Descent data, prestacks and stacks in groupoids over the fppf site

Let $p\text{\hspace{0.17em}⁣}:\mathcal{S}\to (Sch\mathrm{/}S{)}_{fppf}$ be a category fibred in groupoids (Categories fibred in groupoids over a site) over the fppf site (Fppf coverings and the fppf site). For a morphism $T^{\mathrm{\prime }}\to T$ of $S$-schemes and an object $x$ of $\mathcal{S}$ over $T$ we write $x{\mathrm{\mid }}_{T^{\mathrm{\prime }}}$ for the value of a chosen pullback along $T^{\mathrm{\prime }}\to T$; different choices are canonically isomorphic and the definitions below do not depend on them.

For an fppf covering $\{T_i\to T{\}}_{i\in I}$ write $T_{ij}=T_i{\times }_T{T}_j$ and $T_{ijk}=T_i{\times }_T{T}_j{\times }_T{T}_k$ (Fibre product of schemes). Descent data for a family of objects $x_i$ of ${\mathcal{S}}_{T_i}$ is a family of isomorphisms ${\phi }_{ij}\text{\hspace{0.17em}⁣}:x_i{\mathrm{\mid }}_{T_{ij}}⟶x_j{\mathrm{\mid }}_{T_{ij}}$ in ${\mathcal{S}}_{T_{ij}}$, one for each ordered pair $(i,j)$, satisfying the cocycle condition ${\phi }_{ik}={\phi }_{jk}\circ {\phi }_{ij}$ over $T_{ijk}$ (after pulling back along the three projections and using the canonical identifications). With the evident notion of morphism — a family of morphisms $x_i\to y_i$ compatible with the ${\phi }_{ij}$ — these data form a category $\mathrm{D}\mathrm{D}(\{T_i\to T\},(x_i))$, and there is a base-change functor ${\mathcal{S}}_T\to \mathrm{D}\mathrm{D}(\{T_i\to T\})$ sending $x$ to $(x{\mathrm{\mid }}_{T_i},\text{canonical isomorphisms})$.

The fibred category $\mathcal{S}$ is a prestack when for all objects $x,y$ of ${\mathcal{S}}_T$ the presheaf $(Sch\mathrm{/}T{)}_{fppf}\to \mathrm{S}\mathrm{e}\mathrm{t},U\mapsto {\mathrm{Mor}}_{{\mathcal{S}}_U}(x{\mathrm{\mid }}_U,y{\mathrm{\mid }}_U)$ is an fppf sheaf (Fppf sheaves of sets and sheafification); equivalently, for every fppf covering the diagram of morphism sets is an equalizer.

The fibred category $\mathcal{S}$ is a stack in groupoids when it is a prestack and every descent datum of objects is effective: for every fppf covering $\{T_i\to T\}$ the base-change functor ${\mathcal{S}}_T\to \mathrm{D}\mathrm{D}(\{T_i\to T\})$ is an equivalence of categories. Effectivity is thus the precise sense in which objects are glued from descent data.

A presheaf of sets $F$ on $(Sch\mathrm{/}S{)}_{fppf}$ determines a category fibred in groupoids ${\mathcal{S}}_F$ whose fibre category over $T$ is the discrete groupoid on the set $F(T)$ (the groupoid with only identity morphisms). Descent data for ${\mathcal{S}}_F$ over a covering $\{T_i\to T\}$ amount to a family of elements of the $F(T_i)$ whose pullbacks agree on all $T_{ij}$, and effectivity amounts to gluing them to an element of $F(T)$; consequently ${\mathcal{S}}_F$ is a stack in groupoids (a stack in setoids) exactly when $F$ is an fppf sheaf. In particular, assuming the Axiom of Choice (The Axiom of Choice) as in Scheme morphisms satisfy fppf descent, every $S$-scheme $X$, whose represented presheaf is then an fppf sheaf, determines the stack in groupoids ${\mathcal{S}}_X$ whose fibre category over $T$ is the discrete groupoid on ${\mathrm{Mor}}_S(T,X)$; this is the Yoneda embedding of schemes into stacks.