In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic.
Throughout the article, we work with a fixed base category C; for example, C can be the category of all schemes over some fixed scheme equipped with some Grothendieck topology.
; this means that one can construct pullbacks along morphisms in C, up to canonical isomorphisms.
; here, the bracket means we canonically identify different Hom sets resulting from different choices of pullbacks.
is a sheaf of sets with respect to the induced Grothendieck topology on
, etc., An object of this category is called a descent datum.
This category is not well-defined; the issue is that the pullbacks are determined only up to canonical isomorphisms; similarly fiber products are defined only up to canonical isomorphisms, despite the notational practice to the contrary.
In practice, one simply makes some canonical identifications of pullbacks, their compositions, fiber products, etc.
that sends an object to the descent datum that it defines.
A statement like this is independent of choices of canonical identifications mentioned early.
These reformulations of the definitions of prestacks and stacks make intuitive meanings of those concepts very explicit: (1) "fibered category" means one can construct a pullback (2) "prestack in groupoids" additionally means "locally isomorphic" implies "isomorphic" (3) "stack in groupoids" means, in addition to the previous properties, a global object can be constructed from local data subject to cocycle conditions.
Note (2) is automatic if G is fibered in groupoids; e.g., an algebraic stack (since all morphisms are cartesian then.)
is the stack associated to a scheme S in the base category C, then the fiber
is, by construction, the set of all morphisms from U to S in C. Analogously, given a scheme U in C viewed as a stack (i.e.,
Firstly, the obvious square does not commute; instead, for each object
In general, a fiber product of F and G over B is a prestack canonically isomorphic to
When B is the base category C (the prestack over itself), B is dropped and one simply writes
from a scheme S in C viewed as a prestack, the fiber product
of prestacks is a scheme in C. In particular, the definition applies to the structure map
, T a scheme viewed as a prestack, the induced projection
has the property P. Let G be an algebraic group acting from the right on a scheme X of finite type over a field k. Then the group action of G on X determines a prestack (but not a stack) over the category C of k-schemes, as follows.
is the classifying prestack of G and its stackification is the classifying stack of G. One viewing X as a prestack (in fact a stack), there is the obvious canonical map over C; explicitly, each object
Then the above canonical map fits into a 2-coequalizer (a 2-quotient): where t: (x, g) → xg is the given group action and s a projection.
given by Let X be a scheme in the base category C. By definition, an equivalence pre-relation is a morphism
Example: Let an algebraic group G act on a scheme X of finite type over a field k. Take
and then for any scheme T over k let By Yoneda's lemma, this determines a morphism f, which is clearly an equivalence pre-relation.
is an injective function ("étale" means the two possible maps
It is similar to the sheafification of a presheaf and is called stackification.
(The details are omitted for now) As it turns out, it is a stack and comes with a natural morphism
In some special cases, the stackification can be described in terms of torsors for affine group schemes or the generalizations.