Originally I was planning to write a post called something like “monoid facts everyone should know”, but I’m going easy on myself and giving you just one fact for now.
If you ask someone for the definition of a monoid, there are two sorts of answers you’ll get:
- it’s a set equipped with an associative multiplication function and a unit element
- it’s (just) a category which happens to only have one object
The category theorist’s lot in life is to minimize the times the word “set” appears in any definition. In this case, the second definition achieves this superficially, but of course not in essence. A category is a set of objects and a set of morphisms. An uncharitable spectator might point out we’re actually moving in the opposite direction!
They’re both good definitions. Of course they define equivalent notions of monoid. Given a set with multiplication, you can invent a one-object category (called the delooping of the monoid) as follows:
- there’s only one object, so it doesn’t have to be anything specific. You don’t even have to give it a name. It is almost no data whatsoever. It’s a phantom.
- the endo-arrows (they’re all endo) are the elements of the monoid. the unit is the identity map.
- composition is given by multiplication.
The most important thing to notice here is: in a general category you can’t just compose any two arrows, they have to be “composable”: the end of one is the beginning of the next. But if there’s only one object, this is automatically true for all arrows. So composition is well-defined for any pair of arrows, which is what multiplications are usually like.
But having two different characterizations of the same thing can often be useful. In this case, the first definition is better suited to internalization, while the second is better for enrichment. Let me explain:
Given a monoidal category C, a monoid object in C ought to be an object of C with a C-map for its multiplication. A topological monoid should be a topological space with continuous multiplication.
A monoid enriched in C should be a one-object C-enriched category. For an example of this, consider the category of abelian groups, Ab, with monoidal structure given by tensor, as our enriching category. This is quite a popular enriching category, e.g. abelian categories. A one-object Ab-enriched category would then be an abelian group A (the unique hom set object) with another operation 
Punchline: C-internal monoids are equivalent to C-enriched monoids. The equivalence is given by an analogous construction: you sit the base C-object on top of a phantom object, use the multiplication C-map as the C-enriched composition operation. So really I could have used either of the above examples for either case.
I really like this stuff because it gives a bridge between internalization and enrichment. I remember when I was first learning category theory that these two big words seemed to be the major paradigms you had to be comfortable with if you had any hope of getting into research. I also found internalization easier than enrichment, so this gave me at least one way to say “yeah, I know how to find examples of C-enriched categories”, as if anybody was going to ask me to do that. It seems like knowing wads of examples (or at least how to find them) is the only way to get truly comfy with a concept.
Joe Moeller 
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