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I believe that's called the Milgram bar construction: R.J. Milgram, The bar construction and abelian $H$-spaces, Illinois J. Math. 11 (1967), 242-250.

From the nLab (although I was the author of these words): A reasonably large class of examples, including the examples of compactly generated spaces and sequential spaces, is given in the article by …

Note: This answer had been sent to Ali in private email (the question had been closed while I was composing it). Now that the question has been reopened, the answer may be given here. (This has been e …

Here is a general claim: in an extensive category $\mathbf{C}$, if $p: E \to B$ is an epimorphism and $E$ is connected (i.e., $\hom(E, -): \mathbf{C} \to \text{Set}$ preserves finite coproducts), then …

Intuitively, I see the product-preservation or indeed finite limit preservation of geometric realization $\hat{R}: [\Delta^{op}, \mathbf{Set}] \to \mathbf{kSpace}$ as lifting (through the forgetful fu …

Why not look at Stasheff's original paper? He does give a point-set model (where $K_{n+2}$ is a compact convex semialgebraic subset of $\mathbb{R}^n$) and describes explicitly the substitution maps $\ …

It doesn't seem to be true. Suppose we take $G = \{-1, 1\}$ with the discrete topology, acting on $X = \mathbb{R}$ by usual multiplication. Here $X^G$ consists of a single point $0$. The coend amounts …

It's a special case of what's called a collage or cograph construction. Recall that a profunctor or bimodule between categories $B$, $A$ is a functor $R: A^{op} \times B \to Set$. The cograph of $R$ i …

Perhaps the mapping class group of $X$? There is an extensive theory for mapping class groups and their computations. The mapping class group of the (2-dimensional) torus is $SL_2(\mathbb{Z})$.

If $X$ is a sober space, you can retrieve $X$ up to homeomorphism from $Cov(X)$. (Nitpick: this is not very good notation; it is very easy to misread it as the category of covering spaces over $X$. I …

I think the reason must be that a pseudomanifold $V$ has singular locus $\Sigma V$ of codimension 2 or greater. (Stratifications of varieties are obtained by letting $X_{k-1}$ be the singular locus of …

Here is one type of example, basically inspired by the manifold-type examples but so general that they are not actually categories of manifolds. Let $B$ be any small category with finite coproducts, a …

What Richard Kent said, although it probably deserves to be expanded on. Suppose for simplicity that $X'$ is obtained from a CW complex $X$ by attaching an $n$-disk $D^n = \{x: |x| \leq 1\}$ along a …

Neil has already given adequate reply; this answer is partly for Simon, and partly for those who do like category theory, and realize that its purpose is to make life simpler, not more complicated! …

Let $G$ be the compact Lie group $SO(n)$. There are some classical constructions of the classifying bundle of $G$ based upon on direct limits of Grassmann and Stiefel manifolds: $$BG \simeq \underse …