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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 25th 2011

    Ben Sprott is interested in the possibilities of using things like the Geometry of Tensor Calculus to present in some way a formal theory of categories. See his MO post here. I suggested starting a discussion at the nForum for anyone who is interested; our software and culture is probably better suited for such a discussion than MO.

    In a comment, David Roberts mentioned the quantum categories of Day-Street, and I notice that there is a bit of discussion on this theme at internal category in a monoidal category as well.

    • CommentRowNumber2.
    • CommentAuthorBen_Sprott
    • CommentTimeJul 2nd 2011

    I want to thank Todd for sharting this post. I am not the right person to start the thread. Perhaps there are other who would like to give their reactions to the idea of a diagrammatic calculus for categories. The best I have done, so far, in having creative ideas in how to present categories was to have the basic intuition that a category is a partial monoid where arrows are the elements and composition in the monoid is arrow composition. As a physicist, I like this presentation because it relates to a person's interaction with the transformations of a system. The data can be accumulated, building up one's knowledge about the structure preserving maps of the system under study. Also, one has a direct way to interpret the data up into a category.
    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJul 2nd 2011

    I’m not sure exactly what you mean by “a diagrammatic calculus for categories”. Do you mean something like: the theory of monoids lives naturally in a monoidal category, which has a string diagram calculus, so is there a context in which the theory of categories lives which has an analogous string diagram calculus?

    • CommentRowNumber4.
    • CommentAuthorBen_Sprott
    • CommentTimeJul 7th 2011
    Hi Mike,
    Yes, I think I do mean that: a context in which the theory of categories lives which has an analogous string diagram calculus. I am a physicist by training and since people have been using monoidal categories (with quite a bit of extra structure) to talk about FdHilb (which I was taught to think of in terms of as vector spaces over Complex numbers), I have wondered about how certain abstractions of this are somewhat more core to basic physics. I guess by this I mean, as you take away all the extra structure on FdHilb (dagger compactness and some other axioms), and approach just the monoidal axioms, these categories become more generally related to physics. I cannot yet give you any results on these ruminations. As an example, any map in FdHilb preserves all the FdHilb structure. In the gross extreme, a functor on FdHilb only preserves its category structure but not necessarily any of the axioms (except the basic category axioms like composition), as in the forgetful functor. Thus, I have tried to see how functors themselves reflect some of the more basic reasoning or structures and even practices of physics in general. This would give a somewhat more solid foundation for categories in physics, I feel. I have seen the diagrammatic calculus for FdHilb, and I have been wondering wondering what the diagrammatic calculus becomes when you remove (or add, I suppose too) various axioms. This thread is about the most extreme "removing of axioms", ie, removing all but the category axioms (such as composition).
    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJul 7th 2011

    Okay, thanks, but I’m still confused, because I think this:

    a context in which the theory of categories lives which has an analogous string diagram calculus

    is in conflict with this:

    wondering what the diagrammatic calculus becomes when you remove (or add, I suppose too) various axioms. This thread is about the most extreme “removing of axioms”, ie, removing all but the category axioms (such as composition).

    It’s easy to say what the diagrammatic calculus of monoidal categories becomes when you remove all axioms except those for a category: you just aren’t allowed to ever have more than one string. So it reduces to a 1-dimensional calculus, where there is a single string with nodes on it, each node labeled by a morphism and each interval between nodes labeled by an object.

    But if we want a context in which we can describe the theory of internal categories, we need more structure than exists in a cartesian monoidal category, not less: we need something to correspond to pullbacks. I do also have an answer to propose to that question, though, which comes from a further generalization: namely, to monoidal fibrations (aka indexed monoidal categories). These have a string+surface diagram calculus, and we can describe the notion of a category “internal to” or “enriched over” such a thing (the difference between internal and enriched starts to evaporate at this level of generality), which in the case of the codomain fibration for a category with pullbacks reduces to the usual notion of internal category. I can say more, if that’s the sort of thing you’re interested in.