Diaconescu cover

We claim that there exists a poset \(D\) with a functor \(f\colon D \to C\), such that:

1. The functor \(f\) is essentially surjective;
2. For all \(x \in D\), the induced functor \(D_{/x} \to C_{/f(x)}\) is essentially surjective.

To this end, let \(\hat{C}\) be a simplicial set whose associated complete Segal anima is \(C\). Define a poset \(P\) as follows: where \((n,\sigma) \leq (n', \sigma')\) if and only if \(n \leq n'\) and \(\sigma = \sigma'\vert_{\Delta^{\{0,1, \dots, n\}}}\). We may define a functor \(N(P) \to \hat{C}\catop\), which sends a sequence \((n_0, \sigma_0) \leq \dots \leq (n_k, \sigma_k)\) to the composite \(\Delta^k \xrightarrow{n_0, \dots, n_k} \Delta^{n_k} \xrightarrow{\sigma_k} \hat{C}\catop\). By passing to realizations, this gives a functor We prove that this satisfies (1) and (2). Essential surjectivity is clear from the \(0\)-simplices. For (2), consider \(P_{(n,\sigma)/} \to (C\catop)_{\sigma(n)/}\) and an arrow \(\alpha\colon\sigma(n)\to c\) in \(C\catop\). Together, \(\sigma\) and \(\alpha\) define a map The inclusion \(\Delta^n\cup_{\{n\}}\Delta^1\hookrightarrow\Delta^{n+1}\) is inner anodyne. Since \(\widehat C\catop\) is a category, we may extend this map to a simplex \(\widetilde\sigma\colon\Delta^{n+1}\to\widehat C\catop\). The object \((n+1,\widetilde\sigma)\) lies above \((n,\sigma)\) and its image represents \(\alpha\). Thus the indicated functor on undercategories is essentially surjective. Passing to opposites gives (2).From now on, fix such an \(f\). For a sieve \(R\hookrightarrow y(x)\) and a map \(\alpha\colon x'\to x\) in \(D\), let \(\langle f(\alpha^*R)\rangle\) denote the sieve on \(f(x')\) generated by the image of \(\alpha^*R\). Declare \(R\) to be a \(\rho\)-cover if \(\langle f(\alpha^*R)\rangle\) is a \(\tau\)-cover of \(f(x')\) for every \(\alpha\). This defines a Grothendieck topology. The maximal sieve is covering and stability under pullback is built into the definition. For transitivity, suppose that \(R\) covers \(x\) and that a sieve \(S\) on \(x\) pulls back to a cover along every map belonging to \(R\). After an arbitrary base change \(x'\to x\), the images under \(f\) of the arrows in the pullback of \(R\) cover \(f(x')\). Over each of them, the image of the corresponding pullback of \(S\) covers. The local character axiom for \(\tau\) then shows that \(\langle f(S|_{x'})\rangle\) covers \(f(x')\).Claim. The composite factors through \(\Shv_{\tau}(C)\). Let \(R\hookrightarrow y(c)\) be a \(\tau\)-covering sieve. It suffices to show that \(f^*R\to f^*y(c)\) becomes an effective epimorphism after \(\rho\)-sheafification. Pull it back along a map \(y(d)\to f^*y(c)\), corresponding to a map \(f(d)\to c\). The resulting sieve \(S\) on \(d\) consists of those arrows \(e\to d\) whose composite \(f(e)\to c\) belongs to \(R\). After any base change \(d'\to d\), the pullback of \(R\) along \(f(d')\to c\) is a \(\tau\)-cover. By (2), every object of its indexing slice is, up to equivalence, the image of an object of \(D_{/d'}\). It follows that the sieve generated by \(f(S|_{d'})\) is this pullback sieve, and hence is a \(\tau\)-cover. Thus \(S\) is a \(\rho\)-cover. This proves the claim and gives a morphism of logoi It remains to prove that \(L_\rho f^*\) detects effective epimorphisms. Let \(X\to Y\) be a map of \(\tau\)-sheaves whose image is an effective epimorphism. To test that \(X\to Y\) is an effective epimorphism, fix \(y(c)\to Y\). By (1), after replacing \(c\) by an equivalent object we may write \(c=f(d)\). The corresponding section of \(f^*Y\) over \(d\) lifts locally through \(f^*X\) after a \(\rho\)-covering sieve \(S\) of \(d\), because the sheafification of \(f^*X\to f^*Y\) is an effective epimorphism. By the definition of \(\rho\), the sieve generated by \(f(S)\) is a \(\tau\)-cover of \(c\). Over this cover the original section lifts through \(X\), so \(X\to Y\) is an effective epimorphism. Hence \(L_\rho f^*\) is conservative on effective epimorphisms, and the corresponding morphism of topoi is surjective.