# Tagged Questions

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-Kahn correspondence etc. Please do not use for questions about geometry of simplices nor about ...

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### Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
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### A theorem of Kan regarding fibrant replacement

Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a ...
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### Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
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### What is a copresheaf on a “precategory”?

Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, §7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and ...
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### Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
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### Associativity of the tensor product of dendroidal sets

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$. ...
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### Question on the relation between the $n$-coskeleton functor and the internal hom-adjunction

(First, let me apologize that I asked an unanswered and related question in the stable case.) The category $\operatorname{sSet_+}$ of pointed simplicial sets is symmetric monoidal closed, i.e. there ...
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### $\Delta$-Complexes Are Hausdorff

I am using the definition of a $\Delta$-complex as given in Hatcher's book here on pg 103. Now on pg. 104, just before the section on Simplicial Homology, Hatcher remarks that if $X$ has a $\Delta$-...
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### How to compute (co)limits of enriched categories?

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This ...
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### Representation of functions in the simplex category

I am using the following characterization of the simplex category $\Delta$: The objects are finite ordinals and the arrows are weakly monotone functions $f:n\rightarrow m$. $0$ is initial and $1$ is ...
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### Computation of the Moore Complex

Given a simplicial abelian group G, one can compute its simplicial homotopy groups as the homology groups of its associated Moore complex. Can someone please show me the explicit computation of the ...
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### Geometric realization of a “simplicial space up to homotopy,” part two

This question is a follow-up, and my initial motivation for asking Is there a sensible way to form the geometric realization of a "simplicial space up to homotopy"? Given that the questions ...
Let ${\mathcal C}$ be a simplicial category. Then there are the following two ways of constructing a simplicial set from ${\mathcal C}$: Form the simplicial nerve \$\text{N}_\Delta({\mathcal C}) := \...