nLab stable (infinity,1)-category

Context

$\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

Models

Stable Homotopy theory

stable homotopy theory

Contents

Idea

A stable (∞,1)-category $C$, is an $\left(\infty ,1\right)$-category with finite limits which is stable under forming loop space objects:

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

$\Omega :C\to C$\Omega : C \to C

is an equivalence with inverse the suspension object functor

$C←C:\Sigma \phantom{\rule{thinmathspace}{0ex}}.$C \leftarrow C : \Sigma \,.

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

Indeed, every $\left(\infty ,1\right)$-category with finite limits has a free stabilization to a stable $\left(\infty ,1\right)$-category $\mathrm{Stab}\left(C\right)$, and the objects of $\mathrm{Stab}\left(C\right)$ are the spectrum objects 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

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 $\left(\infty ,1\right)$-category with a zero object is a pointed $\left(\infty ,1\right)$-category.

Definition

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

$\begin{array}{ccc}\mathrm{ker}\left(g\right)& \to & Y\\ ↓& & {↓}^{g}\\ 0& \to & Z\end{array}$\array{ ker(g) &\to& Y \\ \downarrow && \downarrow^g \\ 0 &\to& Z }

(so that $\mathrm{ker}\left(g\right)\to Y\stackrel{g}{\to }Z$ is a fibration sequence)

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

$\begin{array}{ccc}X& \stackrel{f}{\to }& Y\\ ↓& & ↓\\ 0& \to & \mathrm{coker}\left(f\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X &\stackrel{f}{\to}& Y \\ \downarrow && \downarrow \\ 0 &\to& coker(f) } \,.

An arbitrary commuting square in $C$ of the form

$\begin{array}{ccc}X& \stackrel{f}{\to }& Y\\ ↓& & {↓}^{g}\\ 0& \to & Z\end{array}$\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 $X\to \mathrm{ker}\left(g\right)$ and $\mathrm{coker}\left(f\right)\to Z$. In particular, for every exact triangle there is a canonical morphism $\mathrm{coker}\left(\mathrm{ker}\left(g\right)\to Y\right)\to Z$ and for every coexact triangle there is a canonical morphism $X\to \mathrm{ker}\left(Y\to \mathrm{coker}\left(f\right)\right)$.

Definition

A stable $\left(\infty ,1\right)$-category is a pointed $\left(\infty ,1\right)$-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.

Remark

The notion of stable $\infty$-category should not be confused with that of a 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 stably groupal $\infty$-groupoids, aka infinite loop spaces or ${E}_{\infty }$-spaces.

Constructions in stable $\infty$-categories

Looping and delooping

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

$\begin{array}{ccc}\Omega X& \to & 0\\ ↓& & ↓\\ 0& \to & X\end{array}$\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

$\begin{array}{ccc}X& \to & 0\\ ↓& & ↓\\ 0& \to & \Sigma X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X &\to& 0 \\ \downarrow && \downarrow \\ 0 &\to& \Sigma X } \,.

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

$\Omega :C\to C$\Omega : C \to C
$\Sigma :C\to C$\Sigma : C \to C

which are autoequivalences of $C$ that are inverses of each other.

Stabilization

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

a stable $\left(\infty ,1\right)$-category $\mathrm{Sp}\left(C\right)$ that can be defined as the limit in the (∞,1)-category of (∞,1)-categories

$\mathrm{Sp}\left(C\right):=\mathrm{holim}\left(\cdots \to C\stackrel{\Omega }{\to }C\stackrel{\Omega }{\to }C\right)\phantom{\rule{thinmathspace}{0ex}}.$Sp(C) := holim( \cdots \to C \stackrel{\Omega}{\to} C \stackrel{\Omega}{\to} C ) \,.

For $C=$ Top the $\left(\infty ,1\right)$-category of topological spaces, $\mathrm{Sp}\left(\mathrm{Top}\right)$ is the familiar stable (∞,1)-category of spectra (whose homotopy category is the stable homotopy category) used in stable homotopy theory (which gives stable $\left(\infty ,1\right)$-categories their name).

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

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

Properties

The homotopy category: triangulated categories

The homotopy category $\mathrm{Ho}\left(C\right)$ of a stable $\left(\infty ,1\right)$-category $C$ – its decategorification to an ordinary category – is less well behaved than the original stable $\left(\infty ,1\right)$-category, but remembers a shadow of some of its structure: this shadow is the structure of a triangulated category on $\mathrm{Ho}\left(C\right)$

• the translation functor $T:\mathrm{Ho}\left(C\right)\to \mathrm{Ho}\left(C\right)$ comes from the suspenesion functor $\Sigma :C\to C$;

• the distinguished triangles in $\mathrm{Ho}\left(C\right)$ are pieces of the fibration sequences in $C$.

For details see StabCat, section 3.

Alternately, one can first pass to a stable derivator, and thence to a triangulated category. Any suitably complete and cocomplete $\left(\infty ,1\right)$-category has an underlying derivator, and the underlying derivator of a stable $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-category than does its triangulated homotopy category.

Models

In direct analogy to how a general (∞,1)-category may be presented by model category, a stable $\left(\infty ,1\right)$-categories may be presented by a

or a

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.

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

• a stable $\left(\infty ,1\right)$-category.

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

Stabilization and localization of presheaf $\left(\infty ,1\right)$-categories

Proposition

Let $C$ and $D$ be (∞,1)-categories and $\mathrm{Func}\left(C,D\right)$ the (∞,1)-category of (∞,1)-functors between them.

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

$\mathrm{Stab}\left(\mathrm{Func}\left(C,D\right)\right)\simeq \mathrm{Func}\left(C,\mathrm{Stab}\left(D\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$Stab(Func(C,D)) \simeq Func(C,Stab(D)) \,.

In particular, in the case $D=$ ∞Grpd for which $\mathrm{Func}\left(C,D\right)=\mathrm{Func}\left(C,\infty \mathrm{Grpd}\right)=:{\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)$ is the (∞,1)-category of (∞,1)-presheaves we have with $\mathrm{Stab}\left(\infty \mathrm{Grpd}\right)=\mathrm{Sp}$ (the stable (∞,1)-category of spectra) that

$\mathrm{Stab}\left({\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)\right)\simeq \mathrm{Func}\left(C,\mathrm{Sp}\right)\phantom{\rule{thinmathspace}{0ex}}.$Stab(PSh_{(\infty,1)}(C)) \simeq Func(C,Sp) \,.

This is StabCat, example 10.13 .

Proposition

(“stable Giraud theorem”)

Every stable and presentable (∞,1)-category $C$ is equivalent to an (accessible) left-exact localization of a stabilized $\left(\infty ,1\right)$-presheaf $\left(\infty ,1\right)$-category: there exists a small $\left(\infty ,1\right)$-category $E$ and an adjunction

$C\stackrel{\stackrel{\mathrm{lex}}{←}}{↪}\mathrm{Stab}\left(\mathrm{PSh}\left(E\right)\right)\simeq \mathrm{Func}\left(E,\mathrm{Sp}\right)\phantom{\rule{thinmathspace}{0ex}}.$C \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} Stab(PSh(E)) \simeq Func(E,Sp) \,.

This is StabCat prop 15.9.

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

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

In terms of (stable) model categories, something like an analog of this statement is (Schwede-Shipley, theorem 3.3.3).

References

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

A diagram of the interrelation of all the models for stable $\left(\infty ,1\right)$-categories with a useful list of literature for each are these seminar notes: