Criterion continuous functor | Higher Topos Theory notes

Lemma 6.20.

Let $(C,\tau)$ and $(D,\tau')$ be Grothendieck sites, and let $u\colon C \to D$ be a functor. Assume that every covering sieve $U \hookrightarrow y(X)$ of $X \in C$ is generated by a collection of morphisms $\{f_i\colon U_i \to X\}_{i \in I}$ in $$C$$ satisfying the following conditions:

Pullbacks along $$f_i$$ exist in $$C$$ and are preserved by $u\colon C \to D$ ;
The sieve on $$y(u(X))$$ in $$D$$ generated by the maps $$u(f_i)$$ is a covering sieve.

Then $$u$$ is a continuous functor.

Proof

Let

\Ff

be a

\tau'

-sheaf on

\(D\)

. We need to show that

u^*\Ff

is a

\tau

-sheaf on

\(C\)

. For a covering sieve

U \hookrightarrow y(X)

, let

\Uu = \{f_i \colon U_i \to X\}_{i \in I}

be a collection of morphisms in

\(C\)

generating it satisfying the two conditions. Using Proposition 6.15 and the adjunction

u_! \dashv u^*

, we may equivalently show that the map

(u^*\Ff)(X) \simeq \Hom_{\PSh(D)}(y(u(X)), \Ff) \to \lim_{[n] \in \simp} \Hom_{\PSh(D)}(u_!(\check{C}_n(\Uu)), \Ff)

is an isomorphism. The first assumption on

\Uu

guarantees that the simplicial object

u_!(\check{C}_n(\Uu))

\PSh(D)

agrees with the Čech nerve of the map

\bigsqcup_{i \in I} y(u(U_i)) \to y(u(X))

. The second assumption says that the associated sieve is a covering sieve. So a second application of Proposition 6.15 implies the claim.