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  1. Added how small categories can be thought of as semigroups.


    diff, v22, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 22nd 2020

    I don’t know how others will feel about that, but this functor S:CatSemigroupS: Cat \to Semigroup isn’t full. For example, there are maps S(C)S(0)S(C) \to S(0) where 00 is the empty category.

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeDec 23rd 2020

    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(𝒞)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.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 23rd 2020

    I can certainly believe all that. It should be fully faithful when you restrict to isomorphisms in both the domain and codomain of SS (thinking here of CatCat as a 1-category).

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeDec 23rd 2020

    Added the example of groups, to contrast with the example of monoids.

    diff, v23, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2021

    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 GG can be realized in the following way: there is some category CC with an object XX such that

    Aut(X)=G. Aut(X)=G.

    Then we have this special semigroup

    Γ=End(X) \Gamma=End(X)

    which is called the Mantle of GG. 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 BG(*)G = Aut_{\mathbf{B}G}(*), where ** is the single object of the one-object groupoid associated to GG. Then End(*)=Aut(*)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 CC is defined up to equivalence from GG. And maybe CC 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 GG as the automorphism of an object HH 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 GG 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 :)

    diff, v26, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021

    added pointer to:

    and made “Lie semigroup” redirect here (for the moment, it would deserve its own page, eventually)

    diff, v31, current

    • CommentRowNumber8.
    • CommentAuthorRodMcGuire
    • CommentTimeSep 20th 2021

    Added Example

    A left or right ideal of a monoid MM is a subsemigroup of MM and is only a submonoid if it contains the unit in which case it is MM itself. A monoid MM induces the topos of its right actions on sets - its right M-Set =Set M op= Set^{M^op}. The set of all of MM’s right ideals corresponds to the elements of the truth value object, Ω\Omega, of this topos. The analogous construction holds for left M-Sets =Set M= Set^{M} .

    I hope this is right - I think I understand this. The entry M-Set could use a lot of work.

    diff, v32, current