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  • $\begingroup$ This reminds me of the very first collaboration I've engaged with Misha Gavrilovich on his model categories derived from a model of ZFC. We tried to understand something related to Quillen adjunctions in that context. On the first day I asked him what is a "class", is it a subset of the universe (so $V_{\kappa+1}$ when $\kappa$ is inaccessible), or is it a formula and variables interpreted in the particular model. He opted for the "easy way out", i.e. the former, and at the end what we tried to see turned out somehow trivial (although still interesting). I still believe, though ... (1/2) $\endgroup$ Commented Jan 28, 2021 at 17:45
  • $\begingroup$ ... that if we consider a class as a functor from the category of formulas (even with one parameter), we would have proved the result we set out to prove. But it would have been an order of magnitude harder, if not more. It's not that what we ended up with isn't interesting, just not very, I think, compared to the hypothesis we set out on the first day. It was interesting over all. Anyway, I digress. (2/2) $\endgroup$ Commented Jan 28, 2021 at 17:48