Suppose we have some pointed connected topological space $X$. How can we determine if there exists a space $BX$, called delooping of $X$, such that its space of based loops $\Omega …

I'm trying to understand the relationship between the notions of looping and delooping in category theory and topology. The morphisms in a category with one object have the struct …

Let $X$ be a based space. Then the Moore loop space $MX$ is defined to be the topological monoid whose points are based loops $[0,a] \to X$ where $a \ge 0$ is allowed to vary. Com …

Let $f:X \to Y$ be a homotopy equivalence of pointed topological spaces. Then, is the induced map of pointed loop spaces $\Omega (f): \Omega X \to \Omega Y$ a homotopy equivalence …

This seems like it should be a "standard" thing, and I think I remember even seeing it somewhere, but I can't remember where. Let $C$ be a small category. Is there a category $\L …

My question is about the relationship between the free loop space LX of a space X and the (appropriately defined) Borel construction $PX \times_{\Omega X} \Omega X$ which is a homo …

I'm looking for directions to the literature that might contain fairly explicit constructions that might be called (the algebra of functions on) the "derived mapping space" from a …

I've been reading Chen's original works on iterated integrals and in order to consider differential forms on the path space $PM$ of a smooth manifold $M$ he gives $PM$ the followin …

Let $A$ be a commutative $\mathbb{C}$-algebra, and consider $C_{\bullet}(A,A)$ the simplicial Hochschild homology module of $A$ with respect to itself (i.e. $C_{n}(A,A)=A^{\otimes …

I've been reading Galatius's Park City notes on the Madsen-Weiss theorem (available here). On page 8, he states the following theorem. Let $X$ be a space such that $\pi_1(X)$ is …

Hello, The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied. My problem is the following: The "recognition principle" says that every "group-like …

I am trying to understand the topology (in terms of homology groups) of the free loop space $\Lambda M$ of nice spaces (Complete Riemannian connected finite dimensional manifolds $ …

Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact …

Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let $LG:=C^\infty(S^1,G)$ be its smooth loop group. Given an interval $I\subset S^1$, we have the l …

$SU(2)$ is a Lie group, with a Lie algebra $\mathfrak{su}(2)$. I now consider the loop group $ LSU(2) = \{ \gamma: S^1 \to SU(2); \gamma \mathrm{ smooth} \} $. It is well-known th …