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Stream: learning: questions

Topic: Fully faithful cofunctors

Paolo Perrone (Dec 24 2022 at 15:44):

Are there established notion of full, faithful, and essentially surjective (or injective?) cofunctors, mimicking the ones for functors?

Bryce Clarke (Dec 27 2022 at 02:07):

Paolo Perrone said:

Are there established notion of full, faithful, and essentially surjective (or injective?) cofunctors, mimicking the ones for functors?

I don't know about established, but I think there are suitable notions. Consider a cofunctor $\Phi \colon A \nrightarrow B$ with lifting operation $\Phi_{a, b} \colon B(Fa, b) \rightarrow \sum_{x \in F^{-1}\{b\}} A(a, x)$. A cofunctor is full if this function is surjective, and faithful if it is injective. Note that if we decide to view a cofunctor as a span of functors, these properties correspond to the bijective-on-objects leg being full or faithful, respectively. A fully faithful cofunctor is the same as a discrete opfibration.

As for essentially surjective on objects, I don't know if the concept makes a lot of sense, because cofunctors lift isomorphisms, an eso cofunctor is also surjective on objects.

Paolo Perrone (Dec 27 2022 at 11:11):

Thank you! So maybe the interesting notion is "essentially injective".