equivalences in/of $(\infty,1)$-categories
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
is an equivalence with inverse the suspension object functor
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.
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.
In a pointed (∞,1)-category $C$ with zero object $0$, the kernel of a morphism $g : Y \to Z$ is the (∞,1)-pullback
(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
An arbitrary commuting square in $C$ of the form
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 $X\to ker(Y\to coker(f))$.
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.
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.
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
but also that, conversely, every object $X$ has a suspension object $\Sigma X$ defined by the coexact triangle
These arrange into $(\infty,1)$-endofunctors
which are autoequivalences of $C$ that are inverses of each other.
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
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.
Stable $\infty$-categories are naturally enriched (∞,1)-categories over the (∞,1)-category of spectra (Gepner-Haugseng 13).
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.
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 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.
(e.g. Cohn 13, see also Schwede)
A triangulated category linear over a field $k$ can canonically be refined to
a stable $(\infinity,1)$-category.
If $k$ has characteristic 0, then all these three concepts become equivalent.
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 $C$:
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
This is StabCat, example 10.13 .
(“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
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:
Let $\mathbf{H}$ be an (∞,1)-topos and write
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 $\mathbf{H}$:
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):
Let $\mathcal{C}$ be a stable model category that is in addition
then there is a chain of sSet-enriched Quillen equivalences linking $\mathcal{C}$ to the the spectrum-enriched functor category
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)
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:
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:
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).
This may be thought of as a homotopy-theoretic analog of the Freyd-Mitchell embedding theorem for abelian categories.
One way to read this is that formal duals of presentable stable infinity-categories are a model for spaces in (“derived”) noncommutative algebraic geometry.
t-structure on a stable (∞,1)-category, heart of a stable (∞,1)-category
prime spectrum of a symmetric monoidal stable (∞,1)-category
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
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:
For discussion of the stable model category models of stable $\infty$-categories see
The enrichment over spectra is made precise in
Last revised on September 17, 2021 at 13:14:58. See the history of this page for a list of all contributions to it.