is an equivalence with inverse the suspension object functor
Notice that the definition of triangulated categories is involved and their behaviour is bad, whereas the definition of stable -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.
(so that is a fibration sequence)
and the cokernel of is the (∞,1)-pushout
An arbitrary commuting square in of the form
is a triangle in . 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 and . In particular, for every exact triangle there is a canonical morphism and for every coexact triangle there is a canonical morphism .
A stable -category is a pointed -category such that
for every morphism in 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 -category should not be confused with that of a stably monoidal -category. A connection between the terms is that the stable (∞,1)-category of spectra is the prototypical stable -category, while connective spectra (not all spectra) can be identified with stably groupal -groupoids, aka infinite loop spaces or -spaces.
The relevance of the axioms of a stable -category is that they imply that not only does every object have a loop space object defined by the exact triangle
but also that, conversely, every object has a suspension object defined by the coexact triangle
These arrange into -endofunctors
which are autoequivalences of that are inverses of each other.
For Top the -category of topological spaces, is the familiar stable (∞,1)-category of spectra (whose homotopy category is the stable homotopy category) used in stable homotopy theory (which gives stable -categories their name).
Hence stable homotopy theory and homological algebra are both special cases of the theory of stable -categories.
The homotopy category of a stable -category – its decategorification to an ordinary category – is less well behaved than the original stable -category, but remembers a shadow of some of its structure: this shadow is the structure of a triangulated category on
the translation functor comes from the suspenesion functor ;
the distinguished triangles in are pieces of the fibration sequences in .
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 -category has an underlying derivator, and the underlying derivator of a stable -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 -category than does its triangulated homotopy 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.
A triangulated category linear over a field can canonically be refined to
a stable -category.
If has characteristic 0, then all these three concepts become equivalent.
Its stabilization is equivalent to the functor category into the stabilization of :
In particular, consider the case ∞Grpd where (= the stable (∞,1)-category of spectra). One has is the (∞,1)-category of (∞,1)-presheaves, and is the (∞,1)-category of (∞,1)-presheaves of spectra, we get
This is StabCat, example 10.13 .
(“stable Giraud theorem”)
Let be an (∞,1)-category. Then is stable and presentable (∞,1)-category if and only if is equivalent to an accessible left-exact localization of the (∞,1)-category of presheaves of spectra on some small (∞,1)-category , so that there is an adjunction
This is Higher Algebra, Proposition 22.214.171.124.
This is the stable analog of the statement that every (∞,1)-category of (∞,1)-sheaves is a left exact localization of an -category of presheaves.
Let be an (∞,1)-topos and write
for the (∞,1)-category of sheaves of spectra on (with respect to its canonical topology), hence the (∞,1)-category of left exact (∞,1)-functors from the opposite (∞,1)-category of to the (∞,1)-category of spectra.
This exhibits the stabilization of :
This is (Lurie "Spectral Schemes", remark 1.2).
Let be a stable model category that is in addition
This is in (Schwede-Shipley, theorem 3.3.3)
An -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 -enriched category is an -ringoid. It is has a single object then (as a pointed category) it is an A-infinity algebra.
If if in prop. 4 there is just one compact generator , then there is a one-object -enriched category, hence an A-infinity algebra , which is the endomorphisms , and the stable model category is its category of modules:
This is in (Schwede-Shipley, theorem 3.1.1)
This is (Schwede-Shipley 03, theorem 5.1.6).
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 diagram of the interrelation of all the models for stable -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 -categories see
The enrichment over spectra is made precise in