stabilization

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#### Stable Homotopy theory
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## Idea ##

The _stabilization_ of an [[(∞,1)-category]] $C$ with [[finite (∞,1)-limits]]  is the [[free construction|free]]
[[stable (∞,1)-category]] $Stab(C)$ on $C$. This is also called the $(\infty,1)$-category of
[[spectrum objects]]  of $C$, because for the archetypical example where $C = $ [[Top]]
the stabilization is $Stab(Top) \simeq Spec$ the category of [[spectrum|spectra]].

There is a canonical [[forgetful functor|forgetful]] [[(∞,1)-functor]] $\Omega^\infty : Stab(C) \to C$ that remembers of a [[spectrum object]] the underlying object of $C$ in degree 0. Under mild conditions, notably when $C$ is a [[presentable (∞,1)-category]], this functor has a [[left adjoint]] $\Sigma^\infty : C \to Stab(C)$ that _freely stabilizes_ any given object of $C$.

$$
  (\Sigma^\infty \dashv \Omega^\infty)
  :
  Stab(C)
   \stackrel{\overset{\Sigma^\infty}{\leftarrow}}{\underset{\Omega^\infty}{\to}}
  C
  \,.
$$

Going back and forth this way, i.e. applying the corresponding [[(∞,1)-monad]] $\Omega^\infty \circ \Sigma^\infty$ yields the assignment

$$
  X \mapsto \Omega^\infty \Sigma^\infty X
$$

that may be thought of as the **stabilization of an object** $X$.
Indeed, as the notation suggests, $\Omega^\infty \Sigma^\infty X$ may be thought of as the result as $n$ goes to infinity of the operation that forms from $X$ first the $n$-fold [[suspension object]] $\Sigma^n X$ and then from that the $n$-fold [[loop space object]].

## Definition

### Abstract definition

Let $C$ be an [[(∞,1)-category]] with [[finite (∞,1)-limit]] and write
$C_* := C^{{*}/}$ for its [[(∞,1)-category]] of [[pointed objects]], the
[[undercategory]] of $C$ under the [[terminal object]].

On $C_*$ there is the [[loop space object]] [[(infinity,1)-functor]]
$\Omega : C_* \to C_*$, that sends each object $X$ to the [[pullback]] of the
point inclusion ${*} \to X$ along itself. Recall that if a $(\infty,1)$-category is stable, the [[loop space object]] functor is an equivalence.

The _stabilization_ $Stab(C)$ of $C$ is the
[[(∞,1)-limit]] (in the [[(∞,1)-category of (∞,1)-categories]]) of the
tower of applications of the loop space functor

$$
  Stab(C)
  =
  \underset{\leftarrow}{\lim}
  \left(
    \cdots \to C_* \stackrel{\Omega}{\to} C_* \stackrel{\Omega}{\to} C_*
  \right)
  \,.
$$

This is ([StabCat, proposition 8.14](http://arxiv.org/abs/math/0608228)).

The canonical functor from $Stab(C)$ to $C_*$ and then further, via the functor that
forgets the basepoint, to $C$ is therefore denoted

$$
  \Omega^\infty : Stab(C) \to C
  \,.
$$

### Construction in terms of spectrum objects

Concretely, for any $C$ with finite limits, $Stab(C)$ may be constructed as the category of [[spectrum object]]s of $C_*$:

$$
  Stab(C) = Sp(C_*)
  \,.
$$

This is definition 8.1, 8.2 in [StabCat](http://arxiv.org/abs/math/0608228)

### Construction in terms of stable model categories
 {#ConstructionInTermsOfStableModelCategories}

Given a presentation of an [[(∞,1)-category]] by a [[model category]], there is a notion of stabilization of this model category to a [[stable model category]]. That this in turn presents the abstractly defined stabilization of the corresponding [[(∞,1)-category]] is due to ([Robalo 12, prop. 4.14](#Robalo12)).

For classical discussion see also *[[spectrification]]* and the *[[Bousfield-Friedlander model structure]]*.

## Properties

* For $C$ a [[presentable (∞,1)-category|presentable]] $(\infty,1)$-category with finite limits, the functor $\Omega^\infty \colon Stab(C) \to C$ has a [[left adjoint]] $\Sigma^\infty : C \to Stab(C)$
(forming [[suspension spectra]]).

Prop 15.4 (2) of [StabCat](http://arxiv.org/abs/math/0608228).

* stabilization is _not_ in general functorial on all of $(\infty, 1)Cat$. It's failure of being functorial, and approximations to it, are studied in [[Goodwillie calculus]].

\begin{proposition}
In the case of ordinary homotopy types (at least), the stabilization adjunction
$$
  \Sigma^\infty \dashv \Omega^\infty
  \;\colon\;
  Grpd_\infty^{\ast/}
  \rightleftarrows
  Spectra
$$

is a [[comonadic adjunction]] on [[simply connected homotopy type|simply connected]] [[homotopy types]], meaning that simply-connected classical (but [[pointed homotopy types|pointed]]) homotopy types are identified with the $\Sigma^\infty \Omega^\infty$-[[modales]] among [[stable homotopy types]].
\end{proposition}
[[Blomquist & Harper 2016  Thm. 1.8](#BlomquistHarper16); [Hess & Kedziorek 2017 Thm. 3.11](#HessKedziorek17)]

Here $\Sigma^\infty \Omega^\infty$ is the (pointed) [[exponential modality]] in [[linear homotopy type theory]], see [there](!-modality#RealizationInLinearHomotopyTypeTheory) for more.

## Examples

* For $C = $ [[Top]] the stabilization is the category [[Spec]] of [[spectra]]. The functor $\Sigma^\infty : Top_* \to Spec$  is that which forms [[suspension spectra]].

* For $C=Set$, [[the category of sets]], the stabilization is trivial.
An object in $Stab(Set)$ is a sequence of [[pointed sets]] $(E_0, E_1, \ldots)$ together with isomorphisms $\Omega(E_{i+1}) \simeq E_i$.  But $\Omega(X) = \ast \times_X \ast \simeq \ast$ for every [[pointed set]] $X$. So every object is isomorphic to $(\ast, \ast, \ldots)$. The space of [[endomorphisms]] of this object is a [[limit]] of the [[endomorphism]] spaces $Map_{Set_*}(\ast, \ast) \simeq \ast$, which is again [[contractible]].

## Related concepts

[[!include k-monoidal table]]

## References
 {#References}

A general discussion in the context of [[(∞,1)-category theory]] is in

* [[Jacob Lurie]], section 1.4 of: _[[Higher Algebra]]_

* [[Jacob Lurie]], section 1 of: _[[Spectral Schemes]]_

Discussion of stabilization as inversion of smashing with a suspension objects, and the relation between stabilization of [[(∞,1)-categories]] (to [[stable (∞,1)-categories]]) and of [[model categories]] (to [[stable model categories]]) in

* {#Robalo12} [[Marco Robalo]], section 4 of _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))

published as

* {#Robalo15} [[Marco Robalo]], section 2 of _K-theory and the bridge from motives to noncommutative motives_, Advances in Mathematics Volume 269, 10 January 2015 ([doi:10.1016/j.aim.2014.10.011](https://doi.org/10.1016/j.aim.2014.10.011))

with further remarks in

* {#Hoyois15} [[Marc Hoyois]], section 6.1 of _The six operations in equivariant motivic homotopy theory_, Adv. Math. 305 (2017), 197-279 ([arXiv:1509.02145](https://arxiv.org/abs/1509.02145))

Formalization of stabilization in [[dependent linear homotopy type theory]] (see also on the "[[exponential modality]]" [here](!-modality#RealizationInLinearHomotopyTypeTheory)):

* {#Riley22Thesis} [[Mitchell Riley]], §2.1.2 in: *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139)]

On comonadicity of the "[[exponential modality]]" $\Sigma^\infty \Omega^\infty$:

* {#BlomquistHarper16} Jacobson R. Blomquist, John E. Harper, Thm. 1.8 in: *Suspension spectra and higher stabilization* [[arXiv:1612.08623](https://arxiv.org/abs/1612.08623)]

* {#HessKedziorek17} [[Kathryn Hess]], [[Magdalena Kedziorek]], Thm. 3.11 in: *The homotopy theory of coalgebras over simplicial comonads*, Homology, Homotopy and Applications **21** 1 (2019) [[arXiv:1707.07104](https://arxiv.org/abs/1707.07104), [doi:10.4310/HHA.2019.v21.n1.a11](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a11)]

[[!redirects stabilizations]]
[[!redirects stabilization of an (∞,1)-category]]
[[!redirects stabilization of an (infinity,1)-category]]
[[!redirects stabilization of an ∞-category]]
[[!redirects stabilization of an infinity-category]]
[[!redirects stabilization of (∞,1)-categories]]
[[!redirects stabilization of (infinity,1)-categories]]
[[!redirects stabilization of ∞-categories]]
[[!redirects stabilization of infinity-categories]]


stabilization hypothesis

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## Idea

The _Baez-Dolan stabilization hypothesis_ states that for all $k \geq n+2$ a [[k-tuply monoidal n-category]] is "maximally monoidal".  In other words, for $k \geq n + 2$, a $k$-tuply monoidal $n$-category is the same thing as an $(n+2)$-tuply monoidal $n$-category.  More precisely, the natural inclusion $k Mon n Cat \hookrightarrow (n+2) Mon n Cat$ is an [[equivalence of higher categories]].

More generally, we can state a version for [[(m,n)-categories|(n,k)-categories]]: an $(m+2)$-tuply monoidal $(m,n)$-category is maximally monoidal.

## Proof when $n=1$

An aspect of the proof of this when $n=1$ (i.e. that $m+2$-tuply monoidal $(m,1)$-categories are maximally monoidal) was demonstrated in

* [[Carlos Simpson]], _On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's weak $n$-categories_ ([arXiv:math/9810058](http://arxiv.org/abs/math/9810058))

in terms of [[Tamsamani n-categories]].

A proof of the full statement in terms of [[quasi-categories]] is sketched in
[section 43.5](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf#page=187) of

* [[André Joyal]], _Notes on quasi-categories_ ([pdf](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf)).

Probably the first full proof in print is given in

* [[Jacob Lurie]], _[[Ek-Algebras]]_ ,

where it appears in example 1.2.3 as a direct consequence of a more general statement, corollary 1.1.10.

## Proof in general

A proof of the general form for arbitrary $m,n,k$, using iterated $(\infty,1)$-categorical enrichment to define $(\infty,n)$-categories, is in

* {#GepnerHaugseng13} [[David Gepner]], [[Rune Haugseng]], *Enriched ∞-categories via non-symmetric ∞-operads* ([arXiv:1312.3178](http://arxiv.org/abs/1312.3178))

See also

* [[Jacob Lurie]], section 5.1.2 _[[Higher Algebra]]_

* {#Batanin15} [[Michael Batanin]], _An operadic proof of Baez-Dolan stabilization hypothesis_ ([arXiv:1511.09130](http://arxiv.org/abs/1511.09130))

[[!redirects stabilisation hypothesis]]


stabilizer code

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* table of contents
{:toc}

## Idea

In [[quantum information theory]], a *stabilizer code* is a [[quantum error correcting code]] whose [[code subspace]] is the [[fixed subspace]] of an [[abelian group|abelian]] [[subgroup]] $\{-1\} \notin S \subset \mathcal{P}_n$ of the [[Pauli group]] $\mathcal{P}_n$ [[group action|acting]] on a register of $n$ [[q-bits]].

Conversely, $S \subset \mathcal{P}_n$ is then the [[stabilizer subgroup]] of the code space, whence the name of this class of codes.

Most [[quantum error correcting codes]] known are in fact stabilizer codes.

## Properties

### Relation to classical binary codes

Quantum stabilizer codes are closely related to classical [[error correcting codes]], specifically to [[binary linear codes]].

