Owen Lynch on Nostr: This whole "we need higher morphisms so that path composition can be associative" ...

This whole "we need higher morphisms so that path composition can be associative" seems *so* complicated compared to just taking a category enriched in presheaves on the discrete category on the reals, with Day convolution corresponding to addition.

Or, in plainer English, "have a notion of r-morphism for each r, and then an r-morphism and an s-morphism compose to form an (r+s)-morphism." If the r-morphisms from x to y are maps [0,r] -> X, then this gives a strict composition.

If you want to be fancier, you could enrich in presheaves on the category where the objects are real numbers, and a morphism from r to s is a continuous map [0,r] -> [0,s].

In fact, we could have a whole "weighted cubical set" of higher paths, where we considered r_1,...,r_n-cubes. You could compose strictly along cube faces no problem, no higher associativities needed!

Does anyone take this approach to infinity category in practice? Perhaps it doesn't actually buy you that much.