# Questions tagged [simplicial-stuff]

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

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### Homotopy descent and cohomology

I've been reading "Structured Brown representability via concordance" by D.Pavlov (https://dmitripavlov.org/concordance.pdf) and I'm strugeling with a point and was wondering if someone ...
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### Non-degenerate simplexes in a Kan complex

I have the following question on simplicial sets: a non-constant Kan complex has a non-degenerate simplex in every sufficiently large simplicial degree? It's Exercise 8.2.3 (p. 262) of Charles ...
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### Showing that a certain simplicial set has levelwise small cardinality

My question concerns Section 2 of the article "The Simplicial Model of Univalent Foundations (after Voevodsky)" (https://arxiv.org/pdf/1211.2851.pdf). Let $\alpha$ be a strongly inaccessible cardinal....
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### On equivalences of cartesian fibrations

Let $p:X^{\natural} \to S$, $q:Y^{\natural} \to S$ be two cartesian fibrations of simplicial sets (where the marking is given by cartesian edges) and assume that we are given an equivalence of ...
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### Is every simplicial spectrum equivalent to an abelian group simplicial spectrum?

I wonder if every simplicial $S^1$-spectrum stable equivalent to an abelian group simplicial $S^1$-spectrum? It seems that I could use the stable Dold-Kan on the heart and use Postnikov towers?
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### Representing simplicial homotopy classes by empty cubes

I am looking for references concerning the following facts, which I believe to be true: In a Kan complex $K$, can I use "empty cubes" $\partial(\Delta^1)^{(n+1)}$ with the vertex $(0,0,\ldots,0)$ ...
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### Homotopy of functors

Recently I have read two different proposals for a notion of homotopy between functors, and I am curious which contexts each best lend themselves to. The first comes from Ming-Jung Lee's 1972 paper ...
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### Category of spaces/sheaves

Consider the following category $\mathcal C$: An object of $\mathcal C$ is a pair $(X,\mathcal F)$ where $X$ is a space and $\mathcal F$ is a sheaf on $X$. A morphism $(X,\mathcal F)\to(Y,\mathcal G)$...
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### Face maps of cosimplicial abelian groups under Dold-Kan correspondence

This might be a stupid question, but I feel really confused how the cosimplicial Dold-Kan works explicitly... Recall Let $$C_\bullet:=\dots\to C_2\to C_1\to C_0$$ be a chain complex of abelian ...
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### Cofibrant simplicial categories

Reading about simplicial categories, and in particular the model structure in sCat, I found various sources, among which I can mention this one or section 16.2 in Riehl's book “Categorical homotopy ...
Let $C^\bullet$ be a cochain complex in positive degrees, by Dold-Kan correspondence it corresponds to a cosimplicial abelian group $D\colon \Delta\to \mathrm{Ab}$. Let $F$ be a field, let $C^{\... 2answers 232 views ### Pullbacks and fibers in the$\infty$-category of spaces Suppose given a commutative diagram in the$\infty$-category of spaces, as the one depicted below, where all but the bottom-right squares are pullbacks. Is it true that$H$is (equivalent to) the ... 0answers 117 views ### A notion of fibration on bisimplicial sets [I am not trained in this stuff, but have an outside research interest, so sorry if this question is standard.] I am interested in notions of fibrations, or fibrant objects, in bisimplicial sets. In ... 0answers 210 views ### Does$\pi_0$commute with (homotopy) limits? Consider in the category sSet of simplicial sets, let$X$be a$J$-indexed diagram in sSet ($J$a small category), and for each$j\in J$,$X_j$is a Kan complex, then do we have$\pi_0({\rm holim}_{...
If I have a pointed model category then for I can define based loops objects as homotopy pullbacks: $\require{AMScd}$ \begin{CD} \Omega X @>>> *\\ @V V V @VV V\\ * @>>>...