It is a theorem that the category of compactly generated weakly Hausdorff (CGWH) spaces is Quillen equivalent to the category of simplicial sets with the Kan model structure. Howe …

I was reading HTT 2.1.4, and I just totally lost what was going on. Could someone provide some motivation for this section? Why do we want another model structure? I'm sorry for …

I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I wan …

I was wondering, what a $N$-dimensional simplicial space $X$ should be. Of course the degeneracy maps force the spaces to be nonempty in high dimensions. Currently I have two diffe …

Suppose C and D are two ∞-categories (quasi-categories), $F : C \to D$ and $G : C \to D$ are two functors (i.e. 0-simplices in the ∞-category of functors Fun(C,D), wh …

I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. …

I couldn't think of a title for this, but here we go: Fix $p:S\rightarrow T$, a left fibration of simplicial sets, and an edge $f:\Delta^1 \rightarrow T$. Let $t$ be the first ve …

How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the differenc …

Let $\mathbf{C}$ be a Grothendieck site with enough points. Let $p:\mathcal{E}\to \mathcal{F}$ be a map of simplicial presheaves on $\mathbf{C}$. Is it true that $p$ is a local (Ka …

The Kendall tau distance was originally defined as a correlation coefficient. It seems clear to me that every metric function $d$ that is bounded by $b$, induces a correlation coef …

In Lemma 2.1.3.4 of Higher Topos Theory, the statement of the lemma requires that the fibers are not only nonempty but contractible. However, in the proof, I don't see where contr …

Consider a set of N binary-state nodes at "time" t, each of which is a (boolean) transition function of two nodes in the set, evaluated at time t-1. Thus there are N of these boole …

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structu …

It turns out that joins of simplicial sets are fairly easy to define, but hard to manage. In lots of cases, we'd like to compute what a join is, does it look like a horn?, a bound …

Every surface can be triangulated in such a way that at most 7 trianlges meet at one vertex. Every surface can be decomposed in squares such that at every vertex at most 5 suqares …