Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: faithfulness and Cauchy completions

John Baez (Jul 29 2024 at 14:39):

In a paper I'm coauthoring we want to use the following fact, and we'd like a reference for it:

If $C$ and $D$ are $k$-linear categories, $F : C \to D$ is a fully faithful $k$-linear functor, and $D$ is Cauchy complete, then the functor $\overline{C} \to D$ obtained by extending $F$ to the Cauchy completion $\overline{C}$ is also fully faithful.

John Baez (Jul 29 2024 at 14:40):

Does anyone know a reference for this, or a reference for some more general fact of which this quickly falls out as a consequence?

Nathanael Arkor (Jul 29 2024 at 16:33):

The more general fact is that if $\Phi$ is a class of weights, then a functor $f : A \to B$ is fully faithful if and only if the induced functor $\Phi(f) : \Phi(A) \to \Phi(B)$ between their $\Phi$-cocompletions is fully faithful.

I don't actually know an explicit reference for this, but it is an easy consequence of the fact that if $f : A \to B$ is fully faithful, and $g, g' : A \to C$ are functors, then $C(\text{Lan}_f g, \text{Lan}_f g') \cong C(g, g')$, assuming the pointwise left extensions exist.

John Baez (Jul 29 2024 at 20:18):

Thanks. Can I cite your brain?

Jonas Frey (Jul 29 2024 at 21:01):

This is probably related to what @Nathanael Arkor is writing, but my argument would have been that an enriched functor is fully faithful iff the associated adjunction of profunctors is a coreflection, and Cauchy completion is invisible from the profunctor perspective. (but I don't know a reference either)