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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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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 $\pi_0({\rm holim}_{... 0answers 81 views Based loops objects in model categories If I have a pointed model category then for I can define based loops objects as homotopy pullbacks:$\require{AMScd}$\begin{CD} \Omega X @>>> *\\ @V V V @VV V\\ * @>>>... 1answer 68 views Definition of hypercover for simplicial presheaves and hypercovering in$\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$is a coproduct of ... 0answers 135 views Comparing definitions of cotangent complex Consider the following two ways of defining the cotangent complex of a ring map$R \rightarrow A$(Let$P^{\bullet} \rightarrow A$be a polynomial resolution): As the complex$\Omega^1_{P^{\bullet}/R}...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $$R\vee R' := \langle R\cup R'\rangle,$$...
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 ...
197 views

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 $\... 0answers 48 views A variation of the hammock localization Let$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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