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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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        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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        460 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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        44 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
        151 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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        95 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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        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 ...
        6
        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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        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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        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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        1answer
        169 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 ...
        4
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        1answer
        87 views

        Subdivision of simplicial sets but not the barycentric one!

        Suppose $K$ and $L$ are simplicial sets. When should one consider that $K$ is a subdivision of $L$? I ask with a view to defining some notion of `finer' generalising that of 'finer triangulation' of ...
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        114 views

        Why is the Straightening functor the analogue of the Grothendieck construction?

        In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between peseudo functors into the category of ...
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        254 views

        About Kan-Thurston theorem

        The Kan-Thurston theorem obsesses me recently, which is proved by Daniel Kan and William Thurston, Every connected space has the homology of a K(π,1), Topology Vol. 15. pp. 253–258, 1976. Later, there ...

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