Viewing Semigroups as Categories?

Question

I am wondering how to view semigroups as categories. For example, we can easily view monoids as categories with a single object. Unfortunately, semigroups do not necessarily have identities, so the same view does not work with semigroups.

However, I did come across this idea where one adjoins a new element to a semigroup and treats it like its identity; effectively the semigroup gets promoted to a monoid. Of course, this new identity element must satisfy some additional properties to make sure that the original elements of the semigroup continue to behave as before. I am wondering what the additional properties might be.

Answer 1

The main reference on this question is [1]. You may look in particular to Section 1.2, p.22 (Foundations) and p. 95, where the definition of a semigroupoid is given.

The following is a quote from this book.

Given a semigroup S there are two popular ways to make S into a monoid: $S^I$ and $S^\bullet$ (this latter is often written $S^1$). The construction $S^I$ adjoins a new identity $I$ to $S$, even if $S$ already has an identity, whereas $S^\bullet$ adds an identity only when $S$ has no identity. The first is the object part of a functor $\mathbf{Sgp} \to \mathbf{Mon}$, the second is not a functor; even better, $S \to S^I$ is the left adjoint of the forgetful functor from $\mathbf{Mon}$ to $\mathbf{Sgp}$. So (...) the construction $S \to S^I$ is the correct (or canonical) choice. This has the effect that for $G$ a group, $G^I$ is not a group. That's tough luck: from the categorical viewpoint, we cannot avoid this.

Now, another point of view is to consider semigroupoids instead of categories. Quoting [1] p. 95,

A semigroupoid is defined precisely like a category, but one relaxes the axiom demanding identities at each object. We can view a semigroup as a semigroupoid with a unique object.

[1] J. Rhodes, B. Steinberg, The $q$-theory of finite semigroups. Springer Monographs in Mathematics. Springer, New York, 2009. xxii+666 pp. ISBN: 978-0-387-09780-0