]], then it is a [[cohesive topos]].

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * **stably locally connected topos** / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]], [[locally ∞-connected site]]

* [[connected site]]

* [[local site]]

* [[cohesive site]]

## References

* [[Peter Johnstone]]. _Remarks on punctual local connectedness_. Theory and Application of Categories 25.3 (2011): 51-63. ([TAC](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html))

[[!redirects stably locally connected toposes]]
[[!redirects stably locally connected topoi]]
[[!redirects stably locally connected geometric morphism]]

[[!redirects strongly connected topos]]
[[!redirects strongly connected toposes]]
[[!redirects strongly connected topoi]]
[[!redirects strongly connected geometric morphism]]


stack

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The term _stack_, is a traditional synonym for _[[2-sheaf]]_ or often, more restrictively, as a synonym for _[[(2,1)-sheaf]]_ (see there for more details).

This is part of a whole hierarchy of [[higher category theory|higher categorical]] generalizations of the notion of _[[sheaf]]_. A [[(2,1)-sheaf]] / stack is equivalently a [[1-truncated]] [[(∞,1)-sheaf]]/[[∞-stack]].

Generally, then, _[[n-stack]]_ is a synonym for _[[n-sheaf|(n+1)-sheaf]]_, or more restrictively for _[[(n,1)-sheaf|(n+1,1)-sheaf]]_.

More concretely this means that a 1-stack on a [[site]] $S$ (or more generally [[(2,1)-site]] or even [[(2,2)-site]] $S$) is

* a ([[pseudofunctor|pseudo]]-)[[functor]] from  $S^{op}$ to the 2-category [[Cat]] of categories,

* that satisfies [[descent]] for all covers.

If the pseudofunctor takes values in the 2-subcategory [[Grpd]]$\subset Cat$ of groupoids, the stack is sometimes referred to as a stack of groupoids. This is the more commonly occurring case so the term 'stack' has come to mean 'stack of groupoids' in much of the literature.

In some circles the notion of a stack as a generalized [[groupoid]] is almost more familiar than the notion of sheaf as a [[space and quantity|generalized space]]. For instance [[differentiable stacks]] have attracted much attention in the study of [[Lie groupoids]] and [[orbifolds]], while [[diffeological space|diffeological spaces]] are only beginning to be investigated more in [[Lie theory]]. Groupoid stacks are closely related to internal groupoids (see [this MO post](https://mathoverflow.net/questions/93948/link-between-internal-groupoids-and-stacks-on-a-topos)).

An [[algebraic stack]], [[differentiable stack]] etc. is a stack over a site of schemes or differentiable manifolds with additional representability conditions.

## Provisional discussion

The following is "provisional" material on stacks that [[Todd Trimble]] wrote in the course of a discussion with [[Urs Schreiber|Urs]]. Somebody should turn this here into a coherent entry on stacks.

***

(Todd speaking.) I don't really speak "stacks", but in an effort to build a bridge between sheaves and stacks, I'll write down what I thought I understood, and ask someone such as Urs to come in and check. (Warning: I'm treating this edit box almost as a sandbox, in that what I say below is all a bit provisional until we get some discussion going.)

+--{.query}
 Hi Todd, thanks for this. I started making some remarks on the relation between descent $\infty$-categories and pseudofunctors from [[covers]] regarded as [[sieves]] (hence as presheaves) at [[descent and codescent]] in the section titled _Descent in terms of pseudo-functors_.
=--

At the simplest level, let $C$ be a category. As we know, a presheaf on $C$ is just a functor $X: C^{op} \to Set$.

Now let's categorify just once: regard a category $C$ as a bicategory whose local hom-categories are discrete. What I'll call a "pre-stack" is then a homomorphism of bicategories $X: C^{op} \to Cat$. Here I'm following Street's terminology: a homomorphism of bicategories is the "pseudo" version of a weak map of bicategories, as opposed to the "lax" version. So, we have given coherent isomorphisms $X(f)X(g) \to X(f g)$, and so on.

