A sheaf topos from a colimit quotient | Higher Topos Theory notes

Construction 5.89.

Let $$C$$ be a small category with finite colimits, and let $$T$$ be a topos. The category $$C$$ is filtered, so the colimit functor

\colim\colon \Fun(C,T) \to T

preserves colimits and finite limits, since filtered colimits commute with finite limits in a topos. It is therefore a morphism of logoi. Its right adjoint is the constant-diagram functor $X\mapsto\const_X$ . This functor is fully faithful because $$C$$ is weakly contractible, as follows already from the initial object of $$C$$ . Thus $\colim$ is a quotient map.

Choose a small set $\mathcal G$ of generators of $$T$$ , closed under finite products. For $G\in\mathcal G$ , write

y_G\colon C\catop\longrightarrow\Fun(C,T),\qquad X\longmapsto y(X)\otimes G.

A functor $F\colon C\to T$ is constant if and only if it is local with respect to every map $y_G(f)\colon y_G(Y)\to y_G(X)$ , where $f\colon X\to Y$ ranges through the morphisms of $$C$$ and $$G$$ ranges through $\mathcal G$ . Indeed, the corresponding locality map is obtained by applying $\Hom_T(G,-)$ to $F(X)\to F(Y)$ , and the generators detect isomorphisms. Consequently, if

\Sigma_T:=\{\,y_G(f)\mid f\in\Ar(C), G\in\mathcal G\,\},

then the kernel $$K$$ of $\colim$ is the strong saturation $\Sigma_T^s$ .

We denote the monogenic-epigenic factorization of $\colim$ as follows:

\Fun(C,T) \to \Shv(C\catop; T) \to T.

The middle term is the localization at the monogenic part $K^{\mono}$ . We use the notation $\Shv(C\catop;T)$ because, for $T=\An$ , it is the logos of sheaves for the Grothendieck topology in which every morphism of $C\catop$ generates a covering sieve. The notation in general denotes the corresponding $$T$$ -valued sheaf logos.