Cocontinuous functors

In the stacks project a site is defined as in Artin’s notes on Grothendieck topologies, and not as in SGA4. Hence also our notion of a cocontinuous functor u : C —> D between sites C and D is a bit different (than Verdier’s original one). Namely, it means that, given any object U of C, and any covering {V_j —> u(U)}_j in D there should exist a covering {U_i —> U} in C such that the family of morphisms {u(U_i) —> u(U)}_i refines the given family {V_j —> u(U)}_j.

The reason this definition is convenient is twofold. On the one hand, it is easy to check that a functor is cocontinuous, and on the other hand, it is true that a cocontinuous functor u : C —> D gives rise to a morphism of topoi g : Sh(C) –> Sh(D). For example, for a sheaf G on D the sheaf g^{-1}(G) is the sheaf associated to the presheaf U |—> G(u(U)).

Here are two examples

  • Let f : X —> Y be an open continuous map of topological spaces. Then the functor u(U) = f(U) is a cocontinuous functor between the site of opens of X and the site of opens of Y. The induced morphism of topoi Sh(X) —> Sh(Y) is the usual one.
  • Let f : X —> Y be a morphism of schemes. The “forgetful” functor u : (Sch/X)_{fppf} —> (Sch/Y)_{fppf} is cocontinuous and the associated morphism of topoi is the usual morphism of big topoi f_{big} : Sh((Sch/X)_{fppf}) —> Sh((Sch/Y)_{fppf}).

A little less standard are the following examples, which are related to the discussion in the previous post. Suppose that i : X_0 —> X is a closed immersion of schemes defined by a sheaf of ideals of square zero. Consider the functor of sites u : X_{lisse-etale} —> (X_0)_{lisse-etale}, or u : (Sch/X)_{syntomic} —> (Sch/X_0)_{syntomic} given by the rule V |—> V_0 = X_0 \times_X V. Then you can check that u is cocontinuous (in both cases). Hence we obtain a morphisms of topoi

  • g_{lisse-etale} : Sh(X_{lisse-etale}) —> Sh((X_0)_{lisse-etale})
  • g_{syntomic} : Sh((Sch/X)_{syntomic}) —> Sh((Sch/X_0)_{syntomic})

These maps are somehow contracting the topos associated to X onto the topos associated to X_0. Now in the second case the functor u also gives rise to a morphism of topoi in the opposite direction, namely i_{big} (for the syntomic topology), but I think neither i_{big} nor g_{syntomic} is an equivalence of topoi. In the first case, even though u is continuous, it does not define a morphism of topoi in the other direction.

In any case, cocontinuous functors are very useful and often easier to deal with than the better known continuous ones. For more information see the chapter on Sites and Sheaves.

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