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Stream: theory: category theory

Topic: adjoint string on the naturals

sarahzrf (Apr 15 2020 at 23:36):

i just worked out that i'm pretty sure we have this adjoint string $\dots \dashv \mathrm{S}|_3 \dashv \mathrm{pred}|_3 \dashv \mathrm{S}|_2 \dashv \mathrm{pred}|_2 \dashv \mathrm{S}|_1 \dashv \mathrm{pred} \dashv \mathrm{S}$ of monotone functions $\mathbb{N} \to \mathbb{N}$, where $\operatorname{S}$ is successor

sarahzrf (Apr 15 2020 at 23:37):

i dont have a real discussion to start on it, i just thought it was kinda fun :upside_down:

sarahzrf (Apr 15 2020 at 23:38):

$\mathrm{pred}$ is predecessor (except 0 goes to itself), and each $f|_n$ is like $f$ except that on the first $n$ naturals it is the identity

sarahzrf (Apr 15 2020 at 23:38):

so i guess i'm using that notation kinda backwards even insofar as it's only suggestive, huh

sarahzrf (Apr 15 2020 at 23:38):

oops

sarahzrf (Apr 15 2020 at 23:41):

oh, i bet on $\mathbb{Z}$ you could turn it into one that goes off on the right too, each entry holds a lower set of the integers fixed

sarahzrf (Apr 15 2020 at 23:43):

actually i think this has some relation to cantor-schröder-bernstein, or at least the nice graph-y proof of it

sarahzrf (Apr 15 2020 at 23:44):

a while back i was tinkering with how much of an approximation of CSB i could prove constructively, and i got this sequence of approximations to the classically-defined bijections, and they hinged on a kind of "shift but only on part of the connected component" rather reminiscent of these operations

John Baez (Apr 15 2020 at 23:44):

Nice! Long chains of adjunctions are pleasant and a bit spooky.

sarahzrf (Apr 15 2020 at 23:44):

i know, right?

sarahzrf (Apr 15 2020 at 23:46):

actually i was originally playing with just the pred ⊣ S adjunction b/c Ran_{S^op} : PSh(N) → PSh(N) is important

sarahzrf (Apr 15 2020 at 23:46):

and it gives a nice concrete expression for Lan_{S^op}

sarahzrf (Apr 15 2020 at 23:47):

and then i was like "hmm, we even have an adjoint quadruple now... i wonder what Lan_{pred^op} is like... can i get a nicer expression for that?"

sarahzrf (Apr 15 2020 at 23:51):

i guess we have a whole adjoint string on PSh(N) then, $\dots \dashv \mathrm{S}|_3^{\mathrm{op}*} \dashv \mathrm{pred}|_3^{\mathrm{op}*} \dashv \mathrm{S}|_2^{\mathrm{op}*} \dashv \mathrm{pred}|_2^{\mathrm{op}*} \dashv \mathrm{S}|_1^{\mathrm{op}*} \dashv \mathrm{pred}^{\mathrm{op}*} \dashv \mathrm{S}^{\mathrm{op}*} \dashv \operatorname{Ran}_{\mathrm{S}^{\mathrm{op}}}$

John Baez (Apr 15 2020 at 23:57):

Do adjoint functors f: X -> Y, g: Y -> X give adjoint functors between Psh(X) and Psh(Y) as you suggest? Somehow this feels wrong to me, since the left adjoint of f*: Psh(Y) -> Psh(X) (which always exists, regardless of whether f has an adjoint) is a kind of "pushforward" of presheaves, and the right adjoint is another thing that exists regardless of whether f has an adjoint.

Tim Hosgood (Apr 16 2020 at 00:00):

John Baez said:

Do adjoint functors f: X -> Y, g: Y -> X give adjoint functors between Psh(X) and Psh(Y) as you suggest? Somehow this feels wrong to me, since the left adjoint of f*: Psh(Y) -> Psh(X) (which always exists, regardless of whether f has an adjoint) is a kind of "pushforward" of presheaves, and the right adjoint is another thing that exists regardless of whether f has an adjoint.

is looks like it! https://math.stackexchange.com/a/697656/71510 (specifically the comments on the answer)

sarahzrf (Apr 16 2020 at 00:02):

if f ⊣ g then g* ⊣ f*

sarahzrf (Apr 16 2020 at 00:02):

so g* ≅ Lan_f and f* ≅ Ran_g

sarahzrf (Apr 16 2020 at 00:02):

well, i guess it was the first of those two lines you were questioning

sarahzrf (Apr 16 2020 at 00:03):

but yeah precomposition turns adjunctions into adjunctions (altho it flips which way around they go)

sarahzrf (Apr 16 2020 at 00:03):

for any functor categories, even, not just presheaves

John Baez (Apr 16 2020 at 00:03):

Okay, right now I'm guessing this doesn't even depend on using Psh(X) = [X, Set], the same thing would work if we used [X, C] for any fixed category C.

sarahzrf (Apr 16 2020 at 00:03):

heh

sarahzrf (Apr 16 2020 at 00:04):

good timing

John Baez (Apr 16 2020 at 00:08):

Okay. Here's my reasoning. We've got a 2-functor [-, C]: Cat -> Cat^op, and a 2-functor sends adjunctions to adjunctions, so it sends any adjunction in Cat to one in Cat^op. An adjunction in Cat^op is also an adjunction in Cat, but with left and right adjoints switched!