(e.g. [Ball, Centelles & Huber 20, Sec. 2.3](#BallCentellesHuber20))

## References

Stabilizer codes were introduced, independeny, in

* [[Daniel Gottesman]], *Stabilizer Codes and Quantum Error Correction* ([arXiv:quant-ph/9705052](https://arxiv.org/abs/quant-ph/9705052))

* [[Robert Calderbank]], E. M Rains, [[Peter W. Shor]], N. J. A. Sloane, *Quantum Error Correction and Orthogonal Geometry*, Phys. Rev. Lett. 78:405-408, 1997 ([arXiv:quant-ph/9605005](https://arxiv.org/abs/quant-ph/9605005))

following

* [[Peter W. Shor]], *Scheme for reducing decoherence in quantum computer memory*, Phys. Rev. A 52, R2493(R) 1995 ([doi:10.1103/PhysRevA.52.R2493](https://doi.org/10.1103/PhysRevA.52.R2493))

* [[Andrew M. Steane]], *Multiple Particle Interference and Quantum Error Correction*,  	Proc. Roy. Soc. Lond. A452 (1996) 2551 ([arXiv:quant-ph/9601029](https://arxiv.org/abs/quant-ph/9601029))

Review:

* {#BallCentellesHuber20} Simeon Ball, Aina Centelles, Felix Huber, Section 2 of: *Quantum error-correcting codes and their geometries* ([arXiv:2007.05992](https://arxiv.org/abs/2007.05992))

See also

* Wikipedia, *[Stabilizer code](https://en.wikipedia.org/wiki/Stabilizer_code)*

Realization in [[experiment]]:

Realization of [[quantum error correction]] in [[experiment]]:

* D. G. Cory, M. D. Price, W. Maas, [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojchiek H. Zurek]], T. F. Havel, and S. S. Somaroo, *Experimental Quantum Error Correction*, Phys. Rev. Lett. 81, 2152 (1998) ([doi:10.1103/PhysRevLett.81.2152](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.2152))

[[!redirects stabilizer codes]]

[[!redirects quantum stabilizer code]]
[[!redirects quantum stabilizer codes]]


stabilizer group

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###Context###
#### Group Theory
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#### Representation theory
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[[!include representation theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea

Given an [[action]] of a [[group]] on some [[space]], and given a point or (or more generally some subspace), then the _stabilizer group_ of that point (that subspace) is the [[subgroup]] whose action leaves the point (the subspace) fixed, [[invariant]].

The importance of stabilizer subgroups for the general development of [[geometry]] was famously highlighted in ([Klein 1872](#Klein1872)) in the context of what has come to be known the _[[Erlangen program]]_. For more on this aspect see at _[[Klein geometry]]_ and _[[Cartan geometry]]_.

Sometimes (such as in the context of [[Wigner classification]]) stabilizer groups are called _little groups_.

## Definition

### Traditional
 {#TraditionalDefinition}

Given an [[action]] $G\times X\to X$ of a [[group]] $G$ on a set $X$, for every element $x \in X$,  the __stabilizer subgroup__ of $x$ (also called the __isotropy group__ of $x$) is the set of all elements in $G$ that leave $x$ fixed:

$$
  Stab_G(x) = \{g \in G \mid g\circ x = x\}
  \,.
$$

If all stabilizer groups are trivial, then the action is called a _free action_.

### Homotopy-theoretic formulation
 {#GeneralAbstract}

We reformulate the traditional definition [above](#TraditionalDefinition) from the [[nPOV]], in terms of [[homotopy theory]].

A [[group]] [[action]] $\rho\colon G \times X \to X$ is equivalently encoded in its [[action groupoid]] [[fiber sequence]] in [[Grpd]]

$$
  X \to X//G \to \mathbf{B}G
  \,,
$$

where the $X//G$ is the [[action groupoid]] itself, $\mathbf{B}G$ is the [[delooping]] [[groupoid]] of $G$ and $X$ is regarded as a [[0-truncated]] groupoid.

This fiber sequence may be thought of as being the $\rho$-[[associated bundle]] to the $G$-[[universal principal bundle]]. (Here discussed for $G$ a [[discrete group]] but this discussion goes through verbatim for $G$ a [cohesive group](/nlab/show/cohesive+%28infinity,1%29-topos+--+structures#InfinGroups).)

For

$$
  x\colon * \to X
$$

any [[global element]] of $X$, we have an induced element $x\colon * \to X \to X//G$ of the action groupoid and may hence form the first [[homotopy group]] $\pi_1(X//G, x)$. This is the stabilizer group. Equivalently this is the [[loop space object]] of $X//G$ at $x$, given by the [[homotopy pullback]]

$$
  \array{
     Stab_G(x) &\to& *
     \\
     \downarrow && \downarrow^{\mathrlap{x}}
     \\
     * &\stackrel{x}{\to}& X//G
  }
  \,.
$$

This characterization immediately generalizes to stabilizer [[∞-groups]] of [∞-group actions](/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#GroupRepresentations). This we discuss [below](#ForInfinityGroupActions)

### For $\infty$-group actions
 {#ForInfinityGroupActions}

Let $\mathbf{H}$ be an [[(∞,1)-topos]] and $G \in \infty Grp(G)$ be an [[∞-group]] object in $\mathbf{H}$. Write $\mathbf{B}G \in \mathbf{H}$ for its [[delooping]] object.

By the discussion at _[[∞-action]]_ we have the following.

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

For $X \in \mathbf{H}$ any object, an [[∞-action]] of $G$ on $X$ is equivalently an object $X/G$ and a [[homotopy fiber sequence]] of the form

$$
  \array{
    X &\longrightarrow& X//G
    \\
    && \downarrow
    \\
    && \mathbf{B}G
  }
  \,.
$$

=--

Here $X/G$ is the [[homotopy quotient]] of the [[∞-action]]

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

The action as a morphism $X \times G \to X$ is recovered from prop. \ref{InfinityAction} by the [[(∞,1)-pullback]]

$$
  \array{
     X \times G &\to& X
     \\
     \downarrow && \downarrow
     \\
     X &\to& X//G
  }
  \,.
$$

=--

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

Given an [[∞-action]] $\rho$ of $G$ on $X$ as in prop. \ref{InfinityAction},
and given a [[global element]]  of $X$

$$
  x \colon \ast \to X
$$

then the **stabilizer $\infty$-group** $Stab_\rho(x)$ of the $G$-action at $x$ is the [[loop space object]]

$$
  Stab_\rho(x) \coloneqq \Omega_x (X//G)
 \,.
$$

=--

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

Equivalently, def. \ref{StabilizerInInfinityTopos}, gives the [[loop space object]] of the [[1-image]] $\mathbf{B}Stab_\rho(x)$ of the morphism

$$
  \ast \stackrel{x}{\to} X \to X//G
  \,.
$$

As such the [[delooping]] of the stabilizer $\infty$-group sits in a [[1-epimorphism]]/[[1-monomorphism]] factorization $\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X//G$ which combines with the homotopy fiber sequence of prop. \ref{InfinityAction} to a diagram of the form

$$
  \array{
    \ast &\stackrel{x}{\longrightarrow}& X &\stackrel{}{\longrightarrow}& X//G
    \\
    \downarrow^{\mathrlap{epi}} && & \nearrow_{\mathrlap{mono}} & \downarrow
    \\
    \mathbf{B} Stab_\rho(x)
    &=&
    \mathbf{B} Stab_\rho(x)
    &\longrightarrow&
    \mathbf{B}G
  }
 \,.
$$

In particular there is hence a canonical homomorphism of $\infty$-groups

$$
  Stab_\rho(x) \longrightarrow G
  \,.
$$

However, in contrast to the classical situation, this morphism is not in general a monomorphism anymore, hence the stabilizer $Stab_\rho(x)$ is not a _sub_-group of $G$ in general.

=--

## Examples

### For a group acting on itself

For $G$ any [[∞-group]] in an [[(∞,1)-topos]] $\mathbf{H}$, its (right) [[action]] on itself is given by the [[looping and delooping|looping/delooping]] [[fiber sequence]]

$$
  G \to * \stackrel{\rho}{\to} \mathbf{B}G
  \,.
$$

Clearly, for every point $g \in G$ we have $Stab_{\rho}(g) \simeq * \times_* * \simeq *$ is trivial. Hence the action is free.

### Stabilizers of shapes -- Klein geometry
 {#KleinGeometry}

Let $X \to X//G \stackrel{\rho}{\to} \mathbf{B}G$ be an [[∞-action]] of $G$ on $X$.

Let $Y \in \mathbf{H}$ any other object, and regard it as equipped with the trivial $G$-action $Y \to Y \times \mathbf{B}G \to \mathbf{B}G$. There is then an induced [[∞-action]] $\rho_Y$ on the [[internal hom]] $[Y,X]$, the [[conjugation action]], given by internal hom in the [[slice (∞,1)-topos]] over $\mathbf{B}G$:

$$
  [Y,X] \to \underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} \to \mathbf{B}G
  \,.
$$

Now given any $f \colon Y \to X$, then the stabilizer group $Stab_{\rho_Y}(f)$ is the stabilizer of $Y$ "in" $X$ under this $G$-action.

The morphism of $\infty$-groups

$$
  i_f\colon Stab_{\rho_Y}(f) \to G
$$

hence characterizes the ([[higher Klein geometry|higher]]) [[Klein geometry]] induced by the $G$-action and by the shape $f\colon Y \to X$. (See at [Klein geometry -- History](Klein+geometry#History).)

For completeness we notice that:

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

Here  $(\underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} \to \mathbf{B}G )$ is equivalently the [[(∞,1)-pullback]]  $\rho_Y$ in

$$
  \array{
    \underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} &\to& [Y, X//G]
    \\
    \downarrow^{\mathrlap{\rho_Y}}
    &&
    \downarrow^{\mathrlap{[Y, \rho]}}
    \\
    \mathbf{B}G
    &\to&
    [Y, \mathbf{B}G]
  }
  \,,
$$

where the bottom morphism is the [[internal hom]] [[adjunct]] of the [[projection]] $Y \times \mathbf{B}G \to \mathbf{B}G$.

=--

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

We check the Hom adjunction property, that for any $(A//G \stackrel{\alpha}{\to} \mathbf{B}G) \in \mathbf{B}G$ we have

$$
  \mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G})
  \simeq
  \mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X)
$$

with $[Y,X]_{/\mathbf{B}G}$ replaced by the above pullback.

Notice that by the $G$-action on $Y$ being trivial, we have $A \times_{\mathbf{B}G} Y \simeq (A//G \times Y  \stackrel{p_1}{\to} A//G \stackrel{\alpha}{\to} \mathbf{B}G) \in \mathbf{H}_{/\mathbf{B}G}$.

Then use the characterization of [Hom-spaces in a slice](over-%28infinity%2C1%29-category#HamSpacesInASlice) to find $\mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G})$ as the [[homotopy pullback]] on the left of

$$
  \array{
    \mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G})
      &\longrightarrow&
    \mathbf{H}(A//G, [Y,X]//G) &\to& \mathbf{H}(A//G,[Y, X//G])
    \\
    \downarrow && \downarrow^{\mathrlap{\mathbf{H}(A//G,\rho_Y)}}
    &&
    \downarrow^{\mathrlap{\mathbf{H}(A//G,[Y, \rho])}}
    \\
    \ast
      &\stackrel{\vdash \alpha}{\longrightarrow}&
    \mathbf{H}(A//G,\mathbf{B}G)
      &\to&
    \mathbf{H}(A//G,[Y, \mathbf{B}G])
  }
  \,,
$$

Now using the Hom-adjunction in $\mathbf{H}$ itself, the fact that $\mathbf{H}(A//G,-)$ preserves [[homotopy pullbacks]] and the [[pasting law]] this is equivalent to

$$
  \array{
    \mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X)
      &\longrightarrow&
      &\longrightarrow&
    \mathbf{H}(A//G \times Y, X //G)
    \\
    \downarrow && \downarrow
    &&
    \downarrow
    \\
    \ast
      &\stackrel{\vdash \alpha}{\longrightarrow}&
    \mathbf{H}(A//G,\mathbf{B}G)
      &\to&
    \mathbf{H}(A//G \times Y,  \mathbf{B}G)
  }
  \,,
$$

Here the bottom map is indeed the name of $\alpha \circ p_1$ and so again by the pullback characterization of [Hom-spaces in a slice](over-%28infinity%2C1%29-category#HamSpacesInASlice) this pasting diagram does exhibit $\mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X)$ as indicated.

=--

### For the canonical action on a coset space
 {#ForCanonicalActionOnCosetSpace}

Conversely, for any [[homomorphism]] $H \to G$ of [[∞-groups]] given, then the canonical $G$-$\infty$-action for which $H$ is the stabilizer $\infty$-group of a point is the canonical action on (the "[[coset]]") $G/H$.

This follows from def. \ref{StabilizerInInfinityTopos} by observing that the homotopy fiber sequence of prop. \ref{InfinityAction} for the $G$-action on $G/H$ is just

$$
  \array{
    G/H &\stackrel{}{\longrightarrow}& \mathbf{B}H
    \\
    && \downarrow
    \\
    && \mathbf{B}G
  }
$$

so that for any point $x \colon \ast \to G/H$ we have

$$
  Stab(x) \simeq \Omega_{x}(\mathbf{B}H) \simeq \Omega_\ast \mathbf{B}H \simeq H
  \,.
$$

### Stabilizers of coshapes
 {#StabilizersOfCoShapes}

Dually to "stabilizers of shapes", as [above](#KleinGeometry) one may consider stabilizers of "co-shapes".

