The n-Category Café

What I wrote above makes sense for any category enriched over some 2-category.

Part of my point was that where some people are familiar with decorating dual triangulations in algebroids, we might want to enlarge the point of view to more general -oids, those that do not necessarily come from enriching over $\mathrm{Vect}$.

For instance, for 2-bundle holonomy, we want to decorate in a category enriched over a 2-group.

And the only reason why in the first case we explicitly have a Frobenius-property in the game, while in the second we have not, is that in the second case the Frobenius property is automatically satisfied. So we may not realize that it’s there.

Okay, so that’s what I wanted to say. Shouldn’t have used the work “monoidoid”, probably. (Somehow I enjoyed writing it, was chuckling to myself when I did.)

Is there a Universal -oid?

John was quite right, I shouldn’t have use the word monoidoid the way I did.

So, let me try to clarify the situation:

whenever we have any structure $C$ which comes with a kind of product map

we say $C$ is “monoidal”.

The reason is that we can imagine every element $c$ of $C$ (if $C$ is in fact something that can be thought of as consisting of elements) to be like an arrow starting and ending at an abstract point $\bullet$.

The product operation which takes two such elements and sends it to a third can then be thought of as describing the composition of these arrows.

That’s the very starting point of category theory, in a way (not historically, but still): once we talk about composing arrows from $\bullet$ to $\bullet$, there is nothing more natural than considering arrows that go between more kinds of objects, like

Whenever we have a structure like that, with many different objects and such that the arrows

that start and end at one and the same object behave under composition like elements of $C$, then we say that the entire structure is a $C$-oid.

For instance: and algebra is a vector space with a product. The corresponding many-object entity is a category all whose spaces of morphisms between a given pair of objects is a vector space, such that composition is bilinear. This is hence called an algebroid. It is the same thing as a category enriched over vector spaces.

Now, when we have a collection of things and don’t specify any additional structure on it, like a vector space structure, but still have a way to compose two things to get a third, we just say we have a monoid.

Playing with these terms, it turns out that a monoid-oid is just something with many objects and a rule to compose arrows going between them. So it’s a category. Or some enriched category, if you like.

As John rightly pointed out, saying “monoid-oid” is perverse, and I have to apologize for doing so on a family blog like ours.