Now suppose that $C$ also comes equipped with a topology $J$, and let $F$ be a $J$-covering sieve for $c$, so that in particular it's a subfunctor $i: F \hookrightarrow \hom(-, c)$. We want to build a (truncated) simplicial object out of this, and to this end I'll use some yoga which was basically developed in my Cafe post [on the bar construction](http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html) [perhaps this may go partway to addressing your most recent [query](http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html#c021027) there, Urs].

Namely, there is a canonical way of presenting $F$ as a colimit of representables. Officially, it's given by a coend formula, but it's probably more illuminating to think of it in terms of tensor products over $C$:

$$\hom_C(-, -) \otimes_C F(-) \cong F(-)$$

In the long-winded version, this says that $F$ is the coequalizer of a diagram having the form

$$\sum_{c, d} \hom_C(-, c) \times \hom_C(c, d) \times F(d) \stackrel{\to}{\to} \sum_c \hom_C(-, c) \times F(c) \to F(-)$$

where the more visible one of the two parallel arrows involves the contravariant action of $C$ on $F$:

$$\hom(c, d) \times F(d) \to F(c)$$

and the less visible one uses $C$ acting on itself:

$$\hom(-, c) \times \hom(c, d) \times F(d) \to hom(-, d) \times F(d)$$

The point now is that this coequalizer diagram represents the tail end of a simplicial object (with $F(-)$ appearing in dimension -1), which in the notation of the bar construction one could call $B(C, C, F)$. Let me explain this last bit.

The point is that any category $C$ can be regarded as a monad in the bicategory of spans. The underlying span is of course

$$C_0 \stackrel{dom}{\leftarrow} C_1 \stackrel{cod}{\to} C_0$$

and a presheaf $F$ on $C$, as a discrete op-fibration, has an underlying span

$$C_0 \leftarrow F \to 1$$

and is precisely an algebra over the monad $C$. Then, given the data of a monad and an algebra over that monad, one proceeds to build the bar construction as a simplicial object, and I think this is probably the simplicial thingy we want to base the category of descent data on (given a pre-stack $X$).

In fact, if memory serves the category of descent data can be efficiently expressed in bicategorical language as follows. The covering sieve $F$ becomes a homomorphism of bicategories by changing base from $Set$ to $Cat$:

$$C^{op} \stackrel{F}{\to} Set \stackrel{discrete}{\to} Cat$$

and, abbreviating $discrete$ to $d$, it turns out that

$$Desc_F(X) \simeq Nat(d F, X)$$

where the thing on the right side is the category of strong (i.e., pseudo) natural transformations between the indicated bicategory homomorphisms.

In that case, the stack condition on $X$ becomes the statement that the canonical functor

$$X(c) \stackrel{Yoneda}{\cong} Nat(d \hom(-, c), X) \to Nat(d F, X)$$

(where the first equivalence comes from the bicategorical Yoneda lemma, and the second functor is induced from the subfunctor $i: F \to \hom(-, c)$) is an equivalence for all $J$-covering sieves $F$. This formulation connects up nicely, that is, is a straight categorification of what was put down in the entry [[sheaf]].

###Examples

* The stack of $BG$ is a functor $Mfd^{op} \to Gpd$, sending $U\in Mfd$ to the groupoid of $G$-principal bundles over $U$ with $G$-equivariant morphisms; sending $U\xrightarrow{f} V$ to the functor induced by pullbacks of principal bundles via $f$.  Then $BG$ comes from the prestack $BG^p: Mfd^{op} \to Gpd$ sending $U\in Mfd$ to the groupoid of trivial principal bundles over $U$ with $G$-equivariant morphisms (then it is just a $G$-valued function $U\xrightarrow{g} G$; sending $U\xrightarrow{f}V$ to the functor induced by pullbacks of principal bundles via $f$. Then sheafification or stackification will give us $BG$ back.

## Related concepts

* [[sheaf]]

* [[presheaf of groupoids]]

* [[group stack]]

* [[2-sheaf]] / [[(2,1)-sheaf]] /  **stack** / [[model structure for (2,1)-sheaves]]

  * [[representable morphism of stacks]]

  * [[mapping stack]], [[quotient stack]]

  * [[geometric stack]]

    * [[differentiable stack]], [[Deligne-Mumford stack]]

  * [[rigidification of a stack]]

  * [[moduli stack]]

  * [[Picard stack]]

* [[(∞,1)-sheaf]] / [[∞-stack]]

* [[axiom of stack completion]]

Special kinds of stacks include

* [[geometric stacks]];

* [[gerbes]].

