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There is a notion of the étale homotopy type of a (Grothendieck) topos, going back to Artin and Mazur (I think). However, in classic "French" fashion they turned a theorem (in one setting) into a defi …

My best guess at what you are asking is this: Can one define homotopy groups of arbitrary simplicial sets using only simplicial homotopy? I would say no. Indeed, consider $\partial \Delta^2$. It …

The class of morphisms having the right lifting property with respect to $\Lambda^1_k \times \Delta^n \hookrightarrow \Delta^1 \times \Delta^n$ (for all $k \in \{ 0, 1 \}$ and all $n \ge 0$) is strict …

Here is a proof for the case where $X$ is a Hausdorff space. Note that each $U_n$ is also Hausdorff in this case. A standard argument shows that the face operators of $U_\bullet$ are (surjective) loc …

Given a simplicial object $X$ in a locally small category $\mathcal{M}$ and a simplicial set $S$, define the weighted limit $\{ S, X \}$ to be an object in $\mathcal{M}$ equipped with an isomorphism $ …

Let $U$ and $V$ be Segal spaces and let $f : U \to V$ be a Dwyer–Kan equivalence. That means two things: The following diagram is a homotopy pullback square: $$\require{AMScd} \begin{CD} U_1 @>>> U_0 …

The quotation in the title is attributed to Frank Adams and appears in several places: In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not necessar …

If $X$ is finite, then as a mere category $\mathbf{Fin}_{/ X}$ is equivalent to the cartesian power $\mathbf{Fin}^X$. If $X$ is infinite, then $\mathbf{Fin}_{/ X}$ should be thought of as the full sub …

In [Schwänzl and Vogt, Strong cofibrations and fibrations in enriched categories], the authors refer to an earlier preprint, [Schwänzl and Vogt, Cofibration and fibration structures in enriched catego …

Let $X$ be a topological space. Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$. It is not hard to check that $P^X : \textbf{ …

We can detect the difference between the two constructions using the homotopy category. Given any simplicially enriched category $\mathcal{C}$, we can construct an ordinary category $\pi_0 [\mathcal{C …

[Edit: Corrected some false claims and modified questions accordingly.] Let $\mathcal{S}$ be the cocomplete $(\infty, 1)$-category generated by a point. This is conventionally known as the $(\infty, 1 …

It seems there are many subtly different notions of the shape of a topological space (and, more generally, toposes). For instance, Lurie [Higher topos theory] defines this one: Definition 1. The shap …

Yes. In fact, one such example comes from homotopical algebra: Proposition. Let $\mathcal{C}$ be a small homotopical category and let $\gamma : \mathcal{C} \to \operatorname{Ho} \mathcal{C}$ be th …

Here is a general nonsense fact: if $F \dashv U : \mathcal{A} \to \mathcal{B}$ is an adjunction and $U$ preserves epimorphisms, then $F$ preserves projective objects. Epimorphisms in $\textrm{Ch}(R)$ …