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For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.

2 votes
Accepted

Relating two notions of geometric realization

Regarding What is the relation between $|K|$ and $X$, more specifically between $|\sigma|$ and $X_\sigma$ for each simplex $\sigma \in K$? It seems to me that one can build an intermediate space …
John Klein's user avatar
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4 votes
Accepted

realization of maps between classifying spaces of categories

Here's a counterexample. Let $M$ be a discrete monoid which is not a group. Consider the map $$ M = \hom (\Bbb N, M) \to \text{maps}_{*}(B\Bbb N , BM) = \Omega BM $$ ($\Omega BM =$ the based loops of …
John Klein's user avatar
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8 votes
Accepted

Degeneracies for semi-simplicial Kan complexes

The answer to (1) is to be found in Rourke, C. P.; Sanderson, B. J.$\Delta$-sets. I. Homotopy theory. Quart. J. Math. Oxford Ser. (2) 22 (1971), 321–338. It is shown there that a Kan "semi-simpli …
John Klein's user avatar
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7 votes

Is every connected space equivalent to some B(Aut(X))?

If you are willing to relax your wish from having answer of the form $\text{Aut}(X)$ to the more general group-like topological monoid $\text{Aut}_B(E)$ for a suitable fibration $E \to B$, then the a …
John Klein's user avatar
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7 votes

Homotopy equivalence of geometric realizations

Yes. If you are given a simplicial set $X: \Delta^{\text{op}} \to \text{Sets}$, then the the thick realization $||X||$ of $X$ is given by the same formula as the ordinary realization with the excepti …
John Klein's user avatar
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1 vote

(Co)homological characterization of homotopy pullbacks

Here are some comments in the form of an answer, which can be viewed as the Koszul dual to Tyler's approach via the Eilenberg-Moore spectral sequence. Firstly, in the special case when $D$ is a poin …
John Klein's user avatar
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