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Stream: learning: questions

Topic: Intuition for enrichment

Nolan Peter Shaw (Mar 27 2024 at 19:10):

Hi everyone,

I'm new here and greatly appreciate the chance to talk with you all. I have a quick question I'd like to ask about enrichment. My current intuition has been to think of it in a manner similar to a $2$-category/bicategory: enrichment preserves the original category, $C$ (by way of "$1$-cells"), but uses the category $K$ to specify a structure on the morphisms of $C$.

Here's where I'm hoping for a sanity check: by the definition of enrichment, each $hom_C(a,b)$ is treated as an object rather than a set. However, objects are vacuous by nature. Is it correct to imagine the hom-object as specifying how we draw arrows between the morphisms of $a$ and $b$ in light of this, or has my intuition led me astray? Does the structure of enrichment only manifest when looking at composing our "$1$-cells"? Is there a simpler "visual" intuition for enrichment?

Kevin Carlson (aka Arlin) (Mar 27 2024 at 19:16):

I don't think a 2-category is a complete analogy to think from. For instance, a metric space is a kind of enriched category, and there's really no preservation of an "original category" of 1-cells here. You can always recover an original category by mapping the monoidal unit of the enrichment base into the hom-objects, but I don't think that's the best basis to start from.

John Onstead (Mar 27 2024 at 20:07):

I had a similar thought when I was learning enrichment. Indeed, a 2-category is a case of enrichment when the base is itself Cat. However, the trick of drawing arrows between morphisms above only works, I believe, if one can interpret the hom objects as categories whose objects are the "elements" of the hom object (where an "element" is a point in the underlying set of the object, given it is an object in a concrete category), in which case you could interpret the enrichment like if you were doing it in Cat. If not, then the structure within the hom object cannot be specified by 2-cells. An example is a category enriched in some algebraic structure, for instance groups. A category enriched in groups can be visualized as a normal category, but where every pair of objects comes with a rule for "adding together" the morphisms between them to get other morphisms between them. This structure is a different kind as to what could be specified by 2-cells between those arrows. In general, given a concrete category as a base, you can imagine the elements of the underlying sets of the hom objects as the morphisms between objects, but you still need to account for the extra structure on that set in some way.
But as Kevin Carlson said, the best way to think about enrichment is to imagine collections of objects with no inherent notion of "individual morphism" between them. Instead, a good intuition is that every pair of objects has a "relationship structure" with one another, which details some general way that pairs of things can be related, which is then encoded within the hom object between them. Perhaps you want to see how two objects relate through sets of mappings, in which you use normal categories, but perhaps you want to see how they relate through their "distance" to one another, in which you get Lawvere metric spaces, etc. Put another way, an enriched category isn't necessarily some normal category that has then "been enriched", it is a more or less independent construction in which some of the theory works like it does for normal categories. I hope that helps!

Nolan Peter Shaw (Mar 27 2024 at 20:39):

Thank you! That did help to clarify things

Mike Shulman (Mar 27 2024 at 21:19):

However, it is always true that an enriched category $C$ has an underlying ordinary category $C_0$, defined by $C_0(x,y) = V(I,C(x,y))$, where $V$ is the enriching monoidal category and $I$ is its unit object. So in some sense one can think of "a $V$-enrichment" of an ordinary category $D$ as the specification of a $V$-enriched category $C$ such that $C_0 \cong D$.

Jules Hedges (Mar 28 2024 at 12:39):

I'm absolutely not an expert on enriched categories, but I do think of them as a category equipped with a functor $\mathcal C^\mathrm{op} \times \mathcal C \to \mathcal M$ + structure isos - this seems to be very much a minority view and I don't quite understand why

There's a tensor-hom adjunction between enriched categories and monoidal category actions (aka actegories), which is a category equipped with a functor $\mathcal M \times \mathcal C \to \mathcal C$ + structure isos. So I think of enriched categories and actegories as essentially different perspectives on the same idea, but it seems normal to think of actegories as categories equipped with structure, but think of enriched categories as something philosophically distinct from categories

Nolan Peter Shaw (Mar 28 2024 at 12:49):

Okay, that's super useful to know. I saw actegories briefly and catalogued them in my head as "group actions but for categories-look at more later." I'll work on better understanding both these conceptualisations.

