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Stream: learning: questions

Topic: Lax functors as lax modules?

fosco (Jan 26 2021 at 15:41):

Everyone knows that categories are monoids with more than one object, and that a presheaf $F : \mathcal C \to \sf Set$ can thus be regarded as a module on which the monoid $\mathcal C$ is acting; this is evident, in "external" category theory, because if $\mathcal C$ has a single object, then it is a monoid, and a presheaf out of it is precisely a set over which $M$ is acting.

In internal category theory the analogy becomes slightly more nuanced, in the sense that to regard a category as a monoid one has to do a bit more of callisthenics; but in the end, the definition is given in such a way that an internal category in $\cal E$ whose object of objects is a singleton is just a monoid internal to $\cal E$. And thus an "internal presheaf" is just a module over which the monoid/category acts!

I have always believed this construction behaves well under pseudo-ification; what about trying to describe lax functors internally? It is evident that if in $\cal E$ there is enough room, the usual definition of internal functor can be tweaked in such a way that the usual compatibility rules with identity and composition are lax; and I guess one can render them universal in terms of a certain inserter/lax pullback condition...

My question is: does this make sense? Has anyone done/used this definition before?