nLab
stable (infinity,1)-category

Context

(,1)(\infty,1)-Category theory

Stable Homotopy theory

Contents

Idea

A stable (∞,1)-category CC, is a pointed (,1)(\infty,1)-category with finite limits which is stable under forming loop space objects:

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

Ω:CC \Omega : C \to C

is an equivalence with inverse the suspension object functor

CC:Σ. C \leftarrow C : \Sigma \,.

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

Indeed, every (,1)(\infty,1)-category with finite limits has a free stabilization to a stable (,1)(\infty,1)-category Stab(C)Stab(C), and the objects of Stab(C)Stab(C) are the spectrum objects of CC.

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

Definition

In a pointed (∞,1)-category CC with zero object 00, the kernel of a morphism g:YZg : Y \to Z is the (∞,1)-pullback

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

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

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

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

An arbitrary commuting square in CC of the form

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

is a triangle in CC. 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 Xker(g)X\to\ker(g) and coker(f)Zcoker(f)\to Z. In particular, for every exact triangle there is a canonical morphism coker(ker(g)Y)Zcoker(ker(g)\to Y)\to Z and for every coexact triangle there is a canonical morphism Xker(Ycoker(f))X\to ker(Y\to coker(f)).

Definition

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

  • for every morphism in CC 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 E_\infty-spaces.

Constructions in stable \infty-categories

Looping and delooping

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

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

but also that, conversely, every object XX has a suspension object ΣX\Sigma X defined by the coexact triangle

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

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

Ω:CC \Omega : C \to C
Σ:CC \Sigma : C \to C

which are autoequivalences of CC that are inverses of each other.

Stabilization

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

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

Sp(C):=holim(CΩCΩC). Sp(C) := holim( \cdots \to C \stackrel{\Omega}{\to} C \stackrel{\Omega}{\to} C ) \,.

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

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

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

Properties

Enrichment over spectra

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

The homotopy category: triangulated categories

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

  • the translation functor T:Ho(C)Ho(C)T : Ho(C) \to Ho(C) comes from the suspenesion functor Σ:CC\Sigma : C \to C;

  • the distinguished triangles in Ho(C)Ho(C) are pieces of the fibration sequences in CC.

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 (,1)(\infty,1)-category has an underlying derivator, and the underlying derivator of a stable (,1)(\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 (,1)(\infty,1)-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 (,1)(\infty,1)-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.

(e.g. Cohn 13, see also Schwede)

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

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

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

Proposition

Let CC and DD be (∞,1)-categories and Func(C,D)Func(C,D) the (∞,1)-category of (∞,1)-functors between them.

Its stabilization is equivalent to the functor category into the stabilization of CC:

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

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

Stab(PSh (,1)(C))Func(C,Sp). 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 CC is equivalent to an (accessible) left-exact localization of a stabilized (,1)(\infty,1)-presheaf (,1)(\infty,1)-category: there exists a small (,1)(\infty,1)-category EE and an adjunction

ClexStab(PSh(E))Func(E,Sp). 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 (,1)(\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:

Proposition

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

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

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

This exhibits the stabilization of H\mathbf{H}:

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

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

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).

Dold-Kan correspondence

Examples

References

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

This appears in a more comprehensive context of higher algebra as section 1 of

A brief introduction is in

A detailed discussion of how kk-linear dg-categories present kk-linear stable (,1)(\infty,1)-categories is in

A diagram of the interrelation of all the models for stable (,1)(\infty,1)-categories with a useful list of literature for each can be found in these seminar notes:

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

The enrichment over spectra is made precise in

Revised on November 29, 2014 21:24:10 by Ingo Blechschmidt (137.250.162.16)