2 Replete categories
Let \(\mathcal{C}\) be a category with limits. By \(\mathrm{Ab}(\mathcal{C})\) we denote its abelian group objects.
We say \(\mathcal{C}\) is replete if epimorphisms are stable under sequential limits, i.e., if \((F_n)_{n \in \mathbb {N}}\) is an inverse system in \(\mathcal{C}\) such that \(F_{n+1} \to F_n\) is an epimorphism for all \(n \in \mathbb {N}\), then \(\mathrm{lim}\; F_n \to F_n\) is an epimorphism for all \(n \in \mathbb {N}\).
The following examples are all not needed later, but are useful to gain some intuition for 2.1. The motivating example is:
The category of étale sheaves is rarely replete as the following example shows.
Let \(k\) be a field and \(\bar k\) a separable closure of \(k\). Then \(\mathrm{Shv}(\mathrm{Spec}(k)_{\mathrm{et}})\) is replete if and only if \(\bar k\) is a finite extension of \(k\).
The category of fpqc sheaves, i.e., sheaves on the big fpqc site, is replete.
Let \(\mathcal{C}\) be replete and let \((F_n)_{n \in \mathbb {N}} \to (G_n)_{n \in \mathbb {N}}\) be a morphism of inverse systems in \(\mathcal{C}\). If \(F_n \to G_n\) and \(F_{n+1} \to F_{n} \times _{G_n} G_{n+1}\) are epimorphisms for each \(n \in \mathbb {N}\), then \(\lim F_n \to \lim G_n\) is an epimorphism.
If \(\mathcal{C}\) is replete, countable products are exact.
Let \(\mathcal{C}\) be replete. If \((F_n)_{n \in \mathbb {N}}\) is an inverse system in \(\mathrm{Ab}(\mathcal{C})\) with epimorphic transition maps, then \(\mathrm{lim} \; F_n \cong \mathrm{R}\; \mathrm{lim} \; F_n\).
2.1 Coherent objects
An object \(X\) of \(\mathcal{C}\) is compact if the top element of the poset of subobjects of \(X\) is a compact element.
An object \(X\) of \(\mathcal{C}\) is stable if for all compact \(Y\) and morphisms \(Y \to X\), the fiber product \(Y \times _{X} Y\) is compact.
An object \(X\) of \(\mathcal{C}\) is coherent if it is compact and stable.
2.2 Weakly contractible categories
From now on, we assume that pullbacks in \(\mathcal{C}\) preserve epimorphisms. For example this holds in abelian categories and (pre)topoi.
An object \(F\) of \(\mathcal{C}\) is called weakly contractible if every epimorphism \(G \to F\) has a section. We say \(\mathcal{C}\) has enough weakly contractible objects, if every object admits an epimorphism from a weakly contractible one.
Under our assumptions on \(\mathcal{C}\), an object \(F\) is weakly contractible if and only if it is projective.
We say \(\mathcal{C}\) is locally weakly contractible, if each \(X\) in \(\mathcal{C}\) admits an epimorphism from the coproduct of \(Y_i\), where \(Y_i\) is coherent and weakly contractible.
If \(\mathcal{C}\) is locally weakly contractible, it has enough contractible objects. Indeed, the coproduct of projective objects is projective.
The category of sets is locally weakly contractible.
Let \(\mathcal{C}\) have enough weakly contractible objects and \(F \to G\) be a morphism. Then \(F \to G\) is an epimorphism if and only if for every weakly contractible \(Y\) in \(\mathcal{C}\), the induced map \(F(Y) = \mathrm{Hom}(Y, F) \to \mathrm{Hom}(Y, G) = G(Y)\) is surjective.
Choose an epimorphism \(e \colon P \to G\) where \(P\) is weakly contractible and lift that to \(P \to F\).
If \(\mathcal{C}\) has enough weakly contractible objects, it is replete.
This follows from 2.16, the fact that \(\mathrm{Hom}(Y, -)\) commutes with limits and the corresponding statement in the category of sets.