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$\require{AMScd}$ Let $X$ be a topological space. Here is something that might deserve to be called the singular CW complex of $X$. Let us call it $CW(X)$. It is constructed inductively by skeletons. …

It seems that at least a partial answer can be given using the formalism of $\infty$-topoi: if $X$ is a paracompact Hausdorff space which is locally contractible (in the strong sense discussed above) …

After some more digging I found a (somewhat non-explicit) counterexample to the original question. In his paper "un espace metrique lineaire qui n’est pas un retracte absolu" Cauty constructs a metric …

Milnor proved that any paracompact Hausdorff space which is equi-locally convex (and hence in particular locally contractible) is homotopy equivalent to a CW complex. However, unlike being paracompact …

This is not an answer, but too long for a comment, and hopefully of some use. I believe the theorem you mention should be true for any 5-dimensional Poincaré duality space $X$. You can define the li …

First one should separate between the property and being $p$-complete and process of $p$-completion. In the classical setting, the $p$-completion functor is not so well-behaved for general spaces. For …

It can be shown (see Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?) that if $X$ is a locally contractible paracompact Hausdorff space such that the $ …

I think the answer is no. Suppose there exists an enrichment satisfying your requirements, and let $U: Vect_{\mathbb{C}} \to Vect_{\mathbb{R}}$ be the forgetful functor. Let $C \subseteq Map(\mathbb{R …

Fiber products do commute with sequential colimits of closed embedding in CGWH, but one must remember that neither limits nor colimits in CGWH are the same as the corresponding limits and colimit in t …

Here is my guess. To compare spaces with their notion of strict $\infty$-groupoids (in which everything is strict except inverses) Kapranov and Voevodsky use an intermediate category of Kan diagrammat …

For every $\mathbb{Z}G$-bimodule $M$ there exists a $G$-module $U(M)$ such that the Hochschild cohomology of $\mathbb{Z}G$ with coefficients in $M$ is naturally isomorphic to the group cohomology of $ …

I have not yet considered the new answer in full detail, so apologies for not addressing it. I'm coming back to this question after a while, so I thought I'd share some observations which came up duri …

For question (2), there is actually a left Quillen bifunctor $$ \times_{\mathrm{gr}}: \mathrm{Set}_\Delta^{\mathrm{sc}} \times \mathrm{Set}_\Delta^{\mathrm{sc}} \to \mathrm{Set}_\Delta^{\mathrm{sc}} $ …

No. If this were the case then there would be a section $s: B \to E$ to $f$ induced by the $G$-equivariant map $\widetilde{s}:\widetilde{B} \to \widetilde{B} \times {\rm K}(M,n)$ sending $x$ to $(x,0) …

I'm not sure about Waldhausen categories in general, but if you restrict attention to stable $\infty$-categories (with trivial Waldhausen structure in which all maps are cofibrations) then group compl …