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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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        173 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}_{...
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        81 views

        Based loops objects in model categories

        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\\ * @>>>...
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        1answer
        68 views

        Definition of hypercover for simplicial presheaves and hypercovering in $\infty$-topos

        By lemma 4.9 in Dugger-Hollander-Isaksen, a hypercover is defined as an augmented simplicial object $U_\bullet\to X$ in the category of simplicial presheaves such that each $U_n$ is a coproduct of ...
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        0answers
        135 views

        Comparing definitions of cotangent complex

        Consider the following two ways of defining the cotangent complex of a ring map $R \rightarrow A$ (Let $P^{\bullet} \rightarrow A$ be a polynomial resolution): As the complex $\Omega^1_{P^{\bullet}/R}...
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        1answer
        139 views

        Descent in the injective model structure and descent for simplicial presheaves

        In Jardine's book Local Homotopy Theory P.102, a simplicial presheaf $X$ on a site $C$ is said to satisfy descent if some local injective fibrant replacement $ j : X → Z $ is a sectionwise weak ...
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        1answer
        140 views

        Does a homotopy sheaf functor commute with group completion

        Let $\text{sPre}(C)$ be the category of simplicial presheaves with Grothendieck topology $\tau$. Let $\pi_n^{\tau}$ be the $\tau$-homotopy sheaves defined as follow Does $\pi_n^{\tau}$ commute with ...
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        182 views

        Which spaces are most naturally presented simplicially?

        It's often a great technical convenience to know that you can "do homotopy theory" with simplicial sets. But if you really get down to it geometrically, it can be awkward. In general, if I have a CW ...
        3
        votes
        1answer
        78 views

        Is transfinite composition of weak equivalences of simplicial presheaves a weak equivalence?

        In a left Bousfield localization of the projective model structure on the category of simplicial presheaves, what is the condition that transfinite composition preserves weak equivalences? How about ...
        3
        votes
        1answer
        103 views

        Rigidification of marked simplicial sets

        It is well known that there exists a Quillen equivalence, $$\mathfrak{C}: Set_{\Delta} \rightleftarrows Cat_{\Delta}: N, \enspace \mathfrak{C} \dashv N$$ between Joyal's model structure on ...
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        0answers
        169 views

        When is the localization of all hypercovers equivalent to that of ?ech covers?

        In Theorem A.6 of Dugger's paper, it is shown that a few localizations are equivalent to the localization of ?ech covers. The Nisnevich localization at all hypercovers is equivalent to the ...
        6
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        1answer
        221 views

        Faithfully flat descent for modules from the simplicial point of view

        Let $R \rightarrow R'$ be a faithfully flat ring map, let $M$ be an $R$-module, and let $M_n$ be the base change of $M$ to the tensor product of $n + 1$ copies of $R'$ over $R$. One way to formulate ...
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        0answers
        31 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 $$R\vee R' := \langle R\cup R'\rangle,$$...
        14
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        1answer
        492 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
        197 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 $\...
        3
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        0answers
        48 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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