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Stream: deprecated: id my structure

Topic: Systems of objects "orthogonal" to categories

Paolo Perrone (Apr 16 2023 at 16:40):

When a mathematician thinks of real numbers, one usually thinks about a lot of structures at the same time, compatible with each other: $\Bbb{R}$ as a field, as a poset, a metric space, and so on.
This is somewhat opposite to how we look at $\Bbb{R}$ in category theory, where we usually ask, well, $\Bbb{R}$ in what category?
So, is there a way of formalizing the structure formed by all these different instances of $\Bbb{R}$ as objects of the different categories?
I would be tempted to look at particular families of morphisms in the fibration induced by the identity pseudofunctor $\mathrm{Cat}\to\mathrm{Cat}$, but I can't make it precise... Can anyone help? Does this already exist?

I feel like this would be also a nice way of strengthening the link between the categorical notion of category and the philosophical one (at least in the sense of "there are different categories in which we can study $\Bbb{R}$").

My question, of course, is not limited to $\Bbb{R}$, one could ask similar questions about $\Bbb{Z}$, $S^1$, etc.

Michael Stone (Apr 16 2023 at 23:17):

Can you give some examples, maybe set in an ideal world, of what you’d like to be true about such a formalization or of what you’d like to do with one?

Mike Shulman (Apr 17 2023 at 20:53):

Aren't we just talking about objects in the fiber over $\mathbb{R}\in \rm Set$ of forgetful functors to $\rm Set$?