Questions tagged [simplicial-stuff]
For questions about simplicial sets (functors from simplex category to sets), simplicial (co)algebras and simplicial objects in other categories; geometric realization, model structures, Dold-Kan correspondence etc. Please do not use for questions about geometry of simplices nor about triangulations.
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Décalage as a model for a path space object?
I am wondering how one can view a (plain) décalage of a simplicial space (or at least a simplicial set) $X$ as its path space object in the sense of the model category theory. Although this is what I ...
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Prove continuity of the affine extension mapping between geometric simplicial complexes
Let $\Delta_1$ and $\Delta_2$ be geometric simplicial complexes. Let $K_1$ and $K_2$ be their associated abstract simplicial complexes. Let $f: V(K_1) \to V(K_2)$ be a simplicial mapping.
We define ...
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Cosimplicial resolution associated to a monad
Let $\mathcal{C}$ be a category and $\mathbf{T}$ a monad on $\mathcal{C}$ with functor part $T : \mathcal{C} \to \mathcal{C}$ (I would actually like to consider the case where $\mathcal{C}$ is an $(\...
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On the induced morphism between colimits in an $\infty$-category
Let $\mathcal{C}$ be an $\infty$-category (quasicategory), $F\colon K\rightarrow\mathcal{C}$ a diagram in $\mathcal{C}$ and $y$ a colimit of $F$. To be precise, there is a colimit cone $\overline{F}\...
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Reference for a proof on homotopy groups of simplicial abelian groups
In Simplicial Homotopy Theory, Goerss and Jardine wrote :
The simplicial abelian group structure on A induces an abelian group structure
on the set $\pi_n$(A, 0) = [($\Delta^n$, $\partial \Delta^n$), ...
- 302
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Free abelian group functor preserves Kan fibration
Suppose, $\mathbb{Z}$ is the free abelian group functor from simplicial sets to simplicial abelian groups. Then does $\mathbb{Z}$ preserves Kan fibrations i.e. if $X \to Y$ is a Kan fibration between ...
- 155
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Naturality for the Homotopy Fiber Sequence of Mapping Spaces
For a cartesian fibration $p\colon\mathcal{E}\rightarrow\mathcal{C}$ of $\infty$-categories (quasicategories) and objects $x,y$ of $\mathcal{E}$, the induced map $\mathrm{map}_{\mathcal{E}}(x,y)\...
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simplicial commutative rings and derived commutative rings
I met two definitions with regard to the simplicial(or derived) commutative rings. One way is very direct and literal, that is, a 'simplicial ring' is a simplicial object in the category of ring：
\...
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Nerve theorem and cycles deformation
Consider a finite set of point $P\subset[0,1]^{d}$. Let $r>0$, by nerve theorem, we know that, the Čech complex $C^{2r}(P)$ and $B_{2}(P,r)$ are homotopy equivalent.
Furthermore, Proposition 3.2 of ...
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Infinite category structure on SCRing and 'space of commutative squares' in SCRing
When I read this paper 'virtual cartier divisors and blow ups', I often meet with such phrase like 'mapping space of infinite category $SCRing_{A}$'. See lemma 2.3.5 in the above paper: $Map_{SCRing_{...
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What morphism is sent to a monomorphism by the left Kan extension ${\rm Lan}_{\Delta}\colon{\bf Set^\Delta\to\bf Set^{\hat\Delta}}$ along Yoneda?
For any small category $C$, let us write $\hat{C} = \mathbf{Set}_C$ for the presheaf category $\mathbf{Set}^{C^{\mathrm{op}}}$, and $y=y_C\colon C\to \mathbf{Set}_C$ for the Yoneda embedding. Consider ...
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The core of an $\infty$-category and pointwise invertible maps $\Delta^1 \times \partial \Delta^n \cup \{1\} \times \Delta^n \to \underline \hom(B,X)$
I'm currently working through Theorem 3.5.11 of Cisinski's Higher Categories and Homotopical Algebra. The section in which the theorem is inscribed concers the core of an $\infty$-category.
Trying to ...
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How to understand May's proof that counit map is a weak equivalence?
A similar question was asked about 4 years ago here, but received no answers, so I hope it is appropriate to post a new question. I am trying to read the singular homology section in May's Concise ...
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Inclusion of a certain full subcategory of the category of elements of a simplicial set is a final functor
For a simplicial set $X$, let $\text{el X}$ be its category of elements, whose objects are pairs $([n], x\in X_n)$. Let $\text{(el X)}_{nd}$ denote the full subcategory comprising objects $([m], y\in ...
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Simplicial $\pi_0$ as homotopy classes $\Delta^0 \to X$ using that $\pi_0$ is left-Quillen
In Higher Categories and Homotopical Algebra, Cisinksi defines the connected component functor as
$$\pi_0 \colon \mathsf{sSet} \to \mathsf{Set}, \qquad \pi_0(X) = \mathsf{colim}_{n} X_n,$$
the left ...
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