Bruno Gavranović

Does the math check out @ququ7 ?

... here since we "use" the identity morphism from each object. So to summarize, ends are quite interesting beasts and my initial attempt to see them as limits using the diagonal functor doesn't work since the end says something about non-diagonal elements too.7/7

...to every map f:B→C three objects, B§, f§ and C§ to every functor F:𝒞ᵒᵖ×𝒞→𝒟 a functor F§:𝒞§→𝒟 whose limit is then exactly the ∫F. Even more so, it turns out that this ad-hoc category is actually the twisted-arrow category of 𝒞... (as per Mike Shulman's comment...5/n

Furthermore, even in the case that we are in a dagger category and that there is a functor ▵:𝒞→𝒞ᵒᵖ×𝒞, the limit of precomposition of F:𝒞ᵒᵖ×𝒞→𝒟 with ▵ is different than ∫F! The limit of ▵;F only sees and can only talk about the diagonal elements of 𝒞ᵒᵖ×𝒞... 3/n

...express that a functor 𝒞→𝒞 is contravariant. Then I'd try to mix covariant and contravariant functors (in the form of a map ▵:𝒞→𝒞ᵒᵖ×𝒞), which you can't really do (if you're not careful about domains and codomains). 2/n

Spelling out my previous confusion in more detail, I was mixing up contravariant functors and the ᵒᵖ notation, I'd try to say that there was always a functor 𝒞→𝒞ᵒᵖ (but that it might be contravariant), ignoring the fact that a map 𝒞→𝒞ᵒᵖ is just a way for us to ... 1/nhttps://twitter.com/ququ7/status/1222963230248775685 …

Let F:𝒞ᵒᵖ×𝒞→𝒟 be a profunctor. What is the difference between the end of F and a limit of ▵;F:𝒞→𝒟, where ▵:𝒞→𝒞ᵒᵖ×𝒞 is the copy functor?

us from picking a map. Sure, we can say that in this specific 𝒱 there is a unique map between every two objects, but then that means 𝒱 is Set-enriched, rather than 𝒱-enriched. How do I understand this? This question seems in spirit similar to https://twitter.com/bgavran3/status/1193951310283976706 … 4/4

...A, B, and C. But if each object in 𝒱 _isn't a set_, then how can we pick out that composition _map_? A good example is 𝒱=([0, ∞, ≥]), (so enriching gives you a Lawvere metric space) where objects are _natural numbers_, so each hom object is a number, preventing... 3/n

For some category 𝒞 enriched in 𝒱 the composition is defined as a map Hom(B,C)⊗Hom(A,B)→Hom(A,C) in 𝒱. But some category 𝒱 can also be enriched over itself! This is fine if each object in 𝒱 is a set, so that the composition rule can pick out a specific map for each ... 2/n

I'm realizing there's something that's been bugging me about enriched categories and I now might be finding a way to articulate it. I also sense there's a simple solution! The question goes as follows. 1/n

I'm just now applying this to lens constructions in supervised learning and getting interesting ideas

I just realized any map A⊗B→C can be made into a lens in three different ways, whose types are the following: (A⊗B,C) → (1, 1), (A, C) → (1, B), and (1, C) → (1, A⊗B). In other words, any time you have any such lens, perhaps you should consider a one of the other variants

"Virtual proarrow equipment" is on the top when it comes to cool sounding concepts in CT, right there with "profunctor optics" and "tambara modules"

For V=Set we get small categories in both cases, but for V=Cat we get bicategories and double categories. Is there a table of all the results as we consider other bases? Also, to me, it seems like enriched cats are more widespread. Is this true? 2/2