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It has occurred to me, for some reasons, to ponder the following: Let C be a category, and consider the question of whether there is some category D and functor f : C -> D such that 1) f is an initial object in the category Hom(C, D), and 2) any other g : C -> E which is an initial object in Hom(C, E) factors uniquely through f. (Thus, f is an initial object in a particular full subcategory of C/Cat (the subcategory of coslices under C which are initial with respect to any parallel functors))
It seems to me that there should always be such an f, constructible “syntactically” as for many free constructions, but I haven’t checked this rigorously and, at any rate, I’d like to have a better grasp on what exactly it is. Is there some nicer way to think about this?
Hmm, I’m very unused to thinking about limits in functor categories that are not pointwise. Can you give an example of an initial object in a functor category Hom(C,D) that is not simply constant at an initial object of D?
Oh, that’s a good question! Actually, my fundamental interest is in my question in the context of the 2-category LexCat rather than Cat (thus, with C, D, E above taken as lex categories, f, g taken as lex functors, etc.), but I thought looking at Cat first might provide me useful intuition. Let us switch to the LexCat context, though, since that is ultimately what I care about. Here, initial objects are not generally given pointwise; e.g., an initial object in LexCat(C, Set) is given by the global sections functor C(1, -), which is not everywhere 0-valued.
Okay. I still don’t have any intuition for the general case, though, sorry. My intution doesn’t suggest to me that such a “universally initial functor” ought to exist, because it’s trying to be initial in two different ways at once; what sort of “syntax” were you thinking of?
I was thinking of something like constructing D as whatever can be obtained by applications of the following three “constructors”, and nothing else:
1) A constructor which puts in D a copy of everything in C (which we can think of as our lex functor f : C -> D)
2) Also, a constructor which, given any diagram g of shape C in D (i.e., given any lex functor g : C -> D) yields morphisms in D to serve as the components of a natural transformation from f to g [as well as yielding the morphism equalities (i.e., 2-cells) in D that make this transformation actually natural].
3) Also, a constructor which, given any lex g : C -> D and parallel natural transformations n1, n2 : f -> g yields morphism equalities in D to make n1 and n2 equal.
Ah, I see. I think you’ll need to also include some constructors ensuring that D has finite limits, but then it seems like it should work. Seems probably hard to say anything about what the result will look like, though.
Ah, right; I was leaving the constructors giving D finite limits implicit, along with the constructors ensuring that D has compositions of morphisms, transitivity of morphism equalities, etc. But, yes, add in “0) Constructors making D a lex category”. :)
Alas, it would be great if there were a nicer characterization of the result, but perhaps it is not to be…
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