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John Onstead (Jan 20 2024 at 14:11):Normally, when we learn about natural transformations, we don't think too much about naturality or whether it's a property from the stuff, structure, property POV- we just assume that's how 2-cells work in a category of categories! However, my thinking on this matter has recently changed when I learned about alternate tensor products on nCat. In addition to the usual cartesian product, nCat can have many other tensor products depending on n, most famously the Gray tensor products for 2Cat. But there also exists a tensor product with a very humorous name, the "funny tensor product". As Cat is closed under it, it has an adjoint which can be viewed as an internal hom, giving rise to "funny functor categories" Funny(C, D) where the objects are functors C -> D and the morphisms are "unnatural transformations" between them. A natural transformation is then simply an unnatural transformation with the property of naturality!
This was quite interesting to learn, but it left me with a question: what does the naturality property in this context actually "look like"? That is, given any funny functor category Funny(C, D), how do you know which morphisms are actual natural transformations and which ones are "fully" unnatural transformations? How can you pick out the unnatural transformations which are natural?
Nathanael Arkor (Jan 20 2024 at 15:19):Are you looking for a categorical characterisation of the natural transformations, e.g. some universal property associated to ?
Ralph Sarkis (Jan 20 2024 at 15:29):There is a similar question with a well-known answer. There is a subcategory of containing only cartesian natural transformations (those whose naturality squares are all pullbacks), the pasting lemma shows that pullbacks compose, hence so do cartesian natural transformations.
A cartesian natural transformation can be characterized categorically, it is a cartesian morphism in (hence the name).
John Onstead (Jan 20 2024 at 23:28):Is there a universal property associated to [C, D] -> Funny[C, D], perhaps such that it generates an adjunction? Or maybe a natural transformation is "an X morphism" in Funny[C, D] where X is some property- perhaps monomorphism, epimorphism, cartesian morphism with respect to some functor (following from Ralph's answer), etc.? Or maybe there's some commutative diagrams in Funny[C, D] that only natural transformations can participate in? Any of these would be illuminating!
John Onstead (Jan 21 2024 at 06:30):After thinking some more I might have an idea... A functor N: C -> Arr(D) induces a natural transformation between functors Dom o N: C -> D and Cod o N: C -> D. Thus, the functor category [C, Arr(D)] has natural transformations as objects. I am thinking that likewise, a functor U: C -> FAC(D) induces an unnatural transformation, with FAC(D) ~ Funny[2, D] (FAC = Funny Arrow Category with objects morphisms in D, and morphisms any square of morphisms in D, not just commutative ones). Thus, the functor category [C, FAC(D)] has unnatural transformations as objects. There is an obvious inclusion i: [C, Arr(D)] -> [C, FAC(D)] relevant to determining natural from unnatural transformations. I am also wondering about the inclusion j: Arr(D) -> FAC(D), which should be bijective/essentially bijective on objects (since the objects in both categories are the arrows in D), but with the difference being in the morphisms. I believe that the action of the functor Hom(C, -) on this setup sends j: Arr(D) ->inc FAC(D) to i: [C, Arr(D)] -> [C, FAC(D)]. If so, then the problem of selecting natural transformations from unnatural ones reduces to one of selecting commutative squares of morphisms in D from any old square of morphisms in D (IE, determining how to make the subcategory Arr(D) from FAC(D)). Let me know your thoughts- am I on the right track?
John Onstead (Jan 27 2024 at 13:20):I think I have a good follow-up question to ask related to this one, and that involves determining an extranatural or dinatural transformation from an unnatural one. While it is true that extranatural and dinatural transformations are special cases of unnatural ones, they do not form a subcategory of Funny[C, D] unlike the natural transformations since composition isn't always defined for these transformations- that is, an extranatural composed with an extranatural may not be extranatural (though it will of course still be unnatural).
John Onstead (Jan 27 2024 at 13:21):Instead, they give a "paracategory", so my question for today will involve discussing paracategories. A paracategory is a category with composition partially defined,; any paracategory can be thought of as having all the objects along with some arbitrary selection of morphisms within some category (so long as they contain all the identities). The usual way we select morphisms in a category is via a functor F: D -> C, in which case our selection of morphisms are those in the image of F. The similarities continue: the image of a functor need not be a subcategory because composition isn't always defined when looking purely at the image. Thus, can we define any paracategory as the image of some functor?
[My attempt: It seems like we can if we have some category D with "no composition" (which could be some repeated coproduct of the interval category A -> B) and a surjective on objects functor F: D -> C. There's no composition to be preserved, so logically it seems as though such functors should be able to freely select any class of morphism (plus obviously all the identities) within the target category, and thus be a good candidate for being an equivalent notion to paracategories.]
Matteo Capucci (he/him) (Jan 28 2024 at 11:09):Very speculative idea: one might 'complete' a paracategory by formally adding all the missing composites. Dually, one might 'separate' a paracategory by forming a category such that if isn't defined in the original paracategory, and are no longer consecutive in the new category (something like: take the objects to be pairs and morphisms is such that and ).
Then the comparison going from the latter to the former should have the original paracategory as its 'image'.
John Onstead (Jan 28 2024 at 13:25):Good idea! The "completion" of a paracategory seems simple, it's likely the adjoint to the forgetful functor/inclusion U: Cat -> ParaCat. I also like the separation idea!
We have a more or less canonical inclusion [C, D] -> Funny[C, D] which selects a natural transformation from the unnatural transformations. Which category C and functor F does the same for extranatural transformations; that is, the image of what functor F: A -> Funny[C, D] is the paracategory corresponding to the extranatural transformations? Maybe it can be thought of as some separation of Funny[C, D], but then what is the proper procedure for this separation that would work on any Funny[C, D]?