I.e. given a $G$-action on $X$, and given a map $f \colon X \to A$, then one may ask for the stabilizer of $f$ in the canonical $G$-action on $[X,A]$.

For instance if $A$ here is $\mathbf{B}^{n}U(1)_{conn}$, and $f \colon X \to \mathbf{B}^n U(1)_{conn}$ is regarded as a [[prequantum n-bundle]] ,and $[X,A]$ is replaced by its [[differential concretification]], then these stabilizers are the [[quantomorphism n-groups]].

> to be expanded on

## Related concepts

* [[normalizer subgroup]]

* [[centralizer subgroup]]

* [[stable point]]

* [[Klein geometry]], [[Cartan geometry]]

## References

An early occurence of the concept of stabilizer subgroups:

* {#Klein1872} [[Felix Klein]], p. 4 of: _Vergleichende Betrachtungen über neuere geometrische Forschungen_ (1872)

  translation by M. W. Haskell, _A comparative review of recent researches in geometry_ , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. ([retyped pdf](http://math.ucr.edu/home/baez/erlangen/erlangen_tex.pdf), [[KleinRetyped.pdf:file]], [scan of original](http://math.ucr.edu/home/baez/erlangen/erlangen.pdf))

  (the "[[Erlangen program]]").

* {#Bredon72} [[Glen Bredon]], Section I.2 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

[[!redirects stabilizer group]]
[[!redirects stabilizer groups]]
[[!redirects stabiliser group]]
[[!redirects stabiliser groups]]

[[!redirects stabilizer]]
[[!redirects stabiliser]]
[[!redirects stabilizers]]
[[!redirects stabilisers]]

[[!redirects stabilizer ∞-group]]
[[!redirects stabilizer infinity-group]]
[[!redirects stabiliser ∞-group]]
[[!redirects stabiliser infinity-group]]
[[!redirects stabilizer ∞-groups]]
[[!redirects stabilizer infinity-groups]]
[[!redirects stabiliser ∞-groups]]
[[!redirects stabiliser infinity-groups]]

[[!redirects homotopy stabilizer group]]
[[!redirects homotopy stabilizer groups]]

[[!redirects stabilizer subgroup]]
[[!redirects stabiliser subgroup]]
[[!redirects stabilizer subgroups]]
[[!redirects stabiliser subgroups]]

[[!redirects isotropy group]]
[[!redirects isotropy groups]]
[[!redirects isotropy subgroup]]
[[!redirects isotropy subgroups]]

[[!redirects stabilizer 2-group]]
[[!redirects stabilizer 2-groups]]

[[!redirects little group]]
[[!redirects little groups]]


stable (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
=--
#### Stable Homotopy theory
+-- {: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A stable [[(∞,1)-category]] $C$, is a [[pointed (∞,1)-category]] with [[finite (∞,1)-limit]] which is _stable_ under forming [[loop space objects]]:

$C$ has a [[zero object]] and the corresponding loop [[(∞,1)-functor]]

$$
  \Omega : C \to C
$$

is an _equivalence_ with inverse the [[suspension object]] functor

$$
  C \leftarrow C : \Sigma
  \,.
$$

This means that the objects of a stable $(\infty,1)$-category are stable in the sense of [[stable homotopy theory]]: they behave as if they were [[spectrum|spectra]].

Indeed, every $(\infty,1)$-category with finite [[limit]]s has a free [[stabilization]] to a stable $(\infty,1)$-category $Stab(C)$, and the objects of $Stab(C)$ are the [[spectrum object]]s of $C$.

The [[homotopy category of an (∞,1)-category]] of a stable $\infty$-category is a [[triangulated category]].

Notice that the definition of triangulated categories is involved and their behaviour is bad, whereas the definition of stable $\infty$-category is simple and natural.  The complexity and bad behavior of triangulated categories comes from them being the [[decategorification]] of a structure that is natural in [[higher category theory]].

## Definition
 {#Definition}

As with ordinary categories, an  object in a [[(infinity,1)-category]] is a [[zero object]] if it is both [[initial object]] and a [[terminal object]]. An $(\infty,1)$-category with a zero object is a **pointed** $(\infty,1)$-category.

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

In a pointed [[(∞,1)-category]] $C$ with [[zero object]] $0$, the **kernel** of a morphism $g : Y \to Z$ is the [[(∞,1)-pullback]]

$$
  \array{
     ker(g) &\to& Y
     \\
     \downarrow && \downarrow^g
     \\
     0 &\to& Z
  }
$$

(so that $ker(g) \to Y \stackrel{g}{\to} Z$ is a [[fibration sequence]])

and the **cokernel** of $f:X\to Y$ is the [[(∞,1)-pushout]]

$$
  \array{
     X &\stackrel{f}{\to}& Y
     \\
     \downarrow && \downarrow
     \\
     0 &\to& coker(f)
  }
  \,.
$$

An arbitrary commuting square in $C$ of the form

$$
  \array{
     X &\stackrel{f}{\to}& Y
     \\
     \downarrow && \downarrow^g
     \\
     0 &\to& Z
  }
$$

is a **triangle** in $C$. A pullback triangle is called an **exact triangle** and a pushout triangle a **coexact triangle**. By the universal property of pullback and pushout, to any triangle are associated canonical morphisms $X\to\ker(g)$ and $coker(f)\to Z$. In particular, for every exact triangle there is a canonical morphism $coker(ker(g)\to Y)\to Z$ and for every coexact triangle there is a canonical morphism $X\to ker(Y\to coker(f))$.

=--

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

A **stable $(\infty,1)$-category** is a pointed $(\infty,1)$-category such that

* for every morphism in $C$ kernel and cokernel exist;

* every exact triangle is coexact and vice versa, i.e. every morphism is the cokernel of its kernel and the kernel of its cokernel.

=--

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

The notion of stable $\infty$-category should not be confused with that of a [[k-tuply monoidal n-category|stably monoidal]] $\infty$-category.  A connection between the terms is that the [[stable (∞,1)-category of spectra]] is the prototypical stable $\infty$-category, while _connective_ spectra (not all spectra) can be identified with [[k-tuply groupal n-groupoid|stably groupal]] $\infty$-groupoids, aka _infinite loop spaces_ or
$E_\infty$-[[E-infinity space|spaces]].

=--

## Constructions in stable $\infty$-categories

### Looping and delooping

The relevance of the axioms of a stable $(\infty,1)$-category is that they imply that not only does every object $X$ have a [[loop space object]] $\Omega X$ defined by the exact triangle

$$
  \array{
    \Omega X &\to& 0
    \\
    \downarrow && \downarrow
    \\
    0 &\to& X
  }
$$

but also that, conversely, every object $X$ has a [[suspension object]]
$\Sigma X$ defined by the coexact triangle

$$
  \array{
    X &\to& 0
    \\
    \downarrow && \downarrow
    \\
    0 &\to& \Sigma X
  }
  \,.
$$

These arrange into $(\infty,1)$-endofunctors

$$
  \Omega : C \to C
$$
$$
  \Sigma : C \to C
$$
which are [[autoequivalences]] of $C$ that are inverses of each other.

### Stabilization

For every pointed $(\infty,1)$-category with finite [[limit]]s which is not yet stable there is its free [[stabilization]] (see there for more details):

a stable $(\infty,1)$-category $Sp(C)$ that can be defined as the [[limit in a quasi-category|limit]] in the [[(∞,1)-category of (∞,1)-categories]]

$$
  Sp(C) := holim(   \cdots \to C \stackrel{\Omega}{\to} C \stackrel{\Omega}{\to} C )

  \,.
$$

For $C =$ [[Top]] the $(\infty,1)$-category of topological spaces, $Sp(Top)$ is the familiar [[stable (∞,1)-category of spectra]] (whose [[homotopy category of an (∞,1)-category|homotopy category]] is the [[stable homotopy category]]) used in [[stable homotopy theory]] (which gives stable $(\infty,1)$-categories their name).

Moreover, every [[derived category]] of an [[abelian category]] is the triangulated homotopy category of a stable $(\infty,1)$-category.

Hence stable homotopy theory and homological algebra are both special cases of the theory of stable $(\infty,1)$-categories.

## Properties

### Enrichment over spectra

Stable $\infty$-categories are naturally [[enriched (∞,1)-categories]] over the [[(∞,1)-category of spectra]] ([Gepner-Haugseng 13](#GepnerHaugseng13)).

### The homotopy category: triangulated categories
 {#TheTriangulatedHomotopyCategory}

The [[homotopy category of an (∞,1)-category|homotopy category]] $Ho(C)$ of a stable $(\infty,1)$-category $C$ -- its [[decategorification]] to an ordinary [[category]] -- is less well behaved than the original stable $(\infty,1)$-category, but remembers a shadow of some of its structure: this shadow is the structure of a [[triangulated category]] on $Ho(C)$

* the **translation functor** $T : Ho(C) \to Ho(C)$ comes from the [[suspension object|suspension]] functor $\Sigma : C \to C$;

* the **distinguished triangles** in $Ho(C)$ are pieces of the [[fibration sequence]]s in $C$.

For details see [StabCat, section 3](http://arxiv.org/PS_cache/math/pdf/0608/0608228v5.pdf#page=6).

Alternately, one can first pass to a [[stable derivator]], and thence to a triangulated category.  Any suitably complete and cocomplete $(\infty,1)$-category has an underlying [[derivator]], and the underlying derivator of a stable $(\infty,1)$-category is always stable---while the underlying category of any stable derivator is triangulated.  But the derivator retains more useful information about the original stable $(\infty,1)$-category than does its triangulated homotopy category.

### Models
 {#alternativemodels}

In direct analogy to how a general [[(∞,1)-category]] may be [[presentable (∞,1)-category|presented]] by [[model category]], a stable $(\infty,1)$-categories may be presented by any of the following models.

* [[stable model category]];

* stable [[quasicategory]] (as devloped by [[Lurie]] in Chapter 1 of [[Higher Algebra]]);

* pre-triangulated [[spectral category]].

There are further variants and special cases of these models.
The following three concepts are equivalent to each other and special cases of the above models, or equivalent in characteristic 0.

* [[pretriangulated dg-category]]

* [[Frobenius category]]

(e.g. [Cohn 13](#Cohn13), see also [Schwede](#Schwede))

A [[triangulated category]] linear over a field $k$ can canonically be refined to

* an [[enhanced triangulated category]]:

  * a [[pretriangulated dg-category|pretriangulated]] [[dg-category]];

* an [[A-infinity-category]];

* a stable $(\infinity,1)$-category.

If $k$ has characteristic 0, then all these three concepts become equivalent.

### Stabilization and localization of presheaf $(\infty,1)$-categories
 {#StabGiraud}

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

Let $C$ and $D$ be [[(infinity,1)-category|(∞,1)-categories]] and
$Func(C,D)$ the [[(∞,1)-category of (∞,1)-functors]] between them.

Its [[stabilization]] is equivalent to the functor category into the stabilization of $C$:

$$
  Stab(Func(C,D)) \simeq Func(C,Stab(D))
  \,.
$$

In particular, consider the case $D = $ [[∞Grpd]] where $Stab(D) = Stab(\infty Grpd) = Sp$ (= the [[stable (∞,1)-category of spectra]]). One has $Func(C^{op}, D) = Func(C^{op}, \infty Grpd) =: PSh_{(\infty,1)}(C)$ is the [[(∞,1)-category of (∞,1)-presheaves]], and $Func(C^{op},Sp) =: PSh_{(\infty,1)}^{Sp}(C)$ is the [[(∞,1)-category]] of [[(∞,1)-presheaves]] of [[spectra]], we get

$$
  Stab(PSh_{(\infty,1)}(C)) \simeq PSh_{(\infty,1)}^{Sp}(C)
  \,.
$$

=--

This is [StabCat, example 10.13](#StabCat) .

+-- {: .num_prop #StableGiraud}
###### Proposition
**("stable Giraud theorem")**

Let $C$ be an [[(∞,1)-category]]. Then $C$ is stable and [[presentable (∞,1)-category]] if and only if $C$ is equivalent to an [[accessible (infinity,1)-category|accessible]] left-exact [[localization of an (∞,1)-category|localization]] of the [[(∞,1)-category]] of presheaves of spectra on some small [[(∞,1)-category]] $E$, so that there is an adjunction

$$
  C \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow}
  PSh^{Sp}(E) \simeq Stab(PSh(E))
  \,.
$$

=--

This is [Higher Algebra, Proposition 1.4.4.9](#HigherAlgebra).

This is the stable analog of the statement that every
[[(∞,1)-category of (∞,1)-sheaves]] is a left exact localization of
an $(\infty,1)$-category of presheaves.

