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Let $G$ be a topological group. Following Milnor one way of defining the total space of the universal bundle $EG$ of $G$ is to form the infinite join $$ EG = G^{\ast \infty} = G \ast G \ast \dots $$ e …

Let $\mathcal{G}$ be a strict topological $2$-group, i.e. a strict $2$-category with a single object, a space of invertible $1$-morphisms, a space of invertible $2$-morphisms and continuous structure …

Let $PU(n) = U(n)/U(1)$ be the projective unitary group and denote by $BPU(n)$ its classifying space. Consider the algebra $M_n(\mathbb{C})$ as an $n^2$-dimensional Hilbert space equipped with the inn …

This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway: Let $G$ be a simply-connected topological group. In particular, it is an $H$-spac …

Suppose I have a topological category $\mathcal{C}$ of the following form: The object space consists of just two points $p_1, p_2$. The endomorphism space of $p_1$ contains just the identity. The endo …

If the following is true, it is probably well-known to the experts. Nevertheless, I could not find a reference for it. Suppose $P$ and $Q$ are $A_{\infty}$-operads in topological spaces and $X$ is a …

Let $G$ be a topological group. The geometric bar construction $BG = B_{\bullet}(pt, G, pt)$ together with $EG = B_{\bullet}(pt,G,G)$ and the map $EG \to BG$ yields the universal principal $G$-bundle …