Mario Román

@sarah_zrf Ooooh, I see, that's cool!

@sarah_zrf Yay! Is this homotopy.io? I have the composition of lenses written there but can you do animations?

Mario Román (@mroman42) again - "Open diagrams via coend calculus". I **love** this stuff, and not only because it's useful for me - comb diagrams for open games are part of the motivation - the pictures are so pretty

Mario Román (aka @mroman42) - "Profunctor optics, a categorical update". Comprehensive work on optics that came out of last year's ACT school Hopefully everyone was paying attention, this will be on the exam (in my talk tomorrow)

What a nice surprise! Our paper "Profunctor Optics, a Categorical Update" has been accepted for a keynote at Applied Category Theory 2020. When I was first approached to mentor at the ACT 2019, I though I had nothing, and almost declined the offer. But the great team did it!

New preprint time! Games on graphs: A compositional approach, with Elena Di Lavore and @PawSob The punchline: "Open games on open graphs" are strong monoidal functors from the category of open graphs ("syntax") to the category of open games ("semantics") arxiv.org/abs/2006.03493

@star_autonomy Both online editors (TikZiT, Mathcha) and then manually tweaking the tikz code later with the text editor. It takes a bit of patience and has some problems, but sometimes it is easier than pure tikz.

@star_autonomy Hi, Nicolas! something that works is to let distributors compose to form sets of possible meanings; a point tracks one of them. The intuition in arxiv.org/abs/2004.04526 can be useful. There are variations on this. It is in progress; hopefully, we will have something written soon.

@Nadrieril @_julesh_ This! I think I was assuming the monoidal to be a cartesian product so that forgetful preserves it and then EM -> C -> [Kl,Kl] is composition of strong monoidal functors.

I have a technical question about optics, since basically all of the experts on optics are on twitter For Reasons, for a commutative monad T (say on Set) I'd like to talk about mixed optics where the forwards part lives in Kleisli(T) and the backwards part in Alg(T). (1/n)

@_julesh_ I think we talked about this in Tallinn and the proposal was ∫{C ∈ EM} . EM(MS, C × MA) × Kl(UC ⋊ B, T) ≅ ∫{C ∈ EM} . EM(MS, C) × EM(MS, MA) × Kl(UC ⋊ B, T) ≅ EM(MS, MA) × Kl(UMS ⋊ B, T) ≅ C(S,MA) × C(MS × B, MT) Not sure if that is what you want; it is still subcat

What I don't know is whether this is itself a category of optics, or just a full subcategory of one. My plan was to get Kl(M) to act on Alg(M) by S . B = MS x B, but that doesn't define an action because Kl(M) -> Alg(M) isn't strong monoidal. Have I missed a trick? (3/3)

@8ryceClarke Thank you, Bryce! These things owe a lot to the work together with the ACT group :)

Coends, graphically :) For instance, this is how to compose a lens with a continuation.

@sir_deenicus This is a coend, a particular kind of colimit. So, yes, in some sense, it is analogous; but it is not an integral (at least not in any obvious way). This is a nice intro to coend calculus here: arxiv.org/abs/1501.02503

@mroman42 This notation is opaque to me but I'm curious about what the integral notation signifies. Is there some interesting analogy here? Or is something actually being summed?

@coecke That is, one is a monoid, creating a bimonoid with the cartesian structure. But then we also have their adjoints.

@coecke Black comonoid copies and discards white monoid (or any representable functor); black monoid copies and discards white comonoid (or any correpresentable). White-black monoids coincide iff C is cocartesian; white-black comonoids coincide iff C cartesian.

@rwolffoot @PawSob I may be using notation that suggests squashing but it is not intended to mean squashing, 2-cells are still natural transformations.

@PawSob @mroman42 It's explicitly a generalisation (without squinting), isn't it? It's Carboni-Walters II, except @mroman42 has replaced proper 2-cells by an order relation. I guess the question is: is this squashing down of 2-cells legitimate when talking about these higher-order structures?

@_julesh_ The open boundaries are nodes in the monoidal bicategory of profunctors. Unlabelled structures are the monoidal product (white) and the comonoid structure lifted from Cat via Yoneda (black).

@mroman42 Personally I find this style for string diagrams a bit weird.... using things that look like full nodes for labelling the open boundaries, and using unlabelled nodes that look like a bialgebra structure for the nontrivial morphisms

Or actually computing what a cartesian lens is.

Or composing lenses

Just released a new version of DisCoPy! github.com/oxford-quantum… A cool new feature is a drawing function which turns a diagram into TikZ code, ready to copy and paste in a LaTeX article. That's how I wanna draw all my diagrams from now on.

@xgrommx Ends appear naturally here as natural transformations, but in principle, this is more suited for coends.

@mroman42 how about ends? as I can see here u use existential, but how about forall?

@tangled_zans In this case, I am calling continuation just to a function embedded into Optic (!), and then using coends to describe optics.

@mroman42 woah! i really need to learn the coend calculus. what's the link between coends and continuations?