Lemma 6.57.
Let \(T\) be a topos and let \(u\colon C \to T\) be a functor from a small category. Consider the following four statements:
The restriction functor
\[u^*\colon T \to \PSh(C), \qquad X \mapsto \Hom_T(u(-),X)\]is fully faithful.
\(T\) is generated under colimits by the image of \(u\).
Every \(X \in T\) is covered by the image of \(u\), i.e. there exist objects \(c_i \in C\) and an effective epimorphism \(\bigsqcup_i u(c_i) \twoheadrightarrow X\).
The counit \(u_!u^*(X) \to X\) is \(\infty\)-connected for every \(X \in T\).
Then we have implications (1) \(\Rightarrow\) (2) \(\Rightarrow\) (3) \(\Rightarrow\) (4). Moreover, if \(T\) is hypercomplete then all four statements are equivalent.
Proof
To see that (1) implies (2), let \(u_!\colon \PSh(C) \to T\) be the left Kan extension of \(u\). Since \(u^*\) is fully faithful, the counit \(u_!u^* \to \id_T\) is an equivalence. As every presheaf is a colimit of representables and \(u_!(y(c)) = u(c)\), we conclude that every object of \(T\) is a colimit of objects in the image of \(u\).For (2) \(\Rightarrow\) (3), observe that the collection of objects receiving such an effective epimorphism is closed under colimits.For (3) \(\Rightarrow\) (4), the pointwise formula for left Kan extensions gives
\[u_!u^*(X) \simeq \colim_{(c,\alpha) \in C_{/X}} u(c),\]
where \(C_{/X}\) is the category of pairs \((c,\alpha)\) with \(c\in C\) and \(\alpha\colon u(c)\to X\). Write the counit as \[X' := \colim_{(c,\alpha)\in C_{/X}} u(c) \to X\]
induced by the maps \(\alpha\). The chosen cover of \(X\) by objects in the image of \(u\) factors through this counit, so the counit is an effective epimorphism.We now prove inductively that it is \(n\)-connected for every \(n\geq 0\). Assume that the counit is \((n-1)\)-connected for every object of \(T\). Universality of colimits identifies \[X'\times_X X'\iso
\colim_{(c,\alpha),(d,\beta)\in C_{/X}}u(c)\times_Xu(d).\]
Under this identification, descent for the colimit defining \(X'\) identifies the diagonal \(X'\to X'\times_XX'\) with the colimit of the counits \[\colim_{(e,\gamma)\in C_{/\,u(c)\times_X u(d)}}u(e)
\longrightarrow u(c)\times_Xu(d)\]
as \((c,\alpha)\) and \((d,\beta)\) vary. Each of these maps is \((n-1)\)-connected by the induction hypothesis. Since \((n-1)\)-connected morphisms are closed under colimits, the diagonal is \((n-1)\)-connected. Together with effective epimorphy, Theorem 3.22 shows that the counit is \(n\)-connected. Thus it is \(\infty\)-connected.Finally, if \(T\) is hypercomplete and (4) is satisfied, then the counit map \(u_!u^* \to \id\) is a natural isomorphism, and so \(u^*\) is fully faithful, giving (1).