# 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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### Cocartesian edge gives contractible choice of filling in commutative diagram

This appears on p. 192 of Land's Introduction to Infinity-Categories which is the first page of his section on Straightening-Unstraightening. Let $p:\mathscr{E} \to \mathscr{C}$ be a cocartesian ...
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### "Free" resolution of algebra

Given a ring $R$, we have an adjunction of the forgetful functor $CAlg_R\to Set$ and the free algebra functor $Set\to CAlg_R$, giving us an endofunctor $$T:CAlg_R\to CAlg_R, S\mapsto R[S].$$ In ...
1 vote
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### Iterated homotopy pullbacks

I would like a reference for the following fact, which I believe to be true. Consider the simplicial model category of Kan complexes with the Quillen model structure, and suppose given a commutative ...
1 vote
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### HA, 1.1.1.7, Lurie

In this remark, Lurie states that applying proposition HTT 4.3.2.15 twice, we deduce that $\theta$ is a kan fibration. How is this assertion deduced?
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### The value of the left adjoint of the diagonal functor $\delta^*:\mathsf{bisSet}\to\mathsf{sSet}$ at $\Delta^m\boxtimes \Delta^n$

I'm currently reading the proof of Theorem 5.5.7 in Cisinski's book Higher Categories and Homotopical Algebra. There's one detail I don't understand, and I need someone's help. Let us introduce some ...
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### Nerve of category is a functor

Let $\mathcal{C}$ be a category. We define a mapping $M:\Delta^{\text{op}}\rightarrow\textbf{Set}$ as follows. Write $M_0:=\text{Ob}(\mathcal{C})$ and for $n\geq 1$ let $M_n$ be the collection of $n$ ...
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### Slice and Internal Hom; On the Definition of a Map $\underline{\operatorname{Hom}}(A,X/x)\to \underline{\operatorname{Hom}}(A,X)/x$

Recall that the join of two simplicial sets $X,Y$ are defined by $$(X\ast Y)_n=X_n\amalg X_{n-1}\times Y_0\amalg\cdots\amalg X_{0}\times Y_{n-1}\amalg Y_n,$$ with face and degeneracy maps defined ...
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### Motivation for Quasicategories

Recall that a simplicial set $X$ is called a quasicategory if for every $0<k<n$, every map $\Lambda^n_k\to X$ admits an extension to a map $\Delta^n\to X$. It is good to have a concise ...
1 vote
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### Interchange map in simplicial sets is a monomorphism?

In the category of simplicial sets there is an 'interchange' map $$h : (A \times B) * (C \times D) \to (A * C) \times (B * D)$$ given by $$h = (\pi_A * \pi_C, \pi_B * \pi_D)$$ where $\times$ is the ...
1 vote
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### Face maps of simplicial replacement of diagrams

I am trying to wrap my head around the construction of the simplicial replacement of a diagram, e.g. following Rodríguez-González (Realizable Homotopy Colimits, 1.5, https://arxiv.org/pdf/1104.0646....
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### Composition in an $\infty$-category

In the following, I'm following Markus Land, Introduction to Infinity-Categories (p. 81). Let $\mathscr{C}$ be an $\infty$-category and $x,y,z \in \mathscr{C}$ be objects, then I want to understand ...
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### Calculating the simplicial homology of a tetrahedron.

I want to calculate the simplicial homology of the $\Delta$-complex figure on the right: I believe that it consists of $2$ 0-simplices, four $1$-simplices and three $2$-simplices. So I am now in the ...
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### Dold-Kan correspondence, Model structures and Homology

I am fairly new to the concept of model categories, simplicial sets, etc. And so there is some questions, which may be obivuous, that I need to clarify. Consider the cateogry of simplicial abelian ...
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### Computation of a specific $k-$horn

I have been learning about simplicial sets and at some point we define $\Lambda^k_n$ which is the $k-$th horn of $\Delta[n]$. Now we define it has being the boundary minus the image of $d^n_k$. I ...
1 vote
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### A result on a fibration that is an epimorphism of simplicial sets

Consider $p:E\rightarrow B$ to be a fibration of simplicial sets. I want to see that if $p$ is an epimorphism and $E$ is fibrant then $B$ is also fibrant. I have come to the conclusion that to solve ...
1 vote
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### Multiplicative structure on $\mathbb{S}^1$ in $\mathsf{sSet}$

I have recently been tasked with reading up on simplicial sets and I'm trying to define some concrete maps in the category to try to get my head around how things relate to stuff I already know. In ...
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### Composition in quasi-categories

Let $C$ be a quasi-category. Then $C^{\Delta^2}\to C^{\Lambda_1^2}$ is a trivial fibration and we may choose a section $s$. The map $C^{\Lambda_1^2}\xrightarrow s C^{\Delta^2} \to C^{\{0,2\}}$ defines ...
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### Do we distinguish two singular simplices if they have different vertex orders?

We define a $\textbf{singular$n$-simplex}$ in $X$ to be a continuous map $\sigma:\Delta^n\to X$ where $\Delta^n$ is the standard $n$-simplex. Now, as an example, Let $X$ be a singleton $\{p\}$. Then ...
1 vote
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### Bijection induced by Hom and inner Hom

For an $\infty$-category $X$ and a simplicial set $A\in sSet$. And for a category $\mathcal{C}$ we define $\mathcal{C}^\simeq$ whose objects are those of $\mathcal{C}$ and morphisms contain only ...
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### Common subdivision of two simplicial subcomplexes (on the way to topological invariance of simplicial homology)

In Munkres' Elements of Algebraic Topology Chapter 18, he aims to show that given a continuous map $h:|K|\to |L|$, there is a well-defined map $h_*:H_p(K)\to H_p(L)$, given by $f_*\circ(g_*)^{-1}$, ...
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### stable homotopy category Ho(Spectra) of category of spectra

The category of spectra over CW-complexes has as objects sequences $E:= \{E_n \}_{n \in N}$ of CW complexes $E_i$ together with structure maps $S^1 \wedge E_n \to E_{n+1}$. The morphisms $f: E \to F$ ...
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### Categories as simplicial object in Set VS simplicial object in Cat

It is well known that any category allow for the construction of a functor from the simplex 1-category $\Delta_0$. All of the axioms of a category translate simply in the functoriality requirements. ...
1 vote