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Stream: theory: category theory

Topic: adjoint string on the naturals


view this post on Zulip sarahzrf (Apr 15 2020 at 23:36):

i just worked out that i'm pretty sure we have this adjoint string S3pred3S2pred2S1predS\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 NN\mathbb{N} \to \mathbb{N}, where S\operatorname{S} is successor

view this post on Zulip 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:

view this post on Zulip sarahzrf (Apr 15 2020 at 23:38):

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

view this post on Zulip 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

view this post on Zulip sarahzrf (Apr 15 2020 at 23:38):

oops

view this post on Zulip sarahzrf (Apr 15 2020 at 23:41):

oh, i bet on Z\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

view this post on Zulip 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

view this post on Zulip 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

view this post on Zulip John Baez (Apr 15 2020 at 23:44):

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

view this post on Zulip sarahzrf (Apr 15 2020 at 23:44):

i know, right?

view this post on Zulip 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

view this post on Zulip sarahzrf (Apr 15 2020 at 23:46):

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

view this post on Zulip 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?"

view this post on Zulip sarahzrf (Apr 15 2020 at 23:51):

i guess we have a whole adjoint string on PSh(N) then, S3oppred3opS2oppred2opS1oppredopSopRanSop\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}}}

view this post on Zulip 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.

view this post on Zulip 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)

view this post on Zