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Atrium > Mathematics, Physics & Philosophy: Optimal object in category. Metric, objective funct on category objects. Optimization over category?

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    • CommentRowNumber1.
    • CommentAuthorjonatan
    • CommentTimeDec 20th 2015
    Is there notion about optimal object in category (that can be found by some algorithms, or - more importantly - that can be constructed (if unknown) by some algorithms), about metric and objective function over objects of category. I suppose, that metric and objective function is required to elaborate algoritms, that finds the searched object. What algorithms they can be? I suppose - graph and hypergraph algorithms.

    The core problem is this: there is lot of coalgebraic logics, each logic is defined by functor T, all functors T forms the category (as functors usually do). One should find the appropriate category for the required task and that means - find the optimal functors (object of the functor category). More concrete examle is this - there are deontic coalgebraic logics that may and may not contain paradoxes, find (construct) the logic that does not contain paradoxes.

    This can be great for AI as well. AI sometimes requires (e.g. for modelling cognitive and affective agents) agents with distinct personality and cognitive feature (e.g. to model the profile of customer) and then one should be able to select and mix the appropriate logics that do this modelling.
    • CommentRowNumber2.
    • CommentAuthortomr
    • CommentTimeMay 6th 2019

    While googling for the symbolic methods for optimization I have found this: http://math.mit.edu/~nrozen/juvitop/goodwillie-icm.pdf

    The differentiation of the functor is metioned here, it seems to me that the full analysis for the functors are derived by this author.

    So, if there the analysis of functors then: 1) this analysis should consider maximums and minimums for the functors (e.g. pointing the some sumpremums object in the codomain category) and hence - the objects in the domain category for which those extram are achieved. So - the this can be THE theory of the optimal object. 2) this analysis should consider variational integral for the functor and so - in principle one should be able to derive the optimal functor with such tools.

    Of course, I only will start to read this series of papaers and then I will see whether my expectations are satisfied, but it is nice to know that I am not the only one who is trying to use category theory for finding the optimal objects, optimal categories or optimal functors.

    • CommentRowNumber3.
    • CommentAuthortomr
    • CommentTimeMay 6th 2019
    • (edited May 6th 2019)

    Sorry, work by Goodwillie can not be applied for finding symbolic structures (logics, theories, set of expressions in some theory) that would be optimal in some kind of sense.

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