[[!include homotopy n-types - table]]

## References

The concept originates (under the French term _champ_ and for the purpose of defining [[non-abelian cohomology]]) in:

* {#Giraud66} [[Jean Giraud]], _Cohomologie non abélienne_, Columbia University  (1966) [[GoogleBooks](https://books.google.de/books/about/Cohomologie_non_ab%C3%A9lienne.html?id=EQ1CAAAAIAAJ&redir_esc=y)]

  published as: Grundlehren **179**, Springer (1971) [[doi:10.1007/978-3-662-62103-5](https://www.springer.com/gp/book/9783540053071)]

and under the English term _stack_ (for specialization to [[Deligne-Mumford stacks]]/[[orbifolds]]) in:

* {#DeligneMumford69} [[Pierre Deligne]], [[David Mumford]], *The irreducibility of the space of curves of given genus*, Publications Mathématiques de l'IHÉS (Paris) **36**   (1969) 75-109 [[doi:10.1007/BF02684599](https://doi.org/10.1007/BF02684599), [numdam:PMIHES_1969__36__75_0](http://www.numdam.org/item?id=PMIHES_1969__36__75_0)]

and in the context of [[classifying toposes]]:

* [[Jean Giraud]], _Classifying topos_, in: [[William Lawvere]] (ed.) _Toposes, Algebraic Geometry and Logic_, Lecture Notes in Mathematics, vol 274. Springer (1972) ([doi:10.1007/BFb0073964](https://doi.org/10.1007/BFb0073964))

Further early discussion:

* [[Marta Bunge]], [[Robert Pare]], *Stacks and equivalence of indexed categories*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no.4 (1979) [[numdam](http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0)]

* [[Marta Bunge]], *Stack completions and Morita equivalence for categories in a topos*, Cahiers de topologie et géométrie différentielle xx-4 (1979)
401-436 [[MR558106](http://www.ams.org/mathscinet-getitem?mr=558106), [numdam](http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0)]

See also the references at *[[2-sheaf]]* and for considerably more literature see at *[[algebraic stack]]*.

Review:

* [[Ieke Moerdijk]], _Introduction to the language of stacks and gerbes_ [[arXiv:math/0212266](http://arxiv.org/abs/math/0212266)]

  > (including discussion of [[gerbes]])

* [[Jochen Heinloth]], *Notes on differentiable stacks* (2004) [[pdf](https://math.nyu.edu/~tschinke/WS04/pdf/heinloth.pdf), [[Heinloth-DifferentiableStacks.pdf:file]]]

  > (focused on [[differentiable stacks]])

Discussion of stacks in their incarnation (under the [[Grothendieck construction]]) as [[Grothendieck fibrations]]:

*  [[Angelo Vistoli]], _Grothendieck topologies, fibered categories and descent theory_ [[math.AG/0412512](http://arxiv.org/abs/math/0412512), [MR2223406](http://www.ams.org/mathscinet-getitem?mr=2223406)]  in: Fantechi et al. (eds.), _Fundamental algebraic geometry. Grothendieck's [[FGA explained]]_, Mathematical Surveys and Monographs __123__, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)]

A [[model category]] structure on [[presheaves of groupoids]], presenting stacks, by a [[Bousfield localization of model categories|localization]] of a [[model structure on simplicial presheaves]] (modeling [[∞-stacks]] before localization):

* [[John F. Jardine]], *Stacks and the homotopy theory of simplicial sheaves*, Homology, Homotopy and Applications, **3** 2  (2001) 361-384 [[euclid:hha/1139840259](https://projecteuclid.org/euclid.hha/1139840259)]

* {#Hollander01} [[Sharon Hollander]], *A homotopy theory for stacks*, Israel Journal of Mathematics **163** 1 (2008) 93-124 [[arXiv:math.AT/0110247](http://arxiv.org/abs/math.AT/0110247), [doi:10.1007/s11856-008-0006-5](https://doi.org/10.1007/s11856-008-0006-5)]

[[!redirects stack*]]
[[!redirects stacks]]