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Lax functors describe emergent effects

In this post I’ll describe something that’s kind of common knowledge among applied category theorists, that when you describe behaviour via a (pseudo)functor to the category of relations, emergent effects are described by the failure of that functor to be strong, ie. to be an actual functor. This is more or less in Seven Sketches (Fong and Spivak say “generative effects”, which as far as I can tell is an exact synonym for emergent effects), but I’ll write it in one place and in my own words.

Suppose we have a domain-specific category \mathcal C. It’s hopefully monoidal and most likely a hypergraph category, but the basic idea of what I’m saying applies just to the category structure. The morphisms of \mathcal C are open systems of some sort, the objects describe the open boundaries that a system can have, and composition describes coupling two systems together along a common boundary. The standard sledgehammer for building categories like this is decorated cospans, with structured cospans as a closely related new alternative.

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Categorical cybernetics: A manifesto

(Image credit: H.R. Grant / SaurianDandy)

I suddenly became obsessed with cybernetics (exactly) 2 weeks ago when I learned what the word actually means, closely followed by this tweet bring my attention to the following line from the beginning of Kenneth Boulding’s classic (1956) paper General Systems Theory: “The developments of a mathematics of quality and structure is already on the way, even though it is not as far advanced as the “classical” mathematics of quantity and number.” Which sounds like category theory to me. (I think it’s extremely unlikely Boulding had category theory in mind – Eilenberg and Mac Lane’s General Theory of Natural Equivalences was published in 1945; for comparison the first textbook on graph theory was published in 1936.)

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Symmetric polymorphic lenses (maybe)

Whereas ordinary directed lenses go from a source to a view, symmetric lenses relate two different views of an internal source. I’ve been thinking about how to smoosh together symmetric lenses with polymorphic lenses, which are the sort used in functional programming. My actual goal is to invent “symmetric open games” (without relying on non-well-founded recursion like I did in this paper). But with the right definition, I think symmetric polymorphic lenses could be a useful tool to programmers as well.

Normally when I’m playing with an idea like this I mock it up in Haskell and play around with it to check it works, before committing to too much hard work on the proofs. But this would be very difficult (and maybe impossible) to write in Haskell, so the idea of this post is to give enough detail that some folks could try it in Agda or Idris, and give me feedback on whether it feels like something useful. (In the meantime, I will try an extremely fast and loose Haskell version by pretending that pullbacks and pushouts are products and coproducts, and see what happens.)

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Are open games useful for my problem?

Recently open games have reached a state where they realistically might start becoming useful to some people (see this demo). I want to write down the current state of the art, what I believe is possible and what I believe are the limitations of compositional methods. This guides which sorts of situations I believe open games will be applicable to.

First I will say what open games can already do (at least in theory). Then I will say what is awkward and what I think is probably possible with future work, and finally when I think you shouldn’t use open games. This post is written for a hypothetical reader who wants to use game theory as a modelling paradigm.

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Folklore: Monoidal kleisli categories

I’ve been threatening a few times recently to blog about bits of mathematical folklore that I use, i.e. important things that aren’t easy to find in the literature. I’m going to start with an easy one that won’t take me long to write.

Theorem: Given a commutative strong monad on a symmetric monoidal category, the kleisli category is symmetric monoidal in a canonical way.

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Lenses for philosophers

Lens tutorials are the new monad tutorials, I hear. (This is neat, since monads and lenses were both discovered in the year 1958.) The thing is, after independently rediscovering lenses and working on them for a year and a half before Jeremy Gibbons made the connection, I have a very different perspective on them. This post is based on a talk I gave at the 7th international workshop on bidirectional transformations in Nice. My aim is to move fast and break things, where the things in question are your preconceptions about what lenses are and what they can be used for. Much of this will be a history of lenses, which includes at least 9 independent rediscoveries.

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The pre-history of open games

I feel that having Compositional game theory accepted for publication marks the end of the first chapter for me, and marks open games as a proper research topic. I want to record how open games came to be, told through the story of this paper, which took almost 3 years from writing to acceptance. (That’s roughly the same amount of time as the Higher Order Decisions/Games duo, although they definitely felt longer. Their story can wait for another blog post.)

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