Prestack

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. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object. 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. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object. Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., groupoid scheme or the prestack of projectivized vector bundles) may not be stacks. Prestacks may be studied on their own or passed to stacks. Since a stack is a prestack, all the results on prestacks are valid for stacks as well. 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. Let F be a category and suppose it is fibered over C through the functor p : F → C {displaystyle p:F o C} ; this means that one can construct pullbacks along morphisms in C, up to canonical isomorphisms. Given an object U in C and objects x, y in F ( U ) = p − 1 ( U ) {displaystyle F(U)=p^{-1}(U)} , for each morphism f : V → U {displaystyle f:V o U} in C, after fixing pullbacks f ∗ x , f ∗ y {displaystyle f^{*}x,f^{*}y} , we let be the set of all morphisms from f ∗ x {displaystyle f^{*}x} to f ∗ y {displaystyle f^{*}y} ; here, the bracket means we canonically identify different Hom sets resulting from different choices of pullbacks. For each g : W → V {displaystyle g:W o V} over U, define the restriction map from f to g: Hom _ ( x , y ) ( V → f U ) → Hom _ ( x , y ) ( W → f ∘ g U ) {displaystyle {underline {operatorname {Hom} }}(x,y)(V{overset {f}{ o }}U) o {underline {operatorname {Hom} }}(x,y)(W{overset {fcirc g}{ o }}U)} to be the composition where a canonical isomorphism g ∗ ∘ f ∗ ≃ ( f ∘ g ) ∗ {displaystyle g^{*}circ f^{*}simeq (fcirc g)^{*}} is used to get the = on the right. Then Hom _ ( x , y ) {displaystyle {underline {operatorname {Hom} }}(x,y)} is a presheaf on the slice category C / U {displaystyle C_{/U}} , the category of all morphisms in C with target U. By definition, F is a prestack if, for each pair x, y, Hom _ ( x , y ) {displaystyle {underline {operatorname {Hom} }}(x,y)} is a sheaf of sets with respect to the induced Grothendieck topology on C / U {displaystyle C_{/U}} . This definition can be equivalently phrased as follows. First, for each covering family { V i → U } {displaystyle {V_{i} o U}} , we 'define' the category F ( { V i → U } ) {displaystyle F({V_{i} o U})} as a category where: writing p 1 : V i × U V j → V i , p 12 : V i × U V j × U V k → V i × U V j {displaystyle p_{1}:V_{i} imes _{U}V_{j} o V_{i},,p_{12}:V_{i} imes _{U}V_{j} imes _{U}V_{k} o V_{i} imes _{U}V_{j}} , etc.,