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As soon as you do serious homotopy theory in a context outside topological spaces, a formalism for abstract homotopy theory is very helpful. Let's give a few examples: 1) Waldhausen's algebraic K-the …

Recently I became through this mathoverflow question aware of the article Codensity and the ultrafilter monad by Tom Leinster. There he shows that the ultrafilter monad on the category $\mathrm{Set}$ …

Quasi-categories (or $\infty$-categories, as they are often called) are a very convenient setting for doing abstract homotopy theory. One of their amazing features is the following: Given a diagram of …

Here is an example, which I got from a discussion with Karol Szumilo (and maybe he got it from Cisinski?). Consider the notion of a cofibration category, which means essentially that you have weak e …

If $X = (X_n)$ and $Y = (Y_n)$ are spectra, one can define a morphism just to be a collection of maps $x_n \to Y_n$ commuting with the suspensions. Thus the set of morphisms between $X$ and $Y$ is a s …

In general, this is certainly not true. Take for example a space $X$ with an action by a group $G$. As a group acts by isomorphisms, it acts in particular by cofibrations. But the map $X/G \to X_{hG}\ …

I came along the following question while trying to understand and apply some ideas of Dugger's article Universal Homotopy Theories. Suppose, we are given a nice model category $\mathcal{C}$, say le …

Let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be abelian categories and $F: \mathcal{A} \to \mathcal{C}$ and $G: \mathcal{B} \to \mathcal{C}$ be functors between them. It is then possible to defi …