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In order to have enough freedom, I would prefer to consider a basis $\mathcal{U}$ of open subsets of $X$. In this case, considering $\mathcal{U}$ as a partially ordered set (for the inclusion), there …

I have never thought of a counter example, but I would not bet on a positive answer to the first question. However, the answer to the second question is yes: this version of the singular functor is a …

The natural map from the $1$-fiber product $A\times_C B$ to the the $2$-fiber product $A\times^2_C B$ is an equivalence of categories in the case where $f:A\to C$ is an isofibration, which means that, …

This article of C. Berger gives a general principle to compute the number of $n+k$-cells of $K(\pi,n)$ for an abelian group $\pi$ in terms of "generalised Fibonacci numbers" (see section 4.10 in loc. …

Consider an abelian category $A$ (or, more generally, an exact category in the sense of Quillen), then the category of complexes of $A$ is a category of cofibrant objects with the quasi-isomorphisms a …

I complete the answer of Charles Rezk by some precise references: you may find the answer to your questions in my book Les préfaisceaux commes modèles des types d'homotopie, Astérisque 308 (2006). In …

If, given any fixed cofibrant object $A$, there is a funtor $map(A,-)$ from $M$ to simplicial sets which preserves weak equivalences between fibrant objects and commutes with homotopy limits up to can …

I suspect that you already have one, but here is a proof. I will assume that $M$ and $N$ are combinatorial and that $M$ is left proper (otherwise, I don't think that the literature contains a general …

For $X$ a $0$-connected nice space (say, a CW-complex), and for any group $G$, there is a natural bijection of the following shape $$[X,BG]\simeq Hom(\pi_1(X),G)$$ which can be proved roughly as fol …

If we look at the proof that I know of in the case this is true - see below - we need in fact that, for any small $1$-category $I$, $h_\infty(C^I)$ has small (co)limits and that they can be computed …

The work of Akhil Mathew et al. really is a continuation of Thomason's: the goal is to consider algebraic $K$-theory and its sibblings (such as $TC$) and to see them as cohomology theories of (affine) …

This is more like a comment, but the usual space allowed for comments is too small! It seems (after Toën's work) that we cannot describe rational homotopy types in terms of algebra, but rather in ter …

Quillen's original proof (in Homotopical Algebra, LNM 43, Springer, 1967) is purely combinatorial (i.e. does not use topological spaces): he uses the theory of minimal Kan fibrations, the fact that th …

I do not know any open problem related to this, but there is a situation where the theory would look quite different if this problem was solved because it is built on such cohomological localizations …

If $X$ and $Y$ are finite type nilpotent spaces, then $X$ and $Y$ are weakly equivalent if and only if their cochain complexes are quasi-isomorphic as $E_\infty$-algebras. Moreover, assuming $X$ and …