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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 …

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 …

Technically speaking the answer to your question is no, in the sense that the data of $(\alpha,\beta,\gamma,\delta)$ alone does not determine the filtered object $V_0 \subseteq V_1 \subseteq V_2$. How …

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 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 …

1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety i …

As you point out, it indeed seems that in order to get a well-behaved answer one should work in a suitable homotopical setting, for example, that of $\infty$-categories and $\infty$-operads. Using thi …

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 $ …

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}} $ …

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 $ …

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 …

Concerning the first question: the simplicial localization functor $L^H$ induces an equivalence from the relative category of small relative categories to the relative category of small simplicial cat …

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 …

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 …

A particular case of Lurie and Hopkins' ambidexterity theory is that if $G$ is a finite group acting on a $K(n)$-local spectrum $X$ then the norm map $$ X_{hG} \to X^{hG} $$ is a $K(n)$-local equivale …