A more intrinsic [[(∞,1)-topos]]-theoretic version of this statement (not mentioning a choice of [[(∞,1)-site]]) is the following:

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

Let $\mathbf{H}$ be an [[(∞,1)-topos]] and write

$$
  Sh_\infty(\mathbf{H}, Spectra)
  \coloneqq
  Func_{lex}(\mathbf{H}^{op}, Spectra)
$$

for the [[(∞,1)-category]] of _[[sheaves of spectra]]_ on $\mathbf{H}$ (with respect to its [[canonical topology]]), hence the  [[(∞,1)-category]] of [[left exact (∞,1)-functors]] from the [[opposite (∞,1)-category]] of $\mathbf{H}$ to the [[(∞,1)-category of spectra]].

This exhibits the [[stabilization]] of $\mathbf{H}$:

$$
  Stab(\mathbf{H}) \simeq Sh_\infty(\mathbf{H}, Spectra)
  \,.
$$

=--

This is ([[Spectral Schemes|Lurie "Spectral Schemes", remark 1.2]]).

See at _[[sheaf of spectra]]_ and _[[model structure on presheaves of spectra]]_ for more.

### As categories of modules over $A_\infty$-ring(oid)s
 {#AsCategoriesOfModules}

In terms of ([[stable model category|stable]]) [[model category|model categories]], something like an analog of this statement is
([Schwede-Shipley, theorem 3.3.3](#SchwedeShipley)):

+-- {: .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 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))

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)).

+-- {: .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 algebraic geometry]].

=--

### Dold-Kan correspondence

* [[infinity-Dold-Kan correspondence]]

## Examples

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

* [[(∞,1)-category of (∞,1)-modules]]

* [[Fukaya category]], [[Lagrangian cobordism]]

## Related concepts

* [[additive (∞,1)-category]]

* [[Stab(∞,1)Cat]]

* [[stable homotopy theory]]

* [[stable model category]]

* [[spectral category]]

* [[dg-category]], [[dg-nerve]]

* [[A-∞ category]]

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

* [[t-structure on a stable (∞,1)-category]], [[heart of a stable (∞,1)-category]]

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

* [[spectral sequence of a filtered stable homotopy type]]

* [[prime spectrum of a symmetric monoidal stable (∞,1)-category]]

## References

The abstract [[(∞,1)-category theory|(∞,1)-category theoretical]] notion was introduced and studied in

* {#StabCat} [[Jacob Lurie]], _[[Stable Infinity-Categories]]_ [[arXiv:math/0608228](https://arxiv.org/abs/math/0608228)

This appears in a more comprehensive context of [[higher algebra]]

* {#HigherAlgebra} [[Jacob Lurie]], as section 1 of: _[[Higher Algebra]]_

A brief introduction is in

* {#Harpaz2013} [[Yonatan Harpaz]], _Introduction to stable $\infty$-categories_, October 2013 ([[HarpazStableInfinityCategory2013.pdf:file]])

Discussion of how $k$-linear [[dg-categories]]/[[A-infinity categories]] present $k$-linear stable $(\infty,1)$-categories is in

* {#Cohn13} [[Lee Cohn]], _Differential Graded Categories are k-linear Stable Infinity Categories_ ([arXiv:1308.2587](http://arxiv.org/abs/1308.2587))

* {#Faonte13} [[Giovanni Faonte]], _Simplicial nerve of an A-infinity category_ ([arXiv:1312.2127](http://arxiv.org/abs/1312.2127))

A diagram of the interrelation of all models for stable $(\infty,1)$-categories (in the guise of [[enhanced triangulated categories]]) with a list of further literature:

* {#Schwede} [[Stefan Schwede]], [[Markus Hausmann]], _Enhancements of triangulated categories_, Graduate Seminar Topology, Bonn 2019-2020 ([[Schwede_EnhancedSeminar.pdf:file]])

For discussion of the [[stable model category]] models of stable $\infty$-categories see

* {#SchwedeShipley} [[Stefan Schwede]], [[Brooke Shipley]], _Classification of stable model categories_ ([Hopf pdf](http://hopf.math.purdue.edu/Schwede-Shipley/class.final.pdf)) and ([arXiv:math/0108143](http://arxiv.org/abs/math/0108143))

The enrichment over spectra is made precise in

* {#GepnerHaugseng13} [[David Gepner]], [[Rune Haugseng]], _Enriched ∞-categories via non-symmetric ∞-operads_ ([arXiv:1312.3178](http://arxiv.org/abs/1312.3178))

[[!redirects stable (infinity,1)-category]]
[[!redirects stable (infinity, 1)-category]]
[[!redirects stable (infinity,1)-categories]]
[[!redirects stable (infinity, 1)-categories]]
[[!redirects stable (∞,1)-category]]
[[!redirects stable (∞, 1)-category]]
[[!redirects stable (∞,1)-categories]]
[[!redirects stable (∞, 1)-categories]]

[[!redirects stable infinity-category]]
[[!redirects stable infinity-categories]]
[[!redirects stable ∞-category]]
[[!redirects stable ∞-categories]]

[[!redirects stable quasicategory]]
[[!redirects stable quasicategories]]

[[!redirects stable (infinity,1)-topos]]
[[!redirects stable (infinity,1)-toposes]]
[[!redirects stable (∞,1)-topos]]
[[!redirects stable (∞,1)-toposes]]


stable (infinity,1)-category of spectra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
=--
#### Stable Homotopy theory
+-- {: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The collection of [[spectra]] forms an [[(∞,1)-category]] $Sp(\infty Grpd) =  $ [[Spectra]], which is in fact a [[stable (∞,1)-category]]. Indeed, it is the _[[universal property]]_ [[stabilization]] of the $(\infty,1)$-category [[∞Grpd]], equivalently of the [[simplicial localization]] of the category [[Top]] at the [[weak homotopy equivalences]].

$Sp(\infty Grpd)$ plays a role in [[stable homotopy theory]] analogous to the role played by the 1-[[category]] [[Ab]] of [[abelian groups]] in [[homological algebra]], or rather of the [[category of chain complexes]] $Ch_\bullet(Ab)$ of abelian groups.

## Definition

In the context of [[(infinity,1)-category|(∞,1)-categories]] a [[spectrum]] is a [[spectrum object]] in the [[(∞,1)-category]] $L_{whe} Top_*$ of pointed [[topological spaces]].

Recall that [[spectrum objects]] in the [[(infinity,1)-category]] $C$ form a [[stable (∞,1)-category]] $Sp(C)$.

The [[stable (∞,1)-category]] of [[spectrum object]]s in $L_{whe} Top_*$ is the **stable $(\infty,1)$-category of spectra**

$$
  Stab(L_{whe}Top) := Sp(L_{whe}Top_*)
  \,.
$$

## Properties

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

### Monoidal structure

* With the [[smash product of spectra]] $Sp(L_{whe}Top_*)$ becomes a [[symmetric monoidal (∞,1)-category]].

  * an [[algebra in an (infinity,1)-category|algebra object]] in $Sp(L_{whe}Top_*)$ with respect to this monoidal structure is an [[associative ring spectrum]];

  * a [[commutative algebra in an (infinity,1)-category|commutative algebra object]] in $Sp(L_{whe}Top_*)$ with respect to this monoidal structure is a [[commutative ring spectrum]];

### Prime spectrum and Morava K-theory

The [[prime spectrum of a monoidal stable (∞,1)-category]] for [[p-localization|p-local]] and [[finite spectra]] is labeled by the [[Morava K-theories]]. This is the content of the _[[thick subcategory theorem]]_.

### Model category presentation

There are several [[presentable (infinity,1)-category|presentations]] of the $(\infty,1)$-category of spectra by [[model categories of spectra]]. In particular there are [[symmetric smash product of spectra|symmetric]] [[monoidal model categories]] where the [[smash product of spectra]] is presented by an ordinary [[tensor product]], so that [[A-∞ rings]], [[E-∞ rings]] and [[∞-modules]] are presented by 1-categorical [[monoid objects]] and [[module objects]], respectively ("[[brave new algebra]]"). See at:

[[model structure on spectra]], [[symmetric monoidal smash product of spectra]]

* [[symmetric spectrum]], [[model structure on symmetric spectra]]

* [[orthogonal spectrum]], [[model structure on orthogonal spectra]]

* [[S-module]], [[model structure on S-modules]]

## References

The stable $(\infty,1)$-category of spectra is described in chapter 1 of

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

or section 9 of

* [[Jacob Lurie]], [[Stable Infinity-Categories]] .

Its monoidal structure is described in section 4.2

* [[Jacob Lurie]], [[higher algebra|Noncommutative algebra]].

That this is a symmetric monoidal structure is described in section  6 of

* [[Jacob Lurie]], [[higher algebra|Commutative algebra]].

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

[[!redirects (∞,1)-category of spectra]]
[[!redirects (infinity,1)-category of spectra]]

[[!redirects symmetric monoidal (∞,1)-category of spectra]]
[[!redirects symmetric monoidal (infinity,1)-category of spectra]]


stable (infinity,1)-topos > history

< [[stable (infinity,1)-topos]]


stable bundle

> to be merged with _[[stable vector bundle]]_.

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

#Contents#
* table of contents
{:toc}

## Idea

A [[vector bundle]] over a [[Riemann surface]] is called _stable_ (resp. _semistable_) if the ratio of its degree and its [[rank]] is bigger (resp. bigger or equal) than this ratio for any of its subbundles.

This means that it is a [[stable point]] under the action of [[automorphisms]].

## Properties

* [[Narasimhan–Seshadri theorem]]

## References

Many references are at [[moduli space of bundles]]. The notion was introduced by [[David Mumford]] in the context of GIT in 1960-s:

* [[David Mumford]], J. Fogarty, [[Frances Kirwan]], (1994), Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) __34__, 3rd ed. Springer 1994

* [[Gerd Faltings]], _Stable G-bundles and projective connections_, J. Alg. Geometry , 2 (1993) pp. 507–568 [MR1211997](http://www.ams.org/mathscinet-getitem?mr=1211997) [Zbl 0790.14019](http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0790.14019)

See also

* Wikipedia, _[Stable vector bundle](http://en.wikipedia.org/wiki/Stable_vector_bundle)_

[[!redirects stable bundles]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

This page compiles links related to the book collection:

* [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.)

  **Stable categories and structured ring spectra**

  MSRI Book Series

  Cambridge University Press (2022)

  ISBN: 9781009123297

  [[GoogleBooks](https://books.google.de/books/about/Stable_Categories_and_Structured_Ring_Sp.html?id=mN6YzgEACAAJ&redir_esc=y)]

on [[stable homotopy theory]] and [[ring spectra]] ([[higher algebra]], [[brave new algebra]]).

## Contents

On [[homotopical categories]] and [[(∞,1)-categories]]:

* [[Emily Riehl]], _Homotopical categories: from model categories to (∞,1)-categories_, ([arXiv:1904.00886](https://arxiv.org/abs/1904.00886))

On [[model categories of spectra]]:

* [[Daniel Dugger]], _Stable categories and spectra via model categories_ ([pdf](https://pages.uoregon.edu/ddugger/models_chapter.pdf))

* [[Clark Barwick]], _Stable homotopy theory via ∞-categories_.

On [[operad]] and [[algebra over an operad]]:

* [[Michael Mandell]], *Operads and Operadic Algebras in Homotopy Theory  ([arxiv.1912.13047](https://arxiv.org/abs/1912.13047))

On [[ring spectra]]:

* [[Birgit Richter]], _Commutative ring spectra_ ([arXiv:1710.02328](https://arxiv.org/abs/1710.02328), [pdf](https://www.math.uni-hamburg.de/home/richter/rev-commringspectra.pdf))

On [[Bousfield localization of model categories]] ([[Bousfield localization of spectra]]):

* [[Tyler Lawson]], *An introduction to Bousfield localization* ([arXiv:2002.03888](https://arxiv.org/abs/2002.03888))

On [[spectral algebraic geometry]]:

* [[Charles Rezk]], *Spectral algebraic geometry* ([pdf](https://faculty.math.illinois.edu/~rezk/sag-chapter-web.pdf))

category: reference


stable category

There are several related notions dealing with similar notions. Historically the first is

* [[stable homotopy category]] of spectra

and V. Puppe has shown that it has a [[triangulated category|triangulated structure]]. Stability referred to [[stable homotopy theory]] and invertibility of [[suspension]] there.

Then Waldhausen came up with

* [[Waldhausen category]]

with the applications in [[algebraic K-theory]]. His construction of K-theory for Quillen exact categories could be factored in two steps and the intermediate case is the Waldhausen category. From each [[Quillen exact category]] one can produce its **stable category** which is an example of a [[suspended category]] rather than triangulated. In particular, its suspension is not invertible in general it is in the case when the exact category is Frobenius (and in particular when it is abelian). [[Alexander Rosenberg|A. Rosenberg]] has devised a generalization of suspended category to a nonadditive setting.

