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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 …

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 …

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 …

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 …

I'd like to add a quick little explanation to Georges's already sufficient answer (I'm sure this explanation is in Baez's post, but it can be said in a few lines here). One of the morals of the famo …

Here is an example: a product of infinitely many $\mathbb{RP}^\infty$'s. The crucial thing thing to see is that $\mathbb{RP}^\infty$ (or, easier to see, its universal cover $S^\infty$) has a group s …

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 …

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 …

I think the answer to the question as literally stated is "the trivial group", but I think there are related inquiries which get into some deep combinatorics. One way of thinking about the rooted tr …

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 $SL_2(\mathbb{Z})$.

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 $\ …

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 …

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 …

As to "why is the unit interval the canonical interval?", there is an interesting universal property of the unit interval given in some observations of Freyd posted at the categories list, characteriz …

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 …