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This question arises when I am reading Dugger and Isaksen's "Hypercovers in topology". According to Definition 4.1 in that paper, A hypercover of a space $X$ is an augmented simplicial space $U_*\to X$ such that the maps $U_n\to M^X_nU$ are open covering maps for all $n \geq 0$. Here $ M^X_nU$ denotes the nth matching object of $U_*$ computed in the category $s(\mathcal{Top}\downarrow X)$.

On the other hand by Definition A.4 in the same paper, a simplicial space $X_*$ is said to be split, or to have free degeneracies, if there exist subspaces $N_k\hookrightarrow X_k$ such that the canonical map $$ \coprod_{\sigma}N_{\sigma}\to X_k $$ is an isomorphism. Here the variable σ ranges over all surjective maps in $\bf{\Delta}$ of the form $ [k]\to [n]$, $N_{\sigma}$ denotes a copy of $N_n$, and the map $N_{\sigma}\to X_k$ is the one induced by $\sigma^*: X_n\to X_k$. The idea is that the spaces $N_k$ represent the ‘non-degenerate’ part of $X_k$, sitting inside of $X_k$ as a direct summand.

Now Theorem 4.3 of the paper claims that if $U_*\to X$ is a a hypercover then the maps $\text{hocolim}U_*\to |U_*|\to X$ are all weak equivalences. In the proof it says that "the simplicial object $U_*$ is Reedy cofibrant since it has free degeneracies,"

It is clear that the Cech nerve of an open cover has free degeneracies. My question is: is it also true for any hypercover in the topological setting?

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  • $\begingroup$ Don't know the answer, but: if I'm understanding the definition of hypercover, it seems to induce a simplicial object in the topos generated by open Sets of $X$, with an epimorphism property at every level; D+I have arranged their definition the other way around, starting with sSpaces over X and then paring down what kind of Spaces are in each level. $\endgroup$ – Jesse C. McKeown Aug 4 '15 at 20:51
  • $\begingroup$ @JesseC.McKeown That is correct. In topos-theoretic language, my argument amounts to the observation that a sheaf whose espace étalé is Hausdorff is a decidable object, i.e. $A \times A \cong \Delta_A \amalg B$, where $\Delta_A$ is the diagonal; a simplicial sheaf that is degreewise decidable will then have free degeneracies. $\endgroup$ – Zhen Lin Aug 4 '15 at 23:48
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Here is a proof for the case where $X$ is a Hausdorff space. Note that each $U_n$ is also Hausdorff in this case.

A standard argument shows that the face operators of $U_\bullet$ are (surjective) local homeomorphisms, so the degeneracy operators of $U_\bullet$ are open embeddings. On the other hand, degeneracy operators are split monomorphisms, and any split monomorphism in the category of Hausdorff spaces is a closed embedding. Thus, the image of any degeneracy operator of $U_\bullet$ is a "direct summand" (i.e. clopen subspace). It more or less follows that $U_n$ decomposes according to the formula you stated.

Note that the above argument can be applied to simplicial Hausdorff spaces other than hypercovers.

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