Finally there are more complete descriptions of stability in notions of [[stable model category|stable model categories]], studied mainly in 1990s, stable Segal categories of French school and [[stable (infinity,1)-category|stable quasi-categories]] of Joyal and of Lurie.

Should there be stable $(\infinity,n)$-categories at some point ?


stable cohomotopy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[generalized cohomology theory]] which is [[Brown representability theorem|represented]] by the [[sphere spectrum]] is also called _stable cohomotopy_, as it is the [[stable homotopy theory]] version of [[cohomotopy]].

Equivalently, it is the cohomological [[duality|dual]] concept to [[stable homotopy homology theory]].

By the [[Pontryagin-Thom theorem]] this is equivalently [[framed manifold|framed]] [[cobordism cohomology theory]].

## Properties

### As algebraic K-theory over $\mathbb{F}_1$
 {#AsAlgebraicKTheoryOverTheFieldWithOneElement}

The following is known as the _Barratt-Priddy-Quillen theorem_:

+-- {: .num_prop #StableCohomotopyIsKTheoryOfFinSet}
###### Proposition
**([[stable cohomotopy]] is K-theory of [[FinSet]])**

Let $\mathcal{C} = $ [[FinSet]] be a [[skeleton]] of the category of [[finite sets]], regarded as a [[permutative category]]. Then the [[K-theory of a permutative category|K-theory of this permutative category]]

$$
  K(FinSet) \;\simeq\; \mathbb{S}
$$

is represented by the [[sphere spectrum]], hence is stable cohomotopy.

=--

This is due to [Barratt-Priddy 72](#BarrattPriddy72) reproved in [Segal 74, Prop. 3.5](#Segal74). See also [Priddy 73](#Priddy73), [Glasman 13](#Glasman13).

+-- {: .num_remark #StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement}
###### Remark
**([[stable cohomotopy]] as [[algebraic K-theory]] over the [[field with one element]])**

Notice that for $F$ a [[field]], the [[K-theory of a permutative category]] of its [[category of modules]] $F Mod$ is its [[algebraic K-theory]] $K F$ (see [this example](K-theory+of+a+permutative+category#OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules))

$$
  K F \;\simeq\; K(F Mod)
  \,.
$$

Now ([[pointed sets|pointed]]) [[finite sets]] may be regarded as the modules over the "[[field with one element]]" $\mathbb{F}_1$ (see [there](field+with+one+element#Modules)):

$$
  \mathbb{F}_1 Mod
  \;=\;
  FinSet^{\ast/}
$$

If this is understood, example \ref{StableCohomotopyIsKTheoryOfFinSet} says that [[stable cohomotopy]] is the algebraic K-theory of the [[field with one element]]:

$$
  \mathbb{S} \;\simeq\; K \mathbb{F}_1
  \,.
$$

=--

This perspective is highlighted in: [Deitmar 06, p. 2](#Deitmar06); [Guillot 06](#Guillot06); [Mahanta 17](#Mahanta17); [Dundas, Goodwillie* McCarthy 13, II 1.2](#DundasGoodwillieMcCarthy13); [Morava](#MoravaSomeBackground), [Connes& Consani 16](#ConnesConsani16); [Chu, Lorscheid & Santhanam 10, Thm. 5.9](#ChuLorscheidSanthanam10). The latter reference generalizes also to [[equivariant stable Cohomotopy]] and [[equivariant K-theory]].

[[!include Segal completion -- table]]

### The third stable framed bordism group

The [[third stable homotopy group of spheres]] is the [[cyclic group]] of [[order of a group|order]] 24:

$$
  \array{
    \pi_3^s &\simeq& \mathbb{Z}/24
    \\
    [h_{\mathbb{H}}] &\leftrightarrow& [1]
  }
$$

where the generator $[1] \in \mathbb{Z}/24$ is represented by the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$.

Under the [[Pontrjagin-Thom isomorphism]], identifying the [[stable homotopy groups of spheres]] with the [[bordism ring]] $\Omega^{fr}_\bullet$ of [[stable framing|stably framed]] manifolds (see at _[[MFr]]_), this generator is represented by the [[3-sphere]] (with its left-invariant framing induced from the identification with the [[Lie group]] [[SU(2)]] $\simeq$ [[Sp(1)]] )

$$
  \array{
    \pi_3^s & \simeq & \Omega_3^{fr}
    \\
    [h_{\mathbb{H}}] & \leftrightarrow & [S^3]
    \,.
  }
$$

Moreover, the relation $2 4 [S^3] \,\simeq\, 0$ is represented by the [[complement]] of 24 [[open balls]] inside [[generalized the|the]] [[K3]]-manifold ([MO:a/44885/381](https://mathoverflow.net/a/44885/381), [MO:a/218053/381](https://mathoverflow.net/a/218053/381)).

### Kahn-Priddy theorem

The [[Kahn-Priddy theorem]] characterizes a comparison map between stable cohomotopy and [[cohomology]] with [[coefficients]] in the infinite [[real projective space]] $\mathbb{R}P^\infty \simeq B \mathbb{Z}/2$.

### Boardman homomorphisms

#### To ordinary cohomology

Consider the [[unit]] morphism

$$
  \mathbb{S} \longrightarrow H \mathbb{Z}
$$

from the [[sphere spectrum]] to the [[Eilenberg-MacLane spectrum]] of the [[integers]]. For any [[topological space]]/[[spectrum]] postcomposition with this morphism induces [[Boardman homomorphisms]] of [[cohomology groups]] (in fact of [[commutative rings]])

\[
  \label{BoardmandCohomotopyToOrdinaryCohomology}
  b^n
  \;\colon\;
  \pi^n(X)
  \longrightarrow
  H^n(X, \mathbb{Z})
\]

from the [[stable cohomotopy]] of $X$ in degree $n$ to its [[ordinary cohomology]] in degree $n$.

+-- {: .num_prop #BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology}
###### Proposition
**(bounds on ([[cokernel|co-]])[[kernel]] of [[Boardman homomorphism]] from [[stable cohomotopy]] to [[integral cohomology]])**

If $X$ is a [[CW-spectrum]] which

1. is $(m-1)$-[[n-connected object of an (infinity,1)-topos|(m-1)-connected]]

1. of dimension $d \in \mathbb{N}$

then

1. the [[kernel]] of the [[Boardman homomorphism]] $b^n$ (eq:BoardmandCohomotopyToOrdinaryCohomology) for

   $$
     m \leq n\leq d -1
   $$

   is a $\overline{\rho}_{d-n}$-[[torsion subgroup|torsion group]]:

   $$
     \overline{\rho}_{d-n} ker(b^n) \;\simeq\; 0
   $$

1. the [[cokernel]] of the [[Boardman homomorphism]] $b^n$ (eq:BoardmandCohomotopyToOrdinaryCohomology) for

   $$
     m \leq n  \leq d - 2
   $$

   is a $\overline{\rho}_{d-n-1}$-[[torsion subgroup|torsion group]]:

   $$
     \overline{\rho}_{d-n-1} coker(b^n) \;\simeq\; 0
   $$

where

$$
  \overline{\rho}_{i}
   \;\coloneqq\;
   \left\{
   \array{
     1 &\vert& i\leq 1
     \\
     \underoverset{j = 1}{i}{\prod}
       exp\left(
         \pi_j\left( \mathbb{S}\right)
       \right)
     &\vert&
     \text{otherwise}
   }
   \right.
$$

is the [[multiplication|product]] of the [[exponent of a group|exponents]] of the [[stable homotopy groups of spheres]] in [[positive number|positive]] degree $\leq i$.

=--

([Arlettaz 04, theorem 1.2](#Arlettaz04))

#### To topological modular forms

Write $\mathbb{S}$ for the [[sphere spectrum]] and _[[tmf]]_ for the [[connective spectrum]] of [[topological modular forms]].

Since [[tmf]] is an [[E-∞ ring|E-∞]][[ring spectrum]], there is an essentially unique homomorphism of [[E-∞ ring|E-∞]][[ring spectra]]

$$
  \mathbb{S}
  \overset{e_{tmf}}{\longrightarrow}
  tmf
  \,.
$$

Regarded as a morphism of [[generalized homology]]-theories, this is called the [[Hurewicz homomorphism]], or rather the [[Boardman homomorphism]] for $tmf$

+-- {: .num_prop}
###### Proposition
**(Boardman homomorphism in $tmf$ is 6-connected)**

The [[Boardman homomorphism in tmf]]

$$
  \mathbb{S}
  \overset{e_{tmf}}{\longrightarrow}
  tmf
$$

induces an [[isomorphism]] on [[stable homotopy groups]]
(hence from the [[stable homotopy groups of spheres]] to the stable homotopy groups of tmf), up to degree 6:

$$
  \pi_{\bullet \leq 6}(\mathbb{S})
  \underoverset{\simeq}{\pi_{\bullet \leq 6}(e_{tmf})}{\longrightarrow}
  \pi_{\bullet\leq 6}(tmf)
  \,.
$$

=--

([Hopkins 02, Prop. 4.6](Boardman+homomorphism+in+tmf#Hopkins02), [DFHH 14, Ch. 13](Boardman+homomorphism+in+tmf#DFHH14))

## Related concepts

[[!include flavours of cohomotopy -- table]]

\linebreak

* [[Boardman homomorphism]]

* [[Spanier-Whitehead duality]]

\linebreak

[[!include flavours of cobordism cohomology theories -- table]]

## References

The concept of stable Cohomotopy as such:

* {#Adams74} [[Frank Adams]], part III, section 6, p. 204 of: _[[Stable homotopy and generalised homology]]_, 1974

* [[John Rognes]], p. 1 of: _The sphere spectrum_, 2004 ([[RognesSphereSpectrum.pdf:file]])

Discussion of stable Cohomotopy as [[framed manifold|framed]] [[cobordism cohomology theory]]:

* {#Stong68} [[Robert Stong]], Chapter IV, Example 1, p. 40 of _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html))

Discussion of stable Cohomotopy of [[Lie groups]]:

* C. T. Stretch, _Stable cohomotopy and cobordism of abelian groups_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 90, Issue 2 September 1981, pp. 273-278 ([doi:10.1017/S0305004100058734](https://doi.org/10.1017/S0305004100058734))

* Ken-ichi Maruyama, _$e$-invariants on the stable cohomotopy groups of Lie groups_, Osaka J. Math. Volume 25, Number 3 (1988), 581-589 ([euclid:ojm/1200780982](https://projecteuclid.org/euclid.ojm/1200780982))

* Sławomir Nowak, _Stable cohomotopy groups of compact spaces_, Fundamenta Mathematicae 180 (2003), 99-137 ([doi:10.4064/fm180-2-1](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/180/2))

The identification of stable cohomotopy with the [[K-theory of a permutative category|K-theory of the permutative category]] of [[finite sets]] is due to

* {#BarrattPriddy72} [[Michael Barratt]], [[Stewart Priddy]], _On the homology of non-connected monoids and their associated groups_, Commentarii Mathematici Helvetici, December 1972, Volume 47, Issue 1, pp 1–14 ([doi:10.1007/BF02566785](https://link.springer.com/article/10.1007/BF02566785))

* {#Segal74} [[Graeme Segal]], _Categories and cohomology theories_, Topology vol 13, pp.  293-312,  1974  (doi:10.1016/0040-9383(74)90022-6, [pdf](http://ncatlab.org/nlab/files/SegalCategoriesAndCohomologyTheories.pdf))

see also:

* {#Priddy73} [[Stewart Priddy]], _Transfer, symmetric groups, and stable homotopy theory_, in _Higher K-Theories_, Springer, Berlin, Heidelberg, 1973. 244-255 ([pdf](https://link.springer.com/content/pdf/10.1007/BFb0067060.pdf))

* {#Glasman13} [[Saul Glasman]], _The multiplicative Barratt-Priddy-Quillen theorem and beyond_, talk at *AMS Sectional Meeting* **1095** (2013) [[webpage](https://www.ams.org/meetings/sectional/2199_program_ss7.html), [pdf](http://math.mit.edu/~sglasman/bpq-beamer.pdf)]

* [[Pedro Boavida de Brito]], [[Ieke Moerdijk]], Thm. 1.2 in: *Dendroidal spaces, $\Gamma$-spaces and the special Barratt-Priddy-Quillen theorem*, Journal für die reine und angewandte Mathematik **2020** 760 (2020) 229-265 [[arXiv:1701.06459](https://arxiv.org/abs/1701.06459), [doi:10.1515/crelle-2018-0002](https://doi.org/10.1515/crelle-2018-0002)]

