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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 ...
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### 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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