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A comment on Topology, a Categorical Approach. Parent comment
I never worked through the details, and I'm not a category theory partisan, but I think that's what this is: https://ncatlab.org/nlab/show/fully%20formal%20ETCS
I never worked through the details, and I'm not a category theory partisan, but I think that's what this is: https://ncatlab.org/nlab/show/fully%20formal%20ETCS