Enriched categories

This post is part two of a little two-part miniseries about defining the cosmos. To learn what a cosmos is in mathematics, or rather what we want it to be, you can read the first part. There we described a cosmos as a nice place to enrich a category, or a nice place to do enriched category theory, and to quickly recap, an enriched category is a category where we have objects of morphisms instead of just a collection of them, and these objects come from some monoidal category....

I think there are many parts of physics worth studying for mathematicians, and the physical notion of a cosmos may be one of them, but, this post is not about physics. Even though the usual field of study one thinks of when hearing the word “cosmos” is physics, there is also a type of mathematical object with the same name. This type of object does have that name for a reason, which is not clear maybe from the object it self, but from what one can do with and in such an object....