Tagged Questions

41 views

Difference between two concepts of homotopy for simplicial maps?

I learn from Gelfand and Manin's Methods of Homological Algebra, Exercise 2 for I.4 that two maps $f,g\colon X\to Y$ between simplicial sets $X,Y$ are simply homotopic (maybe usually called simplicial ...
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Why aren't *weak* test categories enough?

Short version : What is the motivation behind the local condition in the definition of a test category ? Why do we want the slice categories $A/a$ to be weak test ? Long version : I start ...
31 views

cells of quotient CW complex

Let $X$ be a CW complex and $Y$ a CW subcomplex. If $X$ has no cell of dimension $n$, for some $n>0$, then $X/Y$ has no cell of dimension $n$. Is it true? Why?
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When may the bar resolution be truncated for calculating $\operatorname{hocolim}$?

Suppose $D$ is a small category and $F\colon D\to sSets$ a functor. I want to calculate the simplicial set $$\operatorname{hocolim}F$$ via the bar construction: $\operatorname{hocolim}F$ is the ...
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Skeletality of a simplicial set $X$ vs. highest degree of a non-degenerate simplex

Let $X$ be a simplicial set with a non-degenetate simplex in degree $n$ and suppose that all simplices in higher degrees are degenerate. Is $X$ an $n$-skeletal simplicial set and not ...
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Realization (in the sense of homotopy coherent nerve) of $\partial\Delta^n$

I need some help understanding the working of the homotopy coherent nerve (as described in Lurie's HTT). Let $i < j$, then $\mathrm{Hom}_{\mathfrak{C}\Delta^n}(i, j) \simeq (\Delta^1)^{(j-i-1)}$, ...
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How to construct a quasi-category from a category with weak equivalences?

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a ...
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An equality of some equalizers of simplicial sets.

$\newcommand{\cosimp}[1]{{#1}^\bullet} \newcommand{\sSet}{\mathsf{sSet}} \newcommand{\Set}{\mathsf{Set}}$ [I realize that the post is imposing.However, all the content of the problem is in the gray ...
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I'm struggling with a property of basic localizers, namely that adjoint functors are weak equivalences. Recall that a basic localizer $\mathcal W$ is a subset/subclass of the arrows of the category ...
37 views

Combinatorial definition of the homotopy groups of a quasi category?

The homotopy groups of a Kan complex are defined combinatorially, and they coincide with the homotopy groups of the geometric realization. For a general complex $X$, homotopy isn't an equivalence ...
38 views

Geometric realization of function complexes of simplicial sets

Let $\operatorname{sSet}$ be the category of simplicial sets and $\operatorname{Top}$ the category of (say, compactly generated) topological spaces. There is a pair of adjoint functors called ...
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Why do we ask that condition in Kan complex?

Let $\{X_n\}_{n=0}^\infty$ be simplicial set with faces $d_i:X_n\to X_{n-1}$, a simplicial set is called a Kan Complex if for any $x_0,...,x_{k-1},x_{k+1},...,x_{n+1}$ if $d_i(x_j)=d_{j-1}(x_i)$ for ...
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Elementry collapses implies same homotopy type.

Let $\Delta$ be a simplicial complex, and suppose that $\sigma \in \Delta$ is a proper face of exactly one maximal simplex $\tau \in \Delta$. A simplicial collapse of $\Delta$ is the removal of all ...
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Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
61 views

Does $\pi_0$ preserve infinite products?

The functor $\pi_0\colon \operatorname{sSets}\to \operatorname{Sets}$ has value on a simplicial set $X$ the coequalizer of the two boundary morphisms $d_0,d_1\colon X_1\to X_0$. It is easy to see ...
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Model structure on sSet

Which is the model structure on $\text{sSet}$ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $\text{Cat}$ (of small categories) by the adjunction ...
I am looking for the naive notion of homotopy. For maps $f, g: A\to B$ of simplicial sets $\Delta^{op}\to Sets$, a homotopy $H$ is a simplicial map $H:A\times\Delta^{1}\to B$ such that ...