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Stream: learning: questions

Topic: algebra for a profunctor

Harrison Grodin (Jan 16 2024 at 23:05):

I was reading the nLab page [[algebra for a profunctor]], and I was wondering if anyone had resources I could cite and/or read more about this and related concepts? (I would imagine this is a pretty fundamental idea in the theory of profunctors, but I can't seem to find it mentioned anywhere...)

[Apologies if this is a naive question - I'm new to profunctors.]

Nathanael Arkor (Jan 17 2024 at 08:51):

From the history of the nLab page, it looks like @Sridhar Ramesh introduced the concept, and that it may not appear elsewhere in the literature yet.

John Onstead (Jan 19 2024 at 00:05):

This seems like an interesting concept, but the page is a little confusing because it seems to be defining transformations between functors of different domains. The page states, given an "endo-profunctor" H: C x C^op -> Set and functor X: D -> C, an algebra is an extranatural transformation * -> H(X,X). The first functor is *, which the page lists as *: 1 -> Set, but which I think of as *: D x D^op -> 1 -> Set so the composition yields a bifunctor D x D^op -> Set. The other is listed as H(X,X) which I interpret to be H(X(-), X(-)): D x D^op -> Set since the arguments of X are objects in D. Now that both bifunctors have the same domain and codomain it makes it easier to reason about transformations between them. Is this a good way of thinking about it or am I getting something wrong?
As a side note, I'm still kind of "iffy" on extranatural transformations because they don't seem very "category theory friendly". That is, composition plays a deep and essential role in category theory, and yet extranatural transformations aren't always so easily composable. This is also why the "universal extranatural transformation" definition of end confuses me- in which category can an extranatural transformation be seen as a morphism so that it can play the role of a universal morphism?

Kevin Arlin (Jan 19 2024 at 00:30):

Yes, I think your correction seems right. The end is the universal extranatural transformation in the sense that it's the terminal object of a category, which is actually the same sense in which, say, a limit cone is the universal cone: it's the terminal cone, which is a property as an object, not as a morphism. And indeed cones cannot be composed!

John Onstead (Jan 19 2024 at 02:20):

@Kevin Arlin Then, my question would be, which category is the universal extranatural transformation a terminal object within? Usually, a universal morphism is a terminal or initial object in a comma category. For (conical) limits, it's the terminal object in the comma category (Diagonal/F) (AKA Cone(F)) over the functor category between the index category of our diagram and C. Is there some comma category where the extranatural transformation is terminal?

Kevin Arlin (Jan 19 2024 at 03:21):

The end of $F:C^{\mathrm{op}}\times C\to D$ is the terminal object of the category whose objects are extranatural wedges $x\to F(c,c)$ and whose morphisms are factorizations of wedges. This certainly has the vibes of a comma category but it isn't literally one; I'm not aware of a way to make the aesthetic correspondence formal but it's the same kind of idea as cones.

John Onstead (Jan 19 2024 at 04:02):

Recently I was learning about alternate tensor products on Cat: in addition to the cartesian product (and Gray tensor products for 2Cat) there's one called the "funny tensor product". The internal hom adjoint to the funny tensor product gives a functor category Funny[C, D] that gives as objects functors C -> D, and as morphisms "unnatural transformations" between functors, which I believe are the most general kind of map between functors. Indeed, under the page "paracategory", the nlab states that dinatural and extranatural transformations are special cases of unnatural transformations, but because they don't always compose, they form a paracategory out of funny functor categories instead of forming subcategories of them.

John Onstead (Jan 19 2024 at 04:03):

@Kevin Arlin Still, maybe there's a way to find a comma category (Diag/F) within a funny functor category, and then (somehow) restrict the objects of that comma category to the unnatural transformation that are extranatural, in which case the terminal object among this restricted set (maybe) is the universal one?

Sridhar Ramesh (Jan 19 2024 at 04:20):

Yes, I created this page on nLab fourteen years ago. This was just a generalization that seemed natural to me to describe on nLab at the time, but which as far as I know has seen no use in the literature before or since.

(Originally I created the page with the name of algebras for a "bimodule", rather than for an "endoprofunctor". I was trying to emphasize symmetry in presenting this notion which is equally a generalization of endofunctor algebras and of endofunctor coalgebras. But if we fixate specifically on the profunctor perspective, that perspective breaks the symmetry and in that context this notion should really be called "coalgebras for endoprofunctors").

I'm not sure why the category $D$ has been brought into the definition in revision 7, in case that has been the source of confusion.

At any rate, it may be helpful to see that this notion of a category of algebras is closely related to the notion of an inserter in $\mathrm{Cat}$. Specifically, keeping in mind that any bimodule $: C^{\mathrm{op}} \times C \to \mathrm{Set}$ can be represented in the form $(c_1, c_2) \mapsto \mathrm{Hom}(F(c_1), G(c_2))$ for some pair of parallel functors $F$ and $G$ out of $C$ (there will be many such representations, but a canonical such representation is the one via [[cograph of a profunctor]]), we have that the category of algebras for the bimodule is the same as the inserter from $F$ to $G$. And conversely, any inserter is the category of algebras for the bimodule it induces in this manner.

Harrison Grodin (Jan 19 2024 at 17:25):

Sridhar Ramesh said:

But if we fixate specifically on the profunctor perspective, that perspective breaks the symmetry and in that context this notion should really be called "coalgebras for endoprofunctors"

Can you say more about why this looks like coalgebras? Intuitively, this "feels true" to me, but I'm having trouble parsing the explanation in Section 3. (Is $C(1, X)$ a profunctor $D \nrightarrow C$? If so, how does the composition with $H$ typecheck?) Or maybe there's another way to see it, since that seems to rely heavily on $D$, which perhaps could be avoided?

Harrison Grodin (Jan 19 2024 at 17:29):

Also, I'm in the process of writing something that partially involves this concept. How can I best cite your work - something like "unpublished manuscript" on the nLab this bib entry from the nLab page (thanks @Kevin Arlin)?

Kevin Arlin (Jan 19 2024 at 17:29):

nLab pages contain suggested bibtex code, I think.

John Onstead (Jan 20 2024 at 04:35):

Sridhar Ramesh said:

At any rate, it may be helpful to see that this notion of a category of algebras is closely related to the notion of an inserter in $\mathrm{Cat}$. Specifically, keeping in mind that any bimodule $: C^{\mathrm{op}} \times C \to \mathrm{Set}$ can be represented in the form $(c_1, c_2) \mapsto \mathrm{Hom}(F(c_1), G(c_2))$ for some pair of parallel functors $F$ and $G$ out of $C$ (there will be many such representations, but a canonical such representation is the one via [[cograph of a profunctor]]), we have that the category of algebras for the bimodule is the same as the inserter from $F$ to $G$. And conversely, any inserter is the category of algebras for the bimodule it induces in this manner.

This makes sense, given a category of algebras of an endofunctor F is the inserter of F to the identity, so it is reasonable that the generalization would replace the identity with some other arbitrary functor G. If we replace G with id in the above, we get a bimodule Hom(F(-), ?) which is exactly what it lists under the page. It's also interesting that this construction gives any inserter an "algebraic interpretation". I'm still not exactly clear on how to find F and G from any random given bimodule C^op x C -> Set- I know any functor induces two "representable" profunctors, but I have not yet learned the opposite case of how a profunctor yields two parallel functors as indicated here.