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Stream: learning: questions

Topic: representably co-ff 1-cells

sarahzrf (Dec 23 2020 at 22:08):

Is there a name for 1-cells in a bicategory with the dual property to being representably ff? I.e., the property I'm looking for is that restriction along them is fully faithful as a functor between hom-categories.

sarahzrf (Dec 23 2020 at 22:08):

Also, are there nice characterizations of when this occurs? I have some suspicion that it might be related to density/codensity aka adequacy...

sarahzrf (Dec 23 2020 at 22:11):

One primary reason these are of interest to me is that Kan extending along functors p with this property gives rise to adjoint triples Lan_p ⊣ p* ⊣ Ran_p with ff middle entry p*, and these are nice—for example, we get a canonical transformation χ : Ran_p → Lan_p

sarahzrf (Dec 23 2020 at 22:13):

I've noticed that one class of functors with this property are the terminal functors ! : BM → 1 from 1-object categories—in which case the triple above is colim ⊣ Δ ⊣ lim between a category and the category of M-actions in it

Alexander Campbell (Dec 24 2020 at 00:52):

An interesting characterisation of the "co-fully-faithful" functors in Cat (see https://mathoverflow.net/a/354097/57405): a functor $u \colon I \to J$ is co-fully-faithful if and only if it is "initial for ends", i.e. for any functor $F \colon J^{op} \times J \to C$, the canonical map $\int_{j \in J} F(j,j) \to \int_{i \in I} F(u(i),u(i))$ is an isomorphism.

Alexander Campbell (Dec 24 2020 at 00:55):

In the paper (http://www.tac.mta.ca/tac/volumes/8/n20/8-20abs.html), these functors are called "lax epimorphisms", and the authors characterise them as the "absolutely dense" functors. They say that a functor $u \colon I \to J$ is absolutely dense if each object $j \in J$ is the colimit of the canonical diagram $u/j \to J$ (i.e. $u$ is dense) and moreover these colimits are absolute (i.e. preserved by all functors out of $J$).

sarahzrf (Dec 24 2020 at 04:23):

tyvm!

sarahzrf (Dec 25 2020 at 07:55):

ah nice, that paper gives another more direct condition in terms of connectedness of some sort of categories of factorizations

sarahzrf (Dec 25 2020 at 07:56):

also, interesting classes of examples: any localization is co-ff, and the terminal morphism ! : C → 1 is co-ff exactly when C is connected

Alexander Campbell (Dec 26 2020 at 02:10):

sarahzrf said:

ah nice, that paper gives another more direct condition in terms of connectedness of some sort of categories of factorizations

That condition is another way of stating part 2 of the proposition on "Initiality for ends" in the MO answer to which I linked, where a slick proof is also suggested.