This is, in some sense, the homotopy version of this question.) If $C$ is a category with $iC$ the subcategory of isomorphisms, there is a functor $X \mapsto End(X)$ from $iC$ to …

I recently started learning a little model category theory and in particular I found this nice exercise. I only know a little topology, but this prompted me to wonder how many mod …

In essence my question can be stated as follows: fill in the analogy $$ \text{cup product} \qquad\qquad \leftrightarrow \qquad \text{Samelson product} $$ $$ \updownarrow …

In the category of rational spaces, loop spaces split as products of Eilenberg-Mac Lane spaces and SUSPENSIONS split as wedges of (rational) spheres. I wonder if anything of the f …

An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for …

Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial s …

Consider the following interesting theorem:(7.5.7, p.294 in Topology and Groupoids by Ronald Brown) Gluing theorem for adjunction spaces: Suppose that we have the following commut …

Suppose that $X$, $Y$ and $Z$ are topological spaces, with $A\subset X$, a map $f:A\rightarrow Y$, and a homotopy equivalence $\phi:Y\rightarrow Z$. It seems fair to think that the …

It is well known that for simply-connected rational spaces, every suspension splits as a wedge of rational spheres and every loop space splits as a product of rational Eilenberg- …

Let $\mathcal{D}: C\to Top$ be a diagram of spaces (spaces are "nice", and $C$ is small). Let $X$ denote the homotopy colimit of $\mathcal{D}$ (which is connected) and $\pi(C)$ be …

The $f$-localization I mean is the one described and studied in detail in the book by E. D. Farjoun; $L_f$ is a homotopy idempotent functor which associates to each space $X$ an $ …

The functor that embeds the category of commutative algebras to associative algebras has the left adjoint - the quotient by the commutant ideal. For any dg-algebra $A$ let $A_{Ab}$ …

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 …

Background If $X$ is a based space then the James construction on $X$ is the space $J(X)$ given by $$ X \quad \cup \quad X^{\times 2} \quad \cup \quad X^{\times 3} \quad \cup …

Suppose we are given a diagram of topological spaces. We can restrict ourselves to the diagrams over finite partially ordered sets and let all spaces be good enough (e.g. CW-comple …