Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: enrichment as a structure

Matteo Capucci (he/him) (Feb 02 2022 at 16:51):

One can see enrichment in two ways:

1. You have a set $ob(C)$ and a 'hom-function' $[-,=] : ob(C) \times ob(C) \to \mathcal V$ and identities and composition similarly specified
2. You have a category $C$ and a 'hom-functor' $[-,=] : C^{op} \times C \to \mathcal V$ and identities and composition similarly specified

The main difference is that the categorical structure of $C$ can be 'in the way' of defining $[-,=]$ in the second case (I think). Also the underlying category of 'a category with an enriched hom-functor' might not be the category you started with, i.e. there's seems to be no guarantee that

$$C(x,y) = \mathcal V(I, [x,y])$$

So are the two ways of defining an enrichment equivalent? Is the second one even known?

Matteo Capucci (he/him) (Feb 02 2022 at 16:51):

The first can be converted to the second by setting $C = C_0$, but maybe even $C =$ chaotic category on $ob(C)$ might be feasible (since enrichment can also be specified as a functor $ob(C) \to \mathbb B \mathcal V$ where the domain has the chaotic structure)

Matteo Capucci (he/him) (Feb 02 2022 at 16:52):

The other way around is also quite easy, just forget the morphism part of $[-,=]$

Mike Shulman (Feb 02 2022 at 17:04):

If you take the chaotic category for $C$ then the enriched homs won't be a functor $C^{\rm op} \times C \to V$.

Alexander Campbell (Feb 03 2022 at 03:05):

@Matteo Capucci (he/him) In my paper on Skew-enriched categories, I defined the notion of a "skew-enriched category", which is essentially a category with the extra structure of an enrichment. In general, the original category need not coincide with the "underlying category" of the resulting enriched category (this is what the word "skew" refers to); I called a skew-enriched category "normal" when these do coincide.

John Baez (Feb 03 2022 at 04:44):

Is the "underlying category" of a $V$-enriched category $\mathcal{C}$ defined by taking the set of points $p: 1 \to \mathcal{C}(x,y)$ as the homset from the object $x$ to the object $y$, where $1$ is a terminal object (or monoidal unit) in $V$?

Zhen Lin Low (Feb 03 2022 at 04:45):

The conventional definition uses the monoidal unit, but yes.

John Baez (Feb 03 2022 at 04:46):

I know various usual definitions; I was just wondering which Alexander was using, especially since he put "underlying category" in quotes, suggesting he might not mean the usual thing.

Mike Shulman (Feb 03 2022 at 04:58):

What you can do, however, in addition to taking $C=C_0$ is to let $C$ be the discrete category on ${\rm ob}(C)$.

Mike Shulman (Feb 03 2022 at 04:59):

In general, I guess (2) should be equivalent to giving a category $C$, a $V$-category $D$ (in the usual sense of (1)), and a bijective-on-objects functor $C\to D_0$.

Nathanael Arkor (Apr 21 2022 at 16:54):

I notice enrichment is defined along the lines of (2) in §5 of McDermott–Uustalu's preprint What makes a strong monad?.

Tim Campion (Apr 22 2022 at 00:57):

Viewing a $V$-enriched category as a category plus structure is a generalization of viewing a $V$-tensored category as a category with a module structure over the monoidal category $V$. This is a perspective which is often taken in homotopical contexts.