The resulting interpretation of stable cohomotopy as [[algebraic K-theory]] over the [[field with one element]] is amplified in the following texts:

* {#DundasGoodwillieMcCarthy13} [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], chapter II, section 1.2 of  _The local structure of algebraic K-theory_, Springer (2013) [[doi:10.1007/978-1-4471-4393-2](https://doi.org/10.1007/978-1-4471-4393-2)]

* {#Deitmar06} [[Anton Deitmar]], _Remarks on zeta functions and K-theory over $\mathbb{F}_1$_ ([arXiv:math/0605429](https://arxiv.org/abs/math/0605429))

* {#Guillot06} [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](https://arxiv.org/abs/math/0612327))

* {#Mahanta17} [[Snigdhayan Mahanta]], _G-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem_, J. Homotopy Relat. Struct. 12 (4), 901-930, 2017 ([arXiv:1110.6001](https://arxiv.org/abs/1110.6001))

* {#ChuLorscheidSanthanam10} Chenghao Chu, [[Oliver Lorscheid]], Rekha Santhanam, *Sheaves and K-theory for $\mathbb{F}_1$-schemes*, Advances in Mathematics, **229** 4, (2012) 2239-2286 [[arxiv:1010.2896](https://arxiv.org/abs/1010.2896), [doi:10.1016/j.aim.2011.12.023](https://doi.org/10.1016/j.aim.2011.12.023)]

see also

* {#MoravaSomeBackground} [[Jack Morava]], _Some background on Manin's theorem $K(\mathbb{F}_1) \sim \mathbb{S}$_ ([pdf](http://www.alainconnes.org/docs/Morava.pdf), [[MoravaSomeBackground.pdf:file]])

* {#ConnesConsani16} [[Alain Connes]], [[Caterina Consani]], _Absolute algebra and Segal's Gamma sets_, Journal of Number Theory 162 (2016): 518-551 ([arXiv:1502.05585](https://arxiv.org/abs/1502.05585))

* {#Berman18} [[John D. Berman]], p. 92 of: _Categorified algebra and equivariant homotopy theory_, PhD thesis (2018)  [[arXiv:1805.08745](https://arxiv.org/abs/1805.08745), [pdf](http://www.people.virginia.edu/~jdb8pc/Thesis.pdf)]

The [[Kahn-Priddy theorem]] is due to

* {#KahnPriddy72} [[Daniel Kahn]], [[Stewart Priddy]], _Applications of the transfer to stable homotopy theory_, Bull. Amer. Math. Soc. Volume 78, Number 6 (1972), 981-987 ([Euclid](https://projecteuclid.org/euclid.bams/1183534135))

Discussion of stable Cohomotopy as [[framed manifold|framed]] [[cobordism cohomology theory]]:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 5 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

The relation to [[β-rings]] is discussed in

* E. Vallejo, _Polynomial operations from Burnside rings to representation functors_, J. Pure Appl. Algebra, 65 (1990), pp. 163–190.

* E. Vallejo, _Polynomial operations on stable cohomotopy_, Manuscripta Math., 67 (1990), pp. 345–365

* E. Vallejo, _The free β-ring on one generator, J. Pure Appl. Algebra, 86 (1993), pp. 95–108.

* [Guillot 06](#Guillot06)

see also

* [[Jack Morava]], Rekha Santhanam, _Power operations and absolute geometry_ ([pdf](http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf))

Discussion of [[Boardman homomorphisms]] from stable cohomotopy is in

* {#Arlettaz04} [[Dominique Arlettaz]], _The generalized Boardman homomorphisms_, Central European Journal of Mathematics March 2004, Volume 2, Issue 1, pp 50-56 ([doi:10.2478/BF02475949](https://doi.org/10.2478/BF02475949))

A lift of [[Seiberg-Witten invariants]] to classes in [[circle group]]-[[equivariant stable cohomotopy]] is discussed in

* _A stable cohomotopy refinement of Seiberg-Witten invariants: I_ ([arXiv:math/0204340](http://arxiv.org/abs/math/0204340))

* _A stable cohomotopy refinement of Seiberg-Witten invariants: II_ ([arXiv:math/0204267](http://arxiv.org/abs/math/0204267))

On ([[stable cohomotopy|stable]]) [[motivic cohomology|motivic]] [[Cohomotopy]] of [[schemes]] (as  [[motivic homotopy theory|motivic homotopy classes]] of maps into [[motivic sphere|motivic]] [[Tate spheres]]):

* [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], *Euler class groups and motivic stable cohomotopy*, Journal of the EMS **24** 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)]

* [[Samuel Lerbet]], *Motivic stable cohomotopy and unimodular rows*, Advances in Mathematics **436** 109415 (2024) [[arXiv:2206.11688](https://arxiv.org/abs/2206.11688), [doi:10.1016/j.aim.2023.109415](https://doi.org/10.1016/j.aim.2023.109415)]

[[!redirects stable Cohomotopy]]

[[!redirects stable cohomotopy theory]]
[[!redirects stable Cohomotopy theory]]

[[!redirects stable cohomotopy cohomology theory]]
[[!redirects stable Cohomotopy cohomology theory]]

[[!redirects MFr]]

[[!redirects framed bordism theory]]
[[!redirects framed bordism theories]]

[[!redirects framed cobordism homology theory]]
[[!redirects framed cobordism homology theories]]

[[!redirects framed bordism homology theory]]
[[!redirects framed bordism homology theories]]

[[!redirects framed Cobordism]]

[[!redirects framed bordism ring]]
[[!redirects framed bordism rings]]

[[!redirects framed cobordism ring]]
[[!redirects framed cobordism rings]]

[[!redirects Barrat-Priddy-Quillen theorem]]


stable complex structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _stable complex structure_ in a [[manifold]] is an _[[almost complex structure]]_ on its [[stable tangent bundle]].

## Related concepts

* [[Thom spectrum]], [[cobordism ring]]

## References

* _Stable complex and $Spin^c$-structures_  (Appendix D in "Moment maps, Cobordisms, and Hamiltonian Group Actions" V. L. Ginzburg, V. Guillemin, and Y. Karshon.)([[StableComplexSpinC.pdf:file]])

[[!redirects stable complex structures]]

[[!redirects stable almost complex structure]]
[[!redirects stable almost complex structures]]


stable derivator

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

# Contents
* table of contents
{: toc}

## Idea

Just as a [[derivator]] is a notion which lies in between a [[model category]] or a [[(∞,1)-category]] and its [[homotopy category]], a *stable* (or *triangulated*) *derivator* is a notion which lies in between a [[stable model category]] or a [[stable (∞,1)-category]] and its homotopy [[triangulated category]].

Stable derivators are a useful refinement of triangulated categories, since they contain enough information so that [[homotopy limit]] and colimit constructions can be performed "coherently" and desired maps and objects can be specified by true universal properties.  This resolves many common issues with triangulated categories stemming from the fact that at the level of a homotopy category, certain desirable maps can only be stipulated to exist with some weak properties, but not characterized precisely among the class of maps that have those properties.

## Definition
 {#Definition}

As is the case for [[stable (∞,1)-categories]], being stable is a [[stuff, structure, property|property]] of a derivator rather than a structure (or more precisely, it is a [[property-like structure]]).  In fact, an $(\infty,1)$-category is stable precisely when its underlying derivator is.

Write $\square$ for the "free-living commutative square"
$$\array{ & \overset{}{\to} & \\
  \downarrow && \downarrow\\
  & \underset{}{\to} & }$$
and let $L$ and $R$ denote the upper-left [[span]] and the lower-right [[cospan]] as subcategories of $\square$, with inclusions $i\colon L\to \square$ and $j \colon R\to \square$.  For a [[derivator]] $D$, an object $X\in D(\square)$ is said to be **cartesian** if the unit $X\to j_* j^* X$ is an isomorphism, and dually **cocartesian** if the counit $i_! i^* X\to X$ is an isomorphism.  Since $i$ and $j$ are fully faithful, it follows from a general theorem about derivators that $i_!$ and $j_*$ are also fully faithful; thus being cartesian is equivalent to being of the form $j_* Y$ for some $Y\in D(R)$, and likewise being cocartesian is equivalent to being of the form $i_! Z$.

A derivator $D$ is **stable** (or **triangulated**) if it is [[pointed derivator|pointed]], and moreover an $X\in D(\square)$ is cartesian if and only if it is cocartesian.  Such a square is then called **bicartesian**.

## Properties

### Triangulation

One of the central facts about stable derivators is:

+-- {: .un_theorem}
###### Theorem (Maltsiniotis)
If $D$ is a stable derivator, then each category $D(X)$ is a [[triangulated category]] in a canonical way.
=--

We describe the constructions when $X=1$.  The shift/suspension functor $S\colon D(1)\to D(1)$ is defined by
$$ S = b^* i_! a_* $$
where $a\colon 1\to L$ is the inclusion of the top-left vertex of the span, and $b\colon 1\to \square$ is the inclusion of the bottom-right vertex of the square.  In other words, for an object $X\in D(1)$ its suspension is defined by the [[homotopy pushout]]
$$\array{X & \overset{}{\to} & 0\\
  \downarrow && \downarrow\\
  0 & \underset{}{\to} & S X.}$$
This makes sense in the more general context of any [[pointed derivator]], but the stability axiom guarantees that $S$ is actually an equivalence of categories.  Its inverse being given by the obvious dual "loop space" construction (which in a general pointed derivator is only right adjoint to it).  This provides a motivation for the stability axiom: it is a generalization of the statement that every object is the loop space of its suspension and the suspension of its loop space.

One can also prove that $S$ is also a [[copower]] with the pointed [[circle]] $S^1$ in a suitable sense.  In particular, since every object is isomorphic to a double suspension, it is a cogroup object; thus $D(X)$ is canonically an [[Ab-enriched category]].

Let $Q$ denote the category
$$\array{ & \overset{}{\to} & & \to &\\
  \downarrow && \downarrow && \downarrow\\
  & \underset{}{\to} & & \to &}$$
with inclusions $m,n,p\colon \square \to Q$ of the left and right squares and the outer rectangle, respectively.  An object $X\in D(Q)$ is **bicartesian** if $m^*X$, $n^*X$, and $p^*X$ are all bicartesian.

Now consider a bicartesian object $X\in D(Q)$ of the form:
$$\array{A & \overset{f}{\to} & B & \to & 0\\
  \downarrow && \downarrow^g && \downarrow\\
  0 & \to & C & \underset{h}{\to} & D.}$$
In other words, we stipulate that the restrictions to $D(1)$ along the inclusions of the lower-left and upper-right vertices be zero objects.  Now since $X$ is bicartesian, the outer square
$$\array{A & \overset{}{\to} & 0\\
  \downarrow && \downarrow\\
  0& \underset{}{\to} & D}$$
is bicartesian, and thus induces an isomorphism $D \cong S A$.  Thus, from $X$ we can extract a "triangle"
$$ A \overset{f}{\to} B \overset{g}{\to} C \overset{h}{\to} S A$$
and we define the *distinguished triangles* in $D(1)$ to be those isomorphic to triangles obtained in this way.

One can then prove the axioms of a triangulated category.

+-- {: .un_remark}
###### Remark
If $ A \overset{f}{\to} B \overset{g}{\to} C \overset{h}{\to} S A$ is a distinguished triangle in a triangulated category, then $ A \overset{f}{\to} B \overset{g}{\to} C \overset{-h}{\to} S A$ is also an "exact" triangle in the sense that it induces [[long exact sequences]] in [[homology]] and [[cohomology]], but it is not in general distinguished.  On the other hand, if
$$\array{A & \overset{f}{\to} & B & \to & 0\\
  \downarrow && \downarrow^g && \downarrow\\
  0 & \to & C & \underset{h}{\to} & D.}$$
is bicartesian in an $(\infty,1)$-category, then so is
$$\array{A & \overset{f}{\to} & B & \to & 0\\
  \downarrow && \downarrow^g && \downarrow\\
  0 & \to & C & \underset{-h}{\to} & D.}$$
This seeming paradox is resolved by noticing that although these two bicartesian diagrams have the same object $D$ at their lower-right-hand corner, the different maps $h$ and $-h$ cause these diagrams to induce different isomorphisms $D \cong S A$.  The isomorphism for the latter diagram incorporates an extra minus sign, relative to the first one, causing these two diagrams to both induce the *same* triangle $ A \overset{f}{\to} B \overset{g}{\to} C \overset{h}{\to} S A$ in the homotopy category.
=--

## Related concepts

* [[triangulated category]]

* [[pretriangulated dg-category]], [[enhanced triangulated category]]

* **stable derivator**

* [[stable (∞,1)-category]]

## References

See [[derivator]] for general references about derivators, and also [[pointed derivator]].  References particularly pertaining to the stable version include:

* [[Alex Heller]], "Stable homotopy theories and stabilization" [MR](http://www.ams.org/mathscinet-getitem?mr=1431157)

* [[Denis-Charles Cisinski]] and Amnon Neeman, "Additivity for derivator K-theory", [MR](http://www.ams.org/mathscinet-getitem?mr=2382732)

* Jens Franke, _Uniqueness theorems for certain triangulated categories with an Adams spectral sequence_, [K-theory archive](http://www.math.uiuc.edu/K-theory/0139/)

* [[Moritz Groth]], _Derivators, pointed derivators, and stable derivators_ ([pdf](http://www.math.uni-bonn.de/~mgroth/groth_derivators.pdf))

* [[Ian Coley]], _Stabilization of derivators revisited_, 2018, [arxiv](https://arxiv.org/abs/1802.04343)

[[!redirects stable derivators]]
[[!redirects triangulated derivator]]
[[!redirects triangulated derivators]]


stable Dold-Kan correspondence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Dold-Kan correspondence]] [[stabilization|stabilizes]]
to identify _unbounded_ [[chain complexes]] with certain [[spectra]].

