symmetric monoidal (∞,1)-category of spectra
A semigroup is like a monoid where there might not be an identity element.
The term “semigroup” is standard, but semi-monoid would be more sytematic.
A semigroup is, equivalently,
a set equipped with an associative binary operation.
an associative magma;
the hom-set of a semicategory with a single object.
Some semigroups happen to be monoids; even then, a semigroup homomorphism might not be a monoid homomorphism (because it might not preserve the identity element). Nevertheless, semigroup isomorphisms must be monoid isomorphisms. Thus, the identity element of a monoid forms a property-like structure on the underlying semigroup.
As a monoid is a category with one object, so a semigroup is a semicategory with one object.
Some mathematicians consider semigroups to be a case of centipede mathematics. Category theorists sometimes look with scorn on semigroups, because unlike a monoid, a semigroup is not an example of a category.
However, a semigroup can be promoted to a monoid by adjoining a new element and decreeing it to be the identity. This gives a faithful functor from the category of semigroups to the category of monoids. So, a semigroup can actually be seen as a monoid with extra structure.
Describe this structure.
On the other hand, analysts run across semigroups often in the wild, and don't always want to add formal identities just to turn them into monoids.
Another variant with strong links with category theory is that of inverse semigroups, which Charles Ehresmann showed were closely related to ordered groupoid?s. Inverse semigroups naturally occur when considering partial symmetries of an object.
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
Then we have this special semigroup
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 :)
We can internalize the concept of semigroup in any monoidal category (or even multicategory) $V$ to get a semigroup object in $V$.
There is a strong connection between semigroups and finite automata; see e.g. Krohn-Rhodes theory?.
Semicategories and semigroups are mentioned for instance in section 2 in