nLab stable (infinity,1)-category

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 spectra.

Indeed, every $(\infty,1)$-category with finite limits has a free stabilization to a stable $(\infty,1)$-category $Stab(C)$, and the objects of $Stab(C)$ 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 $(\infty,1)$-category with a zero object is a pointed $(\infty,1)$-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

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

(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 $X\to ker(Y\to coker(f))$.

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.

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 $(\infty,1)$-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 limits 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 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 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).

The homotopy category: triangulated categories

The 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 functor $\Sigma : C \to C$;

• the distinguished triangles in $Ho(C)$ 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 $(\infty,1)$-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

In direct analogy to how a general (∞,1)-category may be 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, see also Schwede)

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

• an enhanced triangulated category:

  • a 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

Proposition

Let $C$ and $D$ be (∞,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 .

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 left-exact 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.

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:

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 (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

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

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 generators;

then there is a chain of 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)

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:

Corollary

If in prop. 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)

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).

Remark

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

Remark

One way to read this is that formal duals of presentable stable infinity-categories are a model for spaces in (“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 theoretical notion was introduced and studied in

• Jacob Lurie, Stable Infinity-Categories [arXiv:math/0608228

This appears in a more comprehensive context of higher algebra

• Jacob Lurie, as section 1 of: Higher Algebra

A brief introduction is in

• Yonatan Harpaz, Introduction to stable $\infty$-categories, October 2013 (pdf)

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

• Lee Cohn, Differential Graded Categories are k-linear Stable Infinity Categories (arXiv:1308.2587)

• Giovanni Faonte, Simplicial nerve of an A-infinity category (arXiv: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:

• Stefan Schwede, Markus Hausmann, Enhancements of triangulated categories, Graduate Seminar Topology, Bonn 2019-2020 (pdf)

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

• Stefan Schwede, Brooke Shipley, Classification of stable model categories (Hopf pdf) and (arXiv:math/0108143)

The enrichment over spectra is made precise in

• David Gepner, Rune Haugseng, Enriched ∞-categories via non-symmetric ∞-operads (arXiv:1312.3178)