## Statement

### In terms of abelian combinatorial spectra

The following theorem was established in ([Kan 63, prop. 5.8](#Kan63))

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

The [[category of unbounded chain complexes]] is [[equivalence of categories|equivalent]] to the
category of [[combinatorial spectra]] [[internalization|internal]] to [[abelian groups]].

=--

Similarly to the unstable case,
the above categories, when interpreted as ∞-categories,
are also equivalent to the ∞-category of module spectra over
the Eilenberg-MacLane ring spectrum of the integers.
See the section _[stable Dold-Kan correspondence](module+spectrum#StableDoldKanCorrespondence)_
at _[[module spectrum]]_.

### In terms of EM-module spectra
 {#InTermsOfEMModuleSpectra}

\begin{proposition}
\label{StableDoldKan}
For $R$ any [[ring]] (or [[ringoid]], even) there is a [[Quillen equivalence]]

$$
  H R ModSpectra
  \;\simeq\;
  Ch_\bullet(R Mod)
$$

between a [[model category]] of  $H R$-[[module spectra]] over the [[Eilenberg-MacLane spectrum]] $H R$ and the [[model structure on unbounded chain complexes]] of ordinary $R$-[[modules]].

This presents a corresponding [[equivalence of (∞,1)-categories]]. If $R$ is a [[commutative ring]], then this is an equivalence of [[symmetric monoidal (∞,1)-categories]].
\end{proposition}

{#DerivedCategory} (Notice that the [[homotopy category of a model category|homotopy category]] of the [[model structure on unbounded chain complexes|model structure on $R$-chain complexes]] is the "[[derived category]]" of [[Mod|$R Mod$]].)

This equivalence on the level of [[homotopy categories]] is due to ([Robinson 87](#Robinson87)). The refinement to a [[Quillen equivalence]] is due to [Shipley 2007, Thm. 1.1] and (#Shipley02)[Schwede & Shipley 2003, theorem 5.1.6](#SchwedeShipley03). See also the discussion at _[[stable model categories]]_. A direct description as an [[equivalence of (∞,1)-categories]] appears as [[Higher Algebra|Lurie HA, Thm. 7.1.2.13]].

For $R = \mathbb{Q}$ the [[rational numbers]], then theorem \ref{StableDoldKan} may be thought of as a stable analog of classical [[rational homotopy theory]], see at _[[rational stable homotopy theory]]_ for more on this.

More generally:

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

For $R$ a [[commutative ring]], then there is a [[zig-zag]] of [[Quillen equivalences]] between a [[model structure for ring spectra]] over $H R$ and [[model structure on dg-algebras]] over $R$.

In particular the induced [[total derived functors]] constitute an [[equivalence of categories|equivalence]] of [[homotopy categories]]:

$$
  H R AlgSpec
   \underoverset
     {\underset{\Theta}{\longrightarrow}}
     {\overset{H}{\longleftarrow}}
     {\simeq}
  dgAlg_R
$$

=--

([Shipley 02, theorem 1.1](#Shipley02))

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

For $A$ any [[dg-algebra]], then there is a [[Quillen equivalence]]

$$
  (H A) ModSpec \simeq_{Quillen} A Mod
$$

between $H A$-[[module spectra]] and [[dg-modules]] over $A$.

Dually, for $E$ an $H \mathbb{Z}$-[[algebra spectrum]], then there is a [[Quillen equivalence]]

$$
  E ModSpec \simeq_{Quillen} (\Theta E) Mod
$$

where $H(-)$ and $\Theta(-)$ are from theorem \ref{HRAlgerasAreRdgAlgebras}.

=--

([Schwede-Shipley 03, theorem 5.1.6](#SchwedeShipley03), [Shipley 02, corollary 2.15](#Shipley02))

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

The [[forgetful functor|forgetful]] [[(∞,1)-functor]] $H R Mod \longrightarrow $ [[Spectra]] preserves [[(∞,1)-limits]], so that (after [[simplicial localization]] $L$) we have an [[(∞,1)-functor]]

$$
  DK
  \;\colon\;
  L_{qi} Ch_\bullet(R)
    \stackrel{\simeq}{\longrightarrow}
  H R Mod
   \stackrel{}{\longrightarrow}
  Spectra
$$

from the [[(∞,1)-category of chain complexes]] to the [[(∞,1)-category of spectra]] which preseres [[(∞,1)-limits]]. In particular therefore a [[presheaf]] of [[chain complexes]] (as it appears in [[abelian sheaf cohomology]]/[[hypercohomology]]) which satisfies [[descent]] (for some given [[(∞,1)-site]] structure, hence which is an [[(∞,1)-sheaf]]/[[∞-stack]] of chain complexes) maps under the stable Dold-Kan correspondence $DK$ to an [[(∞,1)-sheaf of spectra]].

=--

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

For $X$ a [[topological space]] and $R$ a ring, let $C_\bullet(X, R)$ be the standard [[chain complex]] for [[singular homology]] $H_\bullet(X, R)$ of $X$ with coefficients in $R$.

Under the stable Dold-Kan correspondence, prop. \ref{StableDoldKan}, this ought to be identified with the [[smash product]] $(\Sigma^\infty_+ X) \wedge H R$ of the [[suspension spectrum]] of $X$ with the [[Eilenberg-MacLane spectrum]]. Notice that by the general theory of [[generalized homology]] the [[homotopy groups]] of the latter are again singular homology

$$
  \pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R)
  \,.
$$

While the correspondence $(\Sigma^\infty_+ X) \wedge H R \sim C_\bullet(X,R)$ under the above equivalence is suggestive, maybe nobody has really checked it in detail. It is sort of stated as true for instance on p. 15  of ([BCT](#BCT)).

=--

### Monoidal version in terms of EM-module specta

More in detail we have the following statement.

Let $R \coloneqq H \mathbb{Z}$ be the [[Eilenberg-MacLane spectrum]] for the [[integers]].

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

There is a zig-zag of [[monoidal Quillen adjunction|lax monoidal]] [[Quillen equivalences]]

$$
  H \mathbb{Z} Mod
    \stackrel{\overset{Z}{\longrightarrow}}{\underset{U}{\leftarrow}}
  Sp^\Sigma(sAb)
    \stackrel{\overset{L}{\leftarrow}}{\underset{\phi^* N}{\longrightarrow}}
  Sp^\Sigma(Ch_+)
   \stackrel{\overset{D}{\longrightarrow}}{\underset{R}{\leftarrow}}
  Ch_\bullet
  \,,
$$

between [[monoidal model categories]] satisfying the [[monoid axiom in a monoidal model category]]:

* the model structure for $H \mathbb{Z}$-[[module spectra]];

* the [[model structure on symmetric spectrum objects]] in [[simplicial abelian group]]s and in [[chain complex]]es;

* and the [[model structure on chain complexes]] (unbounded).

This induces a [[Quillen equivalence]] between the corresponding [[model structures on monoids]] in these [[monoidal category|monoidal categories]], which on the left is the model structure on $H \mathbb{Z}$-algebra spectra and on the right the [[model structure on dg-algebras]]:

$$
  H \mathbb{Z} Alg \simeq dgAlg_\mathbb{Z}
  \,.
$$

=--

This is due to ([Shipley 02](#Shipley02)). The corresponding [[equivalence of (∞,1)-categories]] for $R$ a commutative rings with the intrinsically defined [[(∞,1)-category]] of [[En-algebra|E1-algebra]] objects on the left appears as ([[Higher Algebra|Lurie HA, prop. 7.1.4.6]]). The equivalence on the level of [[homotopy categories]] was proved in ([Stanley 97, theorem 1.1.4](#Stanley97)).

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

This is a stable version of the [[monoidal Dold-Kan correspondence]]. See there for more details.

=--

## Related concepts

* [[∞-Dold-Kan correspondence]]

* [[rational stable homotopy theory]]

## References

* {#Kan63} [[Daniel Kan]], _Semisimplicial spectra_, Illinois J. Math. Volume 7, Issue 3 (1963), 463-478. ([euclid:ijm/1255644953](http://projecteuclid.org/euclid.ijm/1255644953))

* {#Robinson87} [[Alan Robinson]], *The extraordinary derived category*, Math. Z. **196** 2 (1987) 231-238 [[doi:10.1007/BF01163657](https://doi.org/10.1007/BF01163657)]

* {#SchwedeShipley03} [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_, Topology 42 (2003), 103-153 ([pdf](http://www.math.uic.edu/~bshipley/classTopFinal.pdf), doi:10.1016/S0040-9383(02)00006-X)

* {#Shipley02} [[Brooke Shipley]], _$H \mathbb{Z}$-algebra spectra are differential graded algebras_, Amer. Jour. of Math. 129 (2007) 351-379. [[arXiv:math/0209215](http://arxiv.org/abs/math/0209215), [jstor:40068065](https://www.jstor.org/stable/40068065)]

* {#Jardine91} [[John Frederick Jardine]], _Stable Dold-Kan theory_, section 4.6 of _Generalized Étale cohomology theories_, Modern Birkhäuser classics (1991)

* {#BCT} [[Andrew Blumberg]], [[Ralph Cohen]], [[Constantin Teleman]], _Open-closed field theories, string topology and Hochschild homology_ ([arXiv:0906.5198](http://arxiv.org/abs/0906.5198))

* [[Jacob Lurie]], section 7.1.4 of _[[Higher Algebra]]_

* [[Jacob Lurie]], _[[Deformation Theory]]_

* {#Stanley97} Donald Stanley, _Closed model categories and monoidal categories_, Ph.D. Thesis, University of Toronto (1997) ([pdf](https://tspace.library.utoronto.ca/bitstream/1807/10762/1/NQ27731.pdf))


stable equality

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

# Contents
* table of contents
{: toc}

## Definition ##

In [[constructive mathematics]], a [[set]] $X$ has __stable equality__ if any two [[elements]] of $X$ are [[equality|equal]] if (hence iff) they are [[double negation|not not]] equal.

## Examples ##

* Every [[set]] with [[decidable equality]] has stable equality, but not conversely.

* Every [[set]] $S$ with a [[tight apartness relation]] $\#$ has stable equality, because even in [[constructive mathematics]], given elements $x \in S$ and $y \in S$, $\neg \neg \neg (x \# y) \iff \neg (x \# y)$, and because $\neg (x \# y) \iff x = y$, $\neg \neg (x = y) \iff x = y$, and thus $S$ is stable.

* In particular, the [[Dedekind real numbers]] have stable equality, and any Heyting [[subfield]] of the [[Dedekind real numbers]] has stable equality.

## See also ##

* [[stable relation]]
* [[decidable equality]]
* [[stable equivalence relation]]

## References

* {#BauerSwan18} [[Andrej Bauer]] and [[Andrew Swan]], *Every metric space is separable in function realizability*, 2018, [arxiv](https://arxiv.org/abs/1804.00427)

[[!redirects stable equalities]]


stable equivalence of vector bundles

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_prop #StableEquivalenceOfVectorBundles}
###### Definition
**(stable equivalence of [[topological vector bundles]])**

Let $X$ be a [[topological space]]. Define an [[equivalence relation]] $\sim_{stable}$ on [[topological vector bundles]]
over $X$ by declaring two vector bundles $E_1 E_2 \in Vect(X)$ to be equivalent if there exists a
[[trivial vector bundle]] $X \times k^n$ of some [[rank]] $n$ such that after [[tensor product of vector bundles]]
with this trivial bundle, both bundles become [[isomorphism|isomorphic]]

$$
  \left(
    E_1
      \sim_{stable}
    E_2
  \right)
    \;\Leftrightarrow\;
  \underset{n \in \mathbb{N}}{\exists}
  \left(
     E_1 \otimes_X (X \times k^n)
       \;\simeq\;
     E_2 \otimes_X (X \times k^n)
  \right)
  \,.
$$

If $E_1 \sim_{stable} E_2$ we say that $E_1$ and $E_2$ are _stably equivalent vector bundles_.