Mike Shulman (Mar 28 2024 at 14:18):

Jules Hedges said:

There's a tensor-hom adjunction between enriched categories and monoidal category actions (aka actegories), which is a category equipped with a functor $\mathcal M \times \mathcal C \to \mathcal C$ + structure isos.

I wouldn't say there is a tensor-hom adjunction between those things (unless you're referring to something other than what I think), but that the tensor-hom adjunction of either one allows us to construct the other one. Specifically, if the hom of an enriched category has a left adjoint tensor (or "copower"), that makes it an actegory; whereas if the action of an actegory has a right adjoint, that makes it an enriched category; and the result is an equivalence between copowered enriched categories and "hommed" actegories.

Mike Shulman (Mar 28 2024 at 14:18):

I agree that this perspective is underappreciated!

Mike Shulman (Mar 28 2024 at 14:19):

Probably that's partly because it doesn't apply to all enriched categories and all actegories, but it does apply to most "large" ones that appear in practice. And it's a very important perspective in homotopy theory, since it's easier to "derive" a tensor-hom adjunction than it is an enrichment directly.

Nathanael Arkor (Mar 28 2024 at 15:05):

It might be worth mentioned that the notion of [[locally graded category]] is much closer to the intuition of "structured category" than enriched categories, but subsumes enriched categories, actegories, and powered categories in a very natural way.

Reid Barton (Mar 28 2024 at 15:10):

Jules Hedges said:

I'm absolutely not an expert on enriched categories, but I do think of them as a category equipped with a functor $\mathcal C^\mathrm{op} \times \mathcal C \to \mathcal M$ + structure isos - this seems to be very much a minority view and I don't quite understand why

I don't really think it's a minority view if we consider that Ab-enriched categories are preadditive categories and most mathematicians would probably give a definition of those as an ordinary category with extra structure, e.g. https://stacks.math.columbia.edu/tag/00ZY.

And as Mike already mentioned, in homotopy theory the idea of an enriched category as an ordinary category plus some extra structure (which might even be given in the form of an action rather than an enrichment) is definitely an important one. Maybe it's different in other areas of math (or CS, or physics, or ...) though.

Mike Shulman (Mar 28 2024 at 15:37):

I would say that in mathematics at large, the idea of an enriched category (insofar as it is understood abstractly at all) is definitely that of an ordinary category plus structure. I think it's mainly category theorists who emphasize the view of an enriched category as a different thing from which one can construct an "underlying" ordinary category, because that view is easier to define and work with abstractly.

Kevin Carlson (aka Arlin) (Mar 28 2024 at 17:02):

I would also (again?) say that it just doesn’t always make sense to pretend you’re adding more info to the underlying category. When I define a pseudometric space, I can’t imagine starting with the underlying equivalence relation of indistinguishable points and then adding the nonzero distances. It seems like this preference has a lot to do with the idea that all important categories are concrete, which just isn’t quite true, especially if you implicitly restrict to forgetful functors represented by monoidal units.

Mike Shulman (Mar 28 2024 at 17:21):

It always makes sense mathematically, even if it's not always a useful intuition.

Mike Shulman (Mar 28 2024 at 17:22):

Just like it makes sense mathematically to say that a "set" is a group equipped with a distinguished collection of generators, i.e. to consider the free group functor as "forgetful"... (-:

Mike Shulman (Mar 28 2024 at 17:23):

But with my tongue out of my cheek, I think my point was that outside of category theory, for the enriched categories used by most mathematicians, the enriching categories do tend to be concrete, with forgetful functors represented by the monoidal units.

Mike Shulman (Mar 28 2024 at 17:24):

I mean, obviously non-category-theorists do use metric spaces, but they don't think of them as enriched categories.

Mike Shulman (Mar 28 2024 at 17:24):

The main counterexample I can think of is categories enriched over $G$-sets, or $G$-spaces, where the "forgetful functor" represented by the monoidal unit is actually the fixed-point set rather than the underlying set.

Mike Shulman (Mar 28 2024 at 17:25):

But even there, the enriched-category-theory notion of "underlying category" is usually correct, e.g. for the $G$-set-enriched category of $G$-sets itself, the hom-$G$-set $[X,Y]$ is the set of all functions $X\to Y$, and the $G$-fixed-points are the $G$-equivariant maps, which are the ones you would naively take in "the ordinary category of $G$-sets".