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Stream: learning: questions

Topic: "enriched enriched functors"

Paolo Perrone (Feb 10 2022 at 12:49):

The V-enriched functors between V-enriched categories form a (maybe large) set, not a V-object.
Has anyone attempted to create a V-object for them?

Zhen Lin Low (Feb 10 2022 at 13:20):

V-enriched categories in general, not just functor categories, only have a set (maybe large) of objects. Are you thinking about internal categories?

Reid Barton (Feb 10 2022 at 13:22):

So if I have two rings (seen as Ab-enriched categories with one object) then I get a set of ring homomorphisms between them. How would one get an abelian group of morphisms?

John Baez (Feb 10 2022 at 16:18):

One wouldn't, methinks.

Tobias Fritz (Feb 10 2022 at 17:21):

I'm not sure if this addresses the original question, but there is a nice way to consider the homomorphisms between two rings as an abelian group, namely Sweedler's measuring coring construction. I'm not sure how this works in general, but the case of finite-dimensional algebras $A$ and $B$ over a field $k$ is nice and simple, so let me explain that. In that case, one can define the hom-space to be just the set of linear maps $R \to S$! The point is that it becomes a coalgebra with the comultiplication defined as $\Delta(f)(a_1 \otimes a_2) := f(a_1 a_2)$, and the usual algebra homomorphisms are then just those elements coalgebra $f$ that are "group-like", which means that they satisfy $\Delta(f) = f \otimes f$. (It's very instructive to work this out.) But in general, the coalgebra structure contains more information than that. And one can show that this "measuring coalgebra" has a universal property, something like the terminal object in the category of all coalgebra which "intertwine" $A$ and $B$ in a suitable sense.

So in other words, the category of $k$-algebras is enriched over $k$-coalgebras! And these are in particular vector spaces.

Simon Willerton (Feb 10 2022 at 23:04):

For example, you want generalized metric spaces to form a generalized metric space? Like with a generalization of the Gromov-Haussdorf metric? There's this paper by Andrei Akhvlediani, Maria Manuel Clementino, Walter Tholen which looks at that kind of thing.

• On the categorical meaning of Hausdorff and Gromov distances, I