=--

## Related concepts

* [[topological K-theory]]

[[!redirects stable equivalences of vector bundles]]

[[!redirects stable equivalence class of vector bundles]]
[[!redirects stable equivalence classes of vector bundles]]


stable equivalence relation

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

=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Similar to how equality is stable if its [[double negation]] implies equality, an *equivalence relation* ought is **stable** if its [[double negation]] [[implication|implies]] equality.

## Definition

In [[set theory]], an [[equivalence relation]] $\equiv$ on a set $S$ is **stable** if for all $a \in S$ and $b \in S$, if $\neg\neg(a \equiv b)$ then $a \equiv b$.

In [[type theory]], a ([[proposition]]-valued) [[equivalence relation]] $\equiv$ on a type $T$ is **stable** if for all $a:T$ and $b:T$, there is a function $p(a, b):\neg\neg(a \equiv b) \to (a \equiv b)$.

## Properties

* The negation of every stable equivalence relation is an [[apartness relation]].

## See also

* [[equivalence relation]]
* [[stable equality]]
* [[inequality space]]
* [[univalent equivalence relation]]

[[!redirects stable equivalence relations]]


stable factorization system

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An [[orthogonal factorization system]] $(E,M)$ on a [[category]] $C$ with [[pullbacks]] is called **stable** if also the left class $E$ is [[stability under pullback|stable under pullback]].

## Properties

### In terms of indexed left adjoints
 {#InTermsOfIndexedLeftAdjoints}

For a general (orthogonal) factorization system $(E,M)$, the factorizations show that for all [[objects]] the [[full subcategory|full inclusion]] $M/x \to C/x$ (where $M/x$ consists of morphisms in $M$ with [[target]] $x$) has a
[[left adjoint]], hence is a [[reflective subcategory]].

The factorization system is stable if and only if these left adjoints form an [[indexed functor]] --- that is, they commute with the [[base change|pullback functors]] $f^* \colon C/y \to C/x$.

### Stable reflective factorization systems
 {#StableReflectiveFactorizationSystems}

A [[reflective factorization system]] on a finitely [[complete category]] is stable if and only if its corresponding [[reflector]] preserves [[finite limits]] (is a [[left exact functor]]).

The analogous statement also holds in [[(∞,1)-category theory]], or rather at least in [[locally cartesian closed (∞,1)-categories]]. A discussion of this and formal proof in terms of [[homotopy type theory]] is in ([Shulman](#Shulman)).

A stable reflective factorization system is sometimes called **local**.

## Examples

* In an a [[topos]], [[epimorphism]] are stable under [[pullback]] and hence the [[(epi, mono) factorization system]] in a topos is stable.

  More generally, in an [[(∞,1)-topos]] for all $n \in \mathbb{N}$ the [[(n-epi, n-mono) factorization system]] (see there for more details) is a stable [[orthogonal factorization system in an (∞,1)-category]].

## Related concepts

* [[reflective factorization system]]

* [[closure operator]]

## References

The notion appears for instance in

* [[Max Kelly]], _A note on relations relative to a factorization system_, Lecture Notes in Mathematics, 1991, Volume 1488 (1991)

* Stefan Milius, _Relations in categories_, PhD thesis ([pdf](https://www8.cs.fau.de/ext/milius/thesis/thesis_a4.pdf))

On the relation between stable factorization systems and the [[Beck-Chevalley condition]] of the associated fibrations:

* J. Hughes and [[Bart Jacobs]], _Factorization systems and fibrations: Toward a fibred Birkhoff variety
theorem_, Electr. Notes in Theor. Comp. Sci., 69 (2002)

Discussion of the example [[(epi, mono) factorization system]] in [[toposes]] (for more see at *[[regular category]]*, [here](regular+category#ToposesAreRegular)):

* {#LackSobocínski06} [[Stephen Lack]], [[Pawel Sobocínski]], Lemma 18 in:  *Toposes are adhesive*, in: *Graph Transformations* ICGT 2006,  Lecture Notes in Computer Science **4178**, Springer (2006) 184-198 [[doi:10.1007/11841883_14](https://doi.org/10.1007/11841883_14), [pdf](http://users.ecs.soton.ac.uk/ps/papers/toposesAdhesive.pdf)]

Discussion of reflective stable factorization systems in the context of [[(∞,1)-category theory]] (and with an eye towards [[cohesive homotopy type theory]]):

* {#Shulman} [[Mike Shulman]], _Axiomatic cohesion in HoTT_ (2011) ([web](http://homotopytypetheory.org/2011/11/02/axiomatic-cohesion-in-hott/))

Stable factorisation systems are called **complete factorisation systems** in the following, where they are related to [[comprehension schemes]]:

* [[Clemens Berger]], [[Ralph M. Kaufmann]], *Comprehensive Factorization Systems*, Tbilisi Math. J. 10 (2017), 255-277 ([doi:10.1515/tmj-2017-0112](https://projecteuclid.org/journals/tbilisi-mathematical-journal/volume-10/issue-3/Comprehensive-factorisation-systems/10.1515/tmj-2017-0112.short))

[[!redirects stable factorization systems]]
[[!redirects local factorization system]]
[[!redirects local factorization systems]]


stable form

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[differential geometry]] and [[special holonomy]], a _stable form_ ([Hitchin, def. 1.1](#Hitchin)) is an element $\phi$ in some [[exterior power]] $\wedge^p V$ of some [[vector space]] $V$,such that the [[orbit]] of this element under the [[action]] of the [[general linear group]] $GL(V)$ is _[[open subset|open]]_.

The [[stabilizer groups]] of such stable forms correspond to flavors of [[special holonomy]].

## Related concepts

* [[definite form]]

## References

* {#Hitchin} [[Nigel Hitchin]], _Special holonomy and beyond_, Clay Mathematics Proceedings ([pdf](https://people.maths.ox.ac.uk/hitchin/hitchinlist/clay.pdf))

[[!redirects stable forms]]


stable general linear group

Let $R$ be an associative ring with 1. As usual $GL_n(R)$ will denote the [[general linear group]] of $n\times n$ non-singular matrices over $R$.   There is an embedding of $GL_n(R)$ into $GL_{n+1}(R)$ sending  a matrix $M = (m_{i,j})$ to the matrix $M^\prime$ obtained from $M$ by adding an extra row and column of zeros except that $m^\prime_{n+1,n+1} = 1$.  This gives a nested sequence of groups

  $$GL_1(R)\subset GL_2(R)\subset \ldots \subset GL_n(R)\subset GL_{n+1}(R)\subset \ldots$$

and we write $GL(R)$ for the colimit (union in this case) of these. It will be called the _stable general linear group_ over $R$.


Stable homotopy and generalised homology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

This page collects links related to

* [[John Frank Adams]],

  **Stable homotopy and generalized homology**

  Chicago Lectures in Mathematics, 1974

  The University of Chicago Press 1974

  [ucp:bo21302708](https://www.press.uchicago.edu/ucp/books/book/chicago/S/bo21302708.html)

  [pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/adams-shgh.pdf), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/Adams-SHGH-latex.pdf)

on [[stable homotopy theory]] and [[generalised homology]] theory, with emphasis on [[complex cobordism cohomology theory|complex]] [[cobordism theory]], [[complex oriented cohomology theory]], and the [[Adams spectral sequence]]/[[Adams-Novikov spectral sequence]] (today: "[[chromatic homotopy theory]]").

Consists of three lectures, each meant to be readable on their own, and there is overlap in topics. It's part III that begins with an actual introduction to stable homotopy theory, and so the beginner might prefer to start reading with Part III. Also notice that on p. 87 it says that the material there in part II is to be regarded as superseding part I.

A very detailed and readable account based on these lectures is

* [[Stanley Kochmann]], _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

The big story emerging here was later further developed in

* [[Mike Hopkins]], _[[Complex oriented cohomology theories and the language of stacks]]_, 1999

* [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986/2003

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, 2010

This is about understanding the absolute base space [[Spec(S)]] by [[covering]] it with Spec([[MU]]). See at _[Adams spectral sequences -- As derived descent](Adams%20spectral%20sequence#DefinitionInHigherAlgebra)_.

#Contents#
* table of contents
{:toc}

## Part I

### 2. Cobordism groups

* [[cobordism ring]]

* [[cobordism theory]]

* [[zero-section into Thom space of universal line bundle is weak equivalence]]

### 3. Homology

* [[Novikov operations]]

### 4. The Conner-Floyd Chern classes

* [[Conner-Floyd Chern class]]

### 5. The Novikov operations

### 6. The algebra of all operations

### 7. Scholium on Novikov's operations

### 8. Complex manifolds

* [[Umkehr map]]

## Part II -- Quillen's work on formal groups and complex cobordism
 {#PartII}

### 0. Introduction

### 1. Formal groups

* [[formal group law]]

### 2. Examples from algebraic topology

* [[complex oriented cohomology theory]]

* generalized [[Chern classes]]

### 3. Reformulation

* [[coalgebra]]

* [[Hopf algebra]]

### 4. Calculations in $E$-homology and cohomology

* [[universal coefficient theorem]]

* [[universal complex orientation on MU]] (Lemma 4.6, example 4.7)

### 5. Lazard's universal ring

* [[Lazard ring]]

### 6. More calculations in $E$-cohomology

* [[Hurewicz homomorphism]]

* [[Boardman homomorphism]]

* [[ordinary homology spectra split]]

* [[homology of MU]]

### 7. The structure of Lazard's universal ring $L$

* [[Lazard's theorem]]

### 8. Quillen's theorem

* [[Quillen's theorem on MU]]

### 9. Corollaries

### 10. Various formulae in $\pi_\bullet(MU)$

### 11. $MU_\bullet(MU)$

* [[Landweber-Novikov theorem]]

### 12. Behaviour of the Bott map

### 13. $K_\bullet(K)$

### 14. The Hattori-Stong theorem

### 15. Quillen's idempotent cohomology operations

### 16. The Brown-Peterson spectrum

* [[BP]]

### 17. $KO_\bullet(KO)$

## Part III
 {#PartIII}

### 1. Introduction

* [[smash product]] (of [[pointed topological spaces]])

### 2. Spectra

* [[spectrum]]

* [[Omega spectrum]]

* [[homotopy group of a spectrum]]

* [[suspension spectrum]]

* [[Thom spectrum]]

* [[Eilenberg-MacLane spectrum]]

* [[topological K-theory]]

* [[Thom spectrum]]

* [[cylinder spectrum]]

* [[generalized (Eilenberg-Steenrod) cohomology]]

* [[stable homotopy category]]

* [[cup product]]

* [[CW-spectrum]]

### 3. Elementary properties of the category of CW-spectra

* [[Adams category]]

> There is much to love in his book, but not in the foundational part on CW spectra. ([[Peter May]], [MO comment](http://math.stackexchange.com/a/53783/58526))

### 4. Smash products

* [[smash product of spectra]]

### 5. Spanier-Whitehead duality

* [[Spanier-Whitehead duality]]

### 6. Homology and cohomology

* [[generalized (Eilenberg-Steenrod) cohomology]]

* [[Brown's representability theorem]]

* [[stable cohomotopy]]

* [[stable homotopy groups of spheres]]

* [[topological K-theory]]

### 7. The Atiyah-Hirzebruch spectral sequence

* [[exact couple]]

* [[spectral sequence]]

* [[Atiyah-Hirzebruch spectral sequence]]

### 8. The inverse limit and its derived functors

* [[derived functor]], [[homotopy limit]]

### 9. Products

* [[cup product]]

### 10. Duality in manifolds

* [[ring spectrum]], [[module spectrum]]

* [[orientation in generalized cohomology]]

### 11. Applications in K-theory

### 12. The Steenrod algebra and its dual

* [[Steenrod algebra]]

### 13. A universal coefficient theorem

* [[universal coefficient theorem]]

### 14. A category of fractions

What Adams tries to construct here -- the [[localization]] of the [[stable homotopy category]] at the class of $E$-equivalences -- was later constructed by ([Bousfield 79](Bousfield+localization+of+spectra#Bousfield79)). See at _[[Bousfield localization of spectra]]_.

### 15. The Adams spectral sequence

* [[Adams spectral sequence]]

### 16. Applications to $\pi_\bullet(bu \wedge X)$; modules over $K[x,y]$

### 17. Structure of $\pi_\bullet(bu \wedge bu)$

category: reference

[[!redirects Stable homotopy and generalized homology]]