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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.
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?
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.
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