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        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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        25 views

        Equivalence relations: Cosimplicial semilattice?

        For $n\ge 0$, let $E_n$ be the set of all equivalence relations on $[n]:=\{0,\dotsc,n\}$. Now given two equivalence relations $R,R'\in E_n$, we build their join $$R\vee R' := \langle R\cup R'\rangle,$$...
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        1answer
        472 views

        Simplicial set of permutations

        Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone ...
        2
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        1answer
        194 views

        Simplicial manifold associated to Lie groupoid

        Let $\Gamma=(\Gamma_1\rightrightarrows \Gamma_0), \Gamma’=(\Gamma’_1\rightrightarrows \Gamma’_0)$ be Lie groupoids and $\Gamma_{\bullet} ,\Gamma’_{\bullet}$ be the simplicial manifolds associated to $\...
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        0answers
        46 views

        A variation of the hammock localization

        Let $W \subseteq \mathcal{C}$ be a wide subcategory of a category $\mathcal{C}$. The saturation $\overline{W}$ of $W$ is the class of maps of $\mathcal{C}$ which become isomorphisms in $\mathcal{C}[W^{...
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        103 views

        Mysterious identity in cosimplicial $R$-module with Lie brackets

        I have a cosimplicial $R$-module $\mathscr{A}:=(A_n)_{n\ge 0}$ and on each $A_n$ a Lie bracket $[-,-]_n:A_n\otimes A_n\to A_n$. Denote the cofaces by $d_i:A_n\to A_{n+1}$ and the codegeneracies by $...
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        1answer
        174 views

        On minimal Kan simplicial sets having finite number of simplexes in each dimension

        What are the examples of “tame” minimal Kan simplicial sets having finite number of simplexes in each dimension besides simplicial point and $B(G)\approx K(G,1) $ for a finite group $G$? I believe ...
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        2answers
        368 views

        Dold-Kan correspondence in the category of symmetric spectra

        The Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes of abelian groups is a classical result. It states that the ...
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        49 views

        Computing weak operadic colimits as colimits

        I am trying to reduce the computation of weak operadic colimits to colimits. Let me introduct some notation. Let $q:C^{\otimes} \to N(Fin_*)$ be a symmetric monoidal category. Let $p: K \to C^{\...
        3
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        1answer
        109 views

        Two directed colimits of same spaces with different inclusions

        For any natural number $n$, let $i_{n},j_{n}:X_{n}\rightarrow X_{n+1}$ be a pair of monomorphisms of simplcial sets. Define $$X=\operatorname*{colim}_n \{\cdots X_n \rightarrow_{i_n} X_{n+1}\cdots \}...
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        89 views

        Existence of tensor product of infinity operads

        I am trying to show, or find a reference, for the following fact: "Given O,P two infinity operads [in the sense of Lurie, HA, Definition 2.1.1.10], there always exist a tensor product". In other ...
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        0answers
        80 views

        Techniques for computing homotopy pullbacks

        I am in the context of simplicial sets. I have a square diagram that I want to show to be homotopy pullback. What I can grasp is that it is a pullback up to some equivalences in the middle of my ...
        4
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        1answer
        111 views

        Inner fibrations are Kan fibrations on Map sets

        Firstly, a bit of notation. Let $C$ be a simplicial set. We define, for $x,y \in C$ vertices in $C$ $$Map(x,y) = \{x\}\times_{\Delta^{\{0\}}}Map_{sSet}(\Delta^1,C) \times_{\Delta^{\{1\}}} \{y\} $$ ...
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        42 views

        Some properness condition in simplicial sets

        Suppose that $C \to D$ is a trivial Kan fibration, and $D' \to D$ is an equivalence of simplicial sets. Let $C'$ be their pullback. Is it true that $C' \to D'$ is an equivalence? Recall that a ...
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        2answers
        275 views

        What is an example of a quasicategory with an outer 4-horn which has no filler?

        A quasicategory has fillers for all inner horns $\Lambda^i[n]$ where $n\geq 2$ and $0<i<n$, but it need not have fillers for $i=0$ (or $i=n)$. In particular, for $n=2$ and $n=3$ there are easy ...
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        0answers
        85 views

        Homotopy colimits of simplicial objects

        Given a simplicial combinatorial model category $\mathcal{M}$ and a simplicial diagram $F\colon \Delta^{\mathrm{op}} \rightarrow \mathcal{M}$, is there a nice (i.e. explicitely computable) way of ...

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