John Baez (Apr 16 2020 at 00:09):

I'm trying to make "presheaves" seem as irrelevant as they really are here... any 2-functor Cat -> Cat^op would do this.

Morgan Rogers (he/him) (Apr 16 2020 at 09:37):

sarahzrf said:

a while back i was tinkering with how much of an approximation of CSB i could prove constructively, and i got this sequence of approximations to the classically-defined bijections, and they hinged on a kind of "shift but only on part of the connected component" rather reminiscent of these operations

As a concrete example of CSB failing, ie of a family of toposes where it explicitly fails, you could look at my paper on toposes of monoid actions: I don't spell this out in the paper because it's a bit tangential, but as soon as two non-isomorphic monoids are Morita equivalent, the corresponding objects in the presheaf topos they both generate are retracts of one another which are not isomorphic.

Morgan Rogers (he/him) (Apr 16 2020 at 09:38):

I might put this on StackExchange if I get around to checking the decidability condition you ask for.

Matteo Capucci (he/him) (Apr 16 2020 at 09:49):

sarahzrf said:

i just worked out that i'm pretty sure we have this adjoint string $\dots \dashv \mathrm{S}|_3 \dashv \mathrm{pred}|_3 \dashv \mathrm{S}|_2 \dashv \mathrm{pred}|_2 \dashv \mathrm{S}|_1 \dashv \mathrm{pred} \dashv \mathrm{S}$ of monotone functions $\mathbb{N} \to \mathbb{N}$, where $\operatorname{S}$ is successor

Hahaha this is a funny way to justify transport rule for inequalities :)

Matteo Capucci (he/him) (Apr 16 2020 at 09:50):

Does the same thing work for other functions?

Morgan Rogers (he/him) (Apr 16 2020 at 09:56):

sarahzrf said:

i guess we have a whole adjoint string on PSh(N) then, $\dots \dashv \mathrm{S}|_3^{\mathrm{op}*} \dashv \mathrm{pred}|_3^{\mathrm{op}*} \dashv \mathrm{S}|_2^{\mathrm{op}*} \dashv \mathrm{pred}|_2^{\mathrm{op}*} \dashv \mathrm{S}|_1^{\mathrm{op}*} \dashv \mathrm{pred}^{\mathrm{op}*} \dashv \mathrm{S}^{\mathrm{op}*} \dashv \operatorname{Ran}_{\mathrm{S}^{\mathrm{op}}}$

If these are all distinct (which... they must be, given the action on representables, right?), this answers a question I was wondering about regarding how long an adjoint chain constituting a geometric morphism can be; specifically, as long as you like.
It kind of makes you want to consider a limiting adjunction here, say $(\coprod_{i \in \mathbb{N}}\mathrm{pred}|^{\mathrm{op}*}_{i} \dashv \prod_{i \in \mathbb{N}} S|^{\mathrm{op}*}_i)$ (typesetting this made me realise you skipped an index haha). Do these have further adjoints?

sarahzrf (Apr 16 2020 at 15:32):

Matteo Capucci said:

Does the same thing work for other functions?

not sure what "the same thing" would mean...

sarahzrf (Apr 16 2020 at 15:41):

also hmmmmmmm im now realizing that maybe the key underlying this stuff is that N⁺ is naturally equipped both as a reflective subset of N and as equivalent to N, but those are different equipment

sarahzrf (Apr 16 2020 at 15:43):

elaboration:

sarahzrf (Apr 16 2020 at 15:47):

let $\mathbb{N}^+ = \mathbb{N} \setminus \{0\}$; i.e., the positive natural numbers. then we have

1. $P : \mathbb{N} \to \mathbb{N}^+$ where $P(n) = n + 1$. this is an order isomorphism, so we have $P^{-1} \dashv P$ and $P \dashv P^{-1}$. on its domain, $P^{-1}$ coincides with "pred".

2. $\iota : \mathbb{N}^+ \hookrightarrow \mathbb{N}$ has a left adjoint $r \dashv \iota$ defined by $r(0) = 1$ and $r(n) = n$ when $n > 0$.

sarahzrf (Apr 16 2020 at 15:50):

then $\mathrm{pred} = P^{-1} \circ r \dashv \iota \circ P = \mathrm{S}$

sarahzrf (Apr 16 2020 at 16:01):

hmm, so i think $r \dashv \iota$ extends to the left into an adjoint string too, and that gives rise to the one earlier once you plug alternating Ps and P⁻¹s on it... but i think it goes deeper

sarahzrf (Apr 16 2020 at 16:01):

i think that adjoint string comes from the fact that the 2 structures i mentioned keep nesting inside N⁺

sarahzrf (Apr 16 2020 at 16:01):

:thinking: