Locality and descent
Quite generally, one says that an object in a category of higher category satisfies descent along a given morphism in if it is a -local object, hence if the induced map – the descent morphism
is an equivalence. We may read this as saying that every collection of -data on “descends” down along to .
In the context the hom object is also called the descent object.
While roughly synonyms, typically one speaks of “descent” instead of locality when is a category of presheaves or higher presheaves ((2,1)-presheaves, (∞,1)-presheaves, (∞,n)-presheaves).
In this case, in turn, the objects above are typically representables of a given site (or higher site) and is either the Cech nerve of a covering family with respect to a chosen coverage/Grothendieck topology, or is the colimit of this Čech nerve: the corresponding sieve (the codescent object?).
The descent condition then says that the presheaf satisfies the sheaf-condition (stack-condition, (∞,1)-sheaf/∞-stack-condition, etc.) for this given covering family.
Whether one takes to be the Cech nerve or the corresponding sieve depends on homotopical details of the setup. If is taken to be an (∞,1)-category, then it typically does not matter. But if is instead just a homotopical category presenting the desired higher category, then needs to satisfy some extra conditions (such as cofibrancy) to ensure that is indeed the correct descent object, and not too small.
For instance when working with the injective model structure on simplicial presheaves, every object is cofibrant and we can take to be the sieve. But when working with the projective model structure then (as discussed there) needs to be split, which means that we need to use the Cech nerve and even ensure that the corresponding covering family behaves like a good cover (or, more generally, form a split hypercover).
For ordinary presheaves
For ordinary presheaves, a descent object is a set of matching families
More in detail, let be a site, let be an object, a covering family and the corresponding sieve.
Then for any presheaf on , the descent object with respect to this covering is the hom set
This is discussed in detail at sheaf, so just briefly:
the sieve may be realized as the coequalizer
Accordingly the hom out of this realizes the descent object as the equalizer
Writing this out in components shows that this is the set of matching families.
If the descent morphism
is an isomorphism one says that satisfies the sheaf-condition with respect to the cover . If this morphism is only a monomorphism one says that satisfies the separated presheaf-condition.
For groupoid valued presheaves / pseudofunctors
For Grpd a 2-functor (hence a “pseudofunctor” if is an ordinary category regarded as a 2-category) and for a covering morphism in , the descent object now is a groupoid
If the descent morphism
is an equivalence of groupoids, one says that satisfies the (2,1)-sheaf- or stack-condition with respect to the cover . If it is just a full and faithful functor, one says (sometimes) that satisfies the condition for a separated prestack with respect to this cover.
Similar statements hold for the case of 2-functors with values in Cat. Here one also often talks about a stack-condition, though less ambiguous would be to speak of 2-sheaf-conditions.
By the Grothendieck construction one may identifiy pseudofunctors equivalently with fibered categories (or just categories fibered in groupoids for ) over , and all of the above has analogs in this dual description.
For simplicial presheaves
For strict -category-valued presheaves
In (Street) a proposal for a definition of descent objects for presehaves with values in strict ω-categories was proposed. Additional homotopical conditions to ensure that this gives the right answer were discussed in (Verity).
Let be a category, let be morphisms where the parallel arrows are seen as a diagram, let be an object.
Applying the functor to this sequence gives
If this diagram is for all an equalizer diagram is called codescent object for the diagram .
Let be a diagram where , , satisfying for and (these are the identities characterizing a truncated cosimplicial category).
Then the descent category of has as objects pairs where , such that and and a morphism consists of a morphism such that .
Let , be categories.
See also descent and categroy of descent data?.
Discussion related to the computation of descent objects is also at model structure on cosimplicial simplicial sets.
See also the references at descent.
A definition of descent objects for presheaves with values in strict -categories was proposed in
A discussion of a missing condition on this definition is in