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To me the obvious answer involves sheafification of a presheaf. If you look at the construction of the associated sheaf to a presheaf in, say, Hartshorne it goes through the étalé space construction …

A Lawvere theory is a small category with finite products such that every object is isomorphic to a finite product of copies of a distinguished object x. A model of the theory in a category with finit …

The problem of counting semigroups and monoids of order $n$ up to isomorphism and anti-isomorphism (i.e., contravariant equvialence) is a very classical problem whose answer is conjectured but nobody …

A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Butz and Moerdijk have shown that if the topos has …

Conceptual answer. There can be no such functor. Let $C$ be any concrete category of finite sets and mappings such that the only automorphisms in $C$ are trivial. I claim there is no underlying set …

I think the title says it all. Quillen's Theorem A says that a functor $F\colon C\to D$ induces a homotopy equivalence of classifying spaces if each fiber category $F/d$ with $d$ an object of $D$ is …

Following Martin's suggestion, I will turn my comment into an answer. If $T$ is an Grothendieck topos, then the subobjects of the terminal object form a frame. If $X_T$ is the corresponding locale, …

A retract of a finitely generated free monoid is free even though submonoids need not be free. I don't know about the infinitely generated case. Edit: infinitely generated seems ok. The fg case I sa …

I believe the following is an example of a category whose leanification is discrete but which is not a groupoid. It should be possible to simplify it. There may be details to work out. Let $B$ be t …

The Stone-Cech compactification. Neither, Stone nor Cech was thinking about category theory at the time (since it didn't exist), but of course the Stone-Cech compactification is a left adjoint to the …

In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following. …

Probably you want to look at pushouts of categories along a common set of objects. For example, the free product of monoids is the pushout along the inclusion of the identity. Such things and their wo …

Free groups. If I am not mistaken, they were first introduced by Dyck via the reduced words description. The modern universal property definition only came about later.

Here is another answer. It is in fact equivalent to all the previous answers but is more categorical. Let $X$ be a finite transitive $G$-set and let $\mathcal G=G\ltimes X$ be the corresponding Groth …

No. There is a monoid with trivial group image whose classifying space is a sphere. See Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to …