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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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      votes
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      53 views

      Topological realisation of a stack (explicit description)

      Let $X$ be an Artin stack over $k=\mathbf{C}$. I've heard that $X$ has a topological realisation, but I've not been able to find an explicit decription. My first guess would be: take a smooth cover $...
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      vote
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      59 views

      A homotopy problem for morphisms of dg-algebras

      Let $\mathfrak g$ be a real finite-dimensional Lie algebra, and suppose we are given two morphisms of dg-algebras $f,g: (C^\bullet(\mathfrak g),d_{CE}) \to (\Omega^\bullet(\Delta^n),d)$, such that ...
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      120 views

      Does Ex^∞ send homotopy inverse limits of ∞-categories to homotopy inverse limits of spaces

      Kan's functor $\operatorname{Ex}^{\infty}$ plays the role of total localization or groupoid completion in the theory of $\infty$-categories; specifically, it can be viewed as the left-adjoint of the ...
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      votes
      0answers
      95 views

      Extension of sheaves of $\infty$-algebras

      Let $(\mathcal{C},\tau)$ be a site, whose Grothendieck topology is $\tau$ $F : \mathcal{C}\to D(k)$ a sheaf of $\infty$-$k$-algebras, with $k$ a ring and $D(k)$ the derived category of $k$-modules. ...
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      2answers
      468 views

      Why is Kan's $Ex^\infty$ functor useful?

      I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful ...
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      votes
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      45 views

      Simplicial models for mapping spaces of filtered maps

      Let $S$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. Hence $SIMP(S,K)$ is Kan. Suppose that ${...
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      votes
      1answer
      59 views

      Simplicial models for fibrations between mapping spaces

      Let $S,T$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. We let $|S|$ denote the geometric ...
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      votes
      1answer
      142 views

      Monad, algebras and reflexive coequalizer

      Suppose we have an adjunction of categories $F:M\leftrightarrows N:U$. We define the associated (co)monad $G=F\circ U$. For any object $x\in N$ we define the simplicial resolution of $x$ given by $$ ...
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      votes
      1answer
      153 views

      Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair

      Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let $$AW_{A,B}\colon N_\ast(A\otimes B) \longrightarrow N_\ast(A)\otimes N_\ast(B)$$ and ...
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      votes
      0answers
      97 views

      On the existence of nice hypercovers

      Let $\mathcal{C}$ be a site and $X$ a sheaf of sets on $\mathcal{C}$. Then there exists a hypercover $K_{\bullet}$ of $X$ such that $K_n$ is a coproduct of representable presheaves on $\mathcal{C}$ ...
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      votes
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      36 views

      Contiguity for simplicial maps between simplicial sets

      I begin by recalling the definition of contiguous simplicial maps between abstract simplicial complexes: Definition. Two simplicial maps $\varphi,\psi\colon K \to L$ are said to be contiguous if for ...
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      votes
      1answer
      125 views

      Simplicial localization of the cofibrant-fibrant objects

      Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant ...
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      votes
      1answer
      236 views

      Does the cubical nerve preserve weak equivalences of simplicial sets?

      The finite cartesian powers of $\Delta[1]$ form a cubical object in simplicial sets, inducing a "cubical nerve" functor $N_\Box: sSet \to Set^{\Box^{op}}$. $N_\Box$ is a right Quillen equivalence, ...
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      votes
      0answers
      169 views

      Is there a natural homotopy inverse to the map $∣Sing(X)∣\rightarrow X$

      Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set). Let $∣-∣: SSet \rightarrow C$ be the geometric realization functor. For every $X$ we ...
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      votes
      1answer
      172 views

      Simplicially enriched cartesian closed categories

      In this question I asked whether for a complete and cocomplete cartesian closed category $V$, there can be a complete and cocomplete $V$-category $C$ (with powers and copowers) whose underlying ...

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