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Simonas Tutlys (Apr 30 2023 at 05:57):I recently found the paper Categorical semantics of metric spaces and continuous logic and as often happens it's very exiciting to see that people with actual academic positions are thinking similarly to you.I've been thinking about how to fuzzify category theory to use in categorical logic for structuring continuous stuff (optimization,machine learning) and have some barebones python code for a CT framework in python akin to the Algebraic Julia (which is very ambitious) ecosystem where you're able to replace strict equality with a metric for use in for example optimizing diagrams of machine learning models (those diagrams could be the semantics of a logical sentence so you're optimizing it's truth value).With the paper mentioned you could internalize category theory in a continuous topos with a continuous subobject classifier and formalize what I'm thinking about.But I haven't found any work explicitly mentioning computational trilogy and continuous logic or probability not studied through the curry-howard correspondence but side-by-side on equal footing.So my questions is - are there any?
Reid Barton (Apr 30 2023 at 06:31):This isn't an answer to your actual question but you might also be interested in https://arxiv.org/abs/2107.10543, if you haven't come across it already.
Simonas Tutlys (May 01 2023 at 05:08):Thanks,this seems a less cluttered paper.I have this nagging feeling that's been eating at me for years that there's a deeper connection between optimization theory and category theory than modelling updates by lenses,since for example the relationship some model (in the sense of scientific or ML model) has with whatever phenomena it's trying to model is similar to the syntax-semantics duality in logic - the phenomena is the syntax (raw relationships without any meaning) and the model being semantics since it's interpretable.The problem is that trying to write that down you have a whole category which is 'invisible',it being the phenomena.I'm planning to check into goguens hidden algebras and modal behavioural equivalences,maybe that will give some insight into how to formalize a situation where some part of what you're trying to model is inaccessible.also Interesting that modal logic is modeled by (co)monads,in this case on the states of the phenomena and model categories.If we can define a fuzzy adjunction/(co)monad we can formally define what it is to optimize it.Also,relatedly I have a feeling that (co)limits have an intuitive interpretation in this as well-going by Goguens manifesto,limits being solution sets to problems defined by diagrams is similar to it being a generative model (an approximation of the probability distribution of P(D),D being the variables in the base diagram,the apex being the latent space and the morphisms going from it being the model.problem is that they need to interact somehow,maybe 2-categoricallly) and dually colimits are discriminative models P(A|D) where A is the apex,so that it's building a new model combining the situation in the base diagram D into A.with a formal approach to fuzzy commutativity akin what's in this paper that i'm trying to implement without any formal grounding in the python framework I mentioned we could start thinking about optimizing 'universal' properties in general i think.Maybe I'm beating myself up about a formal specification of what I'm thinking about and just need to code more :)