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Stream: learning: questions

Topic: Extensions along units & counits

sarahzrf (Sep 04 2020 at 03:45):

hey!

I'm wondering about something at a couple of levels of generality:

1. Given categories $C$ and $D$, an object mapping $\operatorname{Ob}(C) \to \operatorname{Ob}(D)$ corresponds by adjunction to a functor $f : \operatorname{Disc}(\operatorname{Ob}(C)) \to D$, and then we can ask about Kan extending this along the adjunction counit $\varepsilon_C : \operatorname{Disc}(\operatorname{Ob}(C)) \to C$ and seeing what kind of functor we get $C \to D$.
2. More generally, for any adjunction between 2-categories, we can ask a similar question—although I haven't thought of interesting examples of where it might go...

Does anyone know if either of these have been written about? Are there sources I might wanna look at?

sarahzrf (Sep 04 2020 at 03:50):

a motivating example for me is where $C$ is the poset $\mathbb N$ and $D$ is the poset of extended reals—then an object mapping $f$ is an arbitrary sequence of extended reals, and the right Kan extension along the counit works out to be the monotone sequence $\mathbb N \to \overline R$ given by $n \mapsto \inf_{m \ge n} f(m)$

sarahzrf (Sep 04 2020 at 03:51):

so taking the sup over all of that gives the lim inf of $f$, and the sup is of course the colimit over the diagram given by this map from $\mathbb N$

sarahzrf (Sep 04 2020 at 03:53):

...it also occurred to me that you can compute the sup by taking the left Kan extension along the unit $\eta_{\mathbb N} : \mathbb N \to \operatorname{coDisc}(\operatorname{Ob}(\mathbb N))$, so that we can compute lim infs by $\operatorname{Lan}_{\eta_{\mathbb N}} \circ \operatorname{Ran}_{\varepsilon_{\mathbb N}}$—but im not totally convinced that using a codiscrete category here in place of the terminal category is anything other than cute

sarahzrf (Sep 04 2020 at 03:54):

i just wonder whether interesting stuff might happen for other adjoint cylinders or something... maybe lawvere has already written about this :thinking:

Morgan Rogers (he/him) (Sep 04 2020 at 09:02):

This is a nice variation on the idea of pushing out or pulling back along units and counits of adjunctions to construct special algebras and coalgebras. I would be curious to know whether this ever produces anything interesting when $D$ is some Grothendieck topos and $C$ is a small, manageable category, so that we can extend along Yoneda afterwards for fun. Maybe I'll check that out some time.

sarahzrf (Sep 04 2020 at 18:52):

the idea of pushing out or pulling back along units and counits of adjunctions to construct special algebras and coalgebras

can you elaborate on this?

sarahzrf (Sep 04 2020 at 18:59):

is the idea something like uh...

some kind of category of algebras is maybe a[n] [op]fibration over your starting category, and so if you have an adjunction involving that starting category, then you'll have a unit or counit in it that induces operations on algebras?

sarahzrf (Sep 04 2020 at 23:21):

hmmmmm, i guess we have some sort of analogy to what i was initially suggesting if we do like...

sarahzrf (Sep 04 2020 at 23:23):

say, take a monoid $M$, consider the counit $\varepsilon_M : FUM \to M$, and consider adjoints to restriction of scalars along that—so, left or right Kan extensions $[FUM^{\mathrm{op}}, \mathrm{Set}] \to [M^{\mathrm{op}}, \mathrm{Set}]$

sarahzrf (Sep 04 2020 at 23:25):

and an action of $FUM$ is basically an "action" of $M$ subject to no laws, so this is sort of like taking a random "action" of $M$ and trying to turn it into an actual action

sarahzrf (Sep 04 2020 at 23:25):

which does feel similar in flavor to taking a random mapping of objects and trying to turn it into an actual functor

sarahzrf (Sep 04 2020 at 23:25):

i know you study monoid actions—is this the kind of example you had in mind?

Dan Doel (Sep 04 2020 at 23:36):

Is this the (co)free functor on an object mapping, like people sometimes talk about in Haskell? Except they actually call it Yoneda and Coyoneda because they resemble the types involved in the (co)Yoneda lemma?

sarahzrf (Sep 04 2020 at 23:37):

i mean, let me know if you can think of a better interpretation of that notion

Dan Doel (Sep 04 2020 at 23:37):

The left extension is the free functor, and the right extension is the cofree functor, and (co)algebras are actual functors.

sarahzrf (Sep 04 2020 at 23:38):

im not sure what the haskell thing is, i never actually got that deep into the category theoretic haskell stuff ironically

sarahzrf (Sep 04 2020 at 23:38):

okay that's a half truth

sarahzrf (Sep 04 2020 at 23:39):

what do you mean here by "(co)algebras"?

Dan Doel (Sep 04 2020 at 23:41):

$Lan_ε(f) ⇒ f$ is an algebra structure on $f$. $f ⇒ Ran_ε(f)$ is a coalgebra structure.

sarahzrf (Sep 04 2020 at 23:41):

also crap right i do think i remember seeing types like uhhh

data Free f a where
  Free :: (a -> r) -> f a -> Free f r

sarahzrf (Sep 04 2020 at 23:41):

oh, i see

Dan Doel (Sep 04 2020 at 23:41):

If you expand them into the (co)limit formulas, they both look like ways of writing the type of the mapping operation on arrows (if it's what I'm thinking of).

sarahzrf (Sep 04 2020 at 23:42):

wait, hold on, there should be a restriction along ε somewhere in there—i assume you're making it implicit, but...

sarahzrf (Sep 04 2020 at 23:42):

er, right, only one way around it would make sense

sarahzrf (Sep 04 2020 at 23:43):

ooh, are you suggesting that $\operatorname{Lan}_\varepsilon \dashv \varepsilon^*$ is monadic?

Dan Doel (Sep 04 2020 at 23:44):

Maybe.

sarahzrf (Sep 04 2020 at 23:45):

i think it is for C a preorder, say

Dan Doel (Sep 04 2020 at 23:46):

There's a related construction on the n-lab.

sarahzrf (Sep 04 2020 at 23:47):

do tell

Dan Doel (Sep 04 2020 at 23:47):

About presheaves as coalgebras.

Dan Doel (Sep 04 2020 at 23:48):

Someone was talking about it here a while back and I tried answering some questions. I'll see if I can find it.

Dan Doel (Sep 04 2020 at 23:53):

I think the construction starts in section 3 here

Dan Doel (Sep 04 2020 at 23:58):

I think it's roughly the same, but presented as a construction on $D$ rather than $\mathsf{Disc}(C) → D$ or something.

Dan Doel (Sep 05 2020 at 00:01):

That would be for $Ran_ε(f)$ of course. Then instead of speaking about that as a functor, it's summing over all the points because $D$ is sufficiently cocomplete for that.

Dan Doel (Sep 05 2020 at 00:15):

Oh, also, when $C$ is an appropriate monoid $M$ you get the famous monad $M × -$ and comonad $[M,-]$.

Morgan Rogers (he/him) (Sep 05 2020 at 09:11):

sarahzrf said:

the idea of pushing out or pulling back along units and counits of adjunctions to construct special algebras and coalgebras

can you elaborate on this?

Okay this must have been a really strange crossing of conceptual wires on my part, because I cannot find what I thought I was talking about anywhere, nor can I hash out anything sensible I might have been thinking of on paper. I eventually remembered where I'd seen pulling back along units: to define universal closure operators on subobject lattices, but that is of limited relevance to algebras for general monads. Sorry for my confusion.

Morgan Rogers (he/him) (Sep 05 2020 at 09:18):

sarahzrf said:

say, take a monoid $M$, consider the counit $\varepsilon_M : FUM \to M$, and consider adjoints to restriction of scalars along that—so, left or right Kan extensions $[FUM^{\mathrm{op}}, \mathrm{Set}] \to [M^{\mathrm{op}}, \mathrm{Set}]$

This is a really pleasant example, yes! In fact, one of the problems that @Jens Hemelaer and I have been thinking about on and off is how to characterise toposes of actions of free monoids. The representing monoids are unique up to Morita equivalence, just because free monoids are nice enough, but it would be nice to have a generalisable topos-theoretic description, since being able to exploit the fact that a topos admits a canonical geometric morphism from a topos of a particular type (in this case, whatever the toposes of actions of free monoids turn out to be) has historically been rather useful.

sarahzrf (Sep 05 2020 at 16:12):

universal closure operators?

Morgan Rogers (he/him) (Sep 05 2020 at 16:40):

They're the third way of defining subtoposes :innocent:
A universal closure operator consists of a closure operator on the lattice of subobjects of each object of a topos which is stable under pullback (and so is determined by such an operator on the subobjects of the subobject classifier). cf slide 35 of these notes.

sarahzrf (Sep 05 2020 at 23:21):

oh, is this the same as a lawvere-tierney topology?

sarahzrf (Sep 05 2020 at 23:21):

oh lol says so next page

sarahzrf (Sep 05 2020 at 23:22):

wait, so what other two ways did you have in mind :thinking:

Morgan Rogers (he/him) (Sep 06 2020 at 08:41):

sarahzrf said:

wait, so what other two ways did you have in mind :thinking:

Grothendieck topology, Lawvere-Tierney topology and universal closure operator. The gap between the latter two is admittedly not very large, but they still fundamentally consist of different data.

sarahzrf (Sep 07 2020 at 21:51):

eh, i guess d:

sarahzrf (Sep 07 2020 at 21:52):

i mean, the gap is basically "the yoneda embedding"

Morgan Rogers (he/him) (Sep 08 2020 at 10:28):

Well, I can define a Lawvere-Tierney topology in any category with a subobject classifier, and a universal closure operator on any category with pullbacks, more or less, and those are different classes of category. They coincide in toposes is all.

sarahzrf (Sep 08 2020 at 17:01):

i mean, the different classes are

• well-powered lex categories (ok i guess u dont need a terminal object maybe)
• well-powered lex categories where Sub is representable

i think the gap is still yoneda :p