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I don’t know how others will feel about that, but this functor $S: Cat \to Semigroup$ isn’t full. For example, there are maps $S(C) \to S(0)$ where $0$ is the empty category.
Hmm. Yeah, that is a good point. By “think of categories as semigroups” I meant that you can recover all the objects, morphisms, and all the information about the composition is included in $S(\mathcal{C})$.
Nonetheless, this is a common construction in semigroup theory. For instance, this is exactly how Brandt groupoid turns into a Brandt semigroup.
I can certainly believe all that. It should be fully faithful when you restrict to isomorphisms in both the domain and codomain of $S$ (thinking here of $Cat$ as a 1-category).
The following ancient “query box”-discussion was still in the entry. Am moving it, hereby, from there to here:
AnonymousCoward: In Categories of Symmetries and Infinite-Dimensional Groups by Yu. A. Neretin (London Mathematical Society Monographs, New Series 16, Oxford Science Publications 1996), the author points out that if we consider an infinite-dimensional group $G$ can be realized in the following way: there is some category $C$ with an object $X$ such that
$Aut(X)=G.$Then we have this special semigroup
$\Gamma=End(X)$which is called the Mantle of $G$. Neretin insists it is a semigroup.
I am at a loss as to why this is a semigroup, and not a monoid…
David Roberts: Well, we can realise $G = Aut_{\mathbf{B}G}(*)$, where $*$ is the single object of the one-object groupoid associated to $G$. Then $End(*) = Aut(*)$ in this category, so this ’Mantle’ is nowhere near being uniquely defined. Is Neretin using the same definition of semigroup as here (it’s the obvious first question - a bit like ’is your computer plugged in and turned on at the wall?’). Unless I’ve got the wrong end of the stick, and this category $C$ is defined up to equivalence from $G$. And maybe $C$ isn’t a category, but only a semicategory?
Edit: Having a look, I find his book: Semigroups in algebra, geometry, and analysis, by Karl Heinrich Hofmann, Jimmie D. Lawson, Ėrnest Borisovich Vinberg. They talk about Ol’shanskiĭ semigroups associated to groups - this might be a place to get started. From the examples discussed, it seems like some of the semigroups they consider are monoids, but that was only after I flicked quickly through the book online.
Toby: When Neretin insists that the mantle is a semigroup, does he also insist that it's not a monoid, or is he just silent about that? After all, it is a semigroup.
We category theorists are strongly attracted to monoids, since they come from categories and semigroups don't. But others consider monoids to be just a special kind of semigroup; as long as it's not a group, they're not going to bother worrying about whether a semigroup is a monoid or not.
I agree with David that the mantle doesn't seem to be well defined; a group should have several mantles (the smallest of which is itself). But if he's talking about a particular way of constructing certain groups, then this way may well come about by first constructing a monoid (the mantle) and then taking the mantle's group of invertible elements.
AnonymousCoward: The notion of a semigroup is (as best as I can tell from closely reading the first chapters) left undefined. I assumed that the endomorphism monoid here is also a semigroup, so there is really nothing lost here (well…partially true; I think viewing the Mantle as a semigroup does play a role when considering morphisms!).
After looking a bit more into Neretin’s writings (e.g. “Infinite-dimensional groups, their mantles, trains, and representations” in Kirillov’s book Topics in Representation Theory) it does seem clear that the mantle of an infinite-dimensional group is not well-defined (there are apparently two different ways to consider it that produce not necessarily equal mantles — one is by considering the group $G$ as the automorphism of an object $H$ in some category and thereby obtaining the mantle as the endomorphism monoid of this object; the other is to consider the closure of sequences of $G$ under a weak-operator norm, or something to that effect).
I was just worried that I was forgetting some special situation when the endomorphisms form a semigroup instead of a monoid.
Also, thank you both Toby and David for your quick and informative replies, I really appreciate it :)
added pointer to:
and made “Lie semigroup” redirect here (for the moment, it would deserve its own page, eventually)
Added Example
A left or right ideal of a monoid $M$ is a subsemigroup of $M$ and is only a submonoid if it contains the unit in which case it is $M$ itself. A monoid $M$ induces the topos of its right actions on sets - its right M-Set $= Set^{M^op}$. The set of all of $M$’s right ideals corresponds to the elements of the truth value object, $\Omega$, of this topos. The analogous construction holds for left M-Sets $= Set^{M}$ .
I hope this is right - I think I understand this. The entry M-Set could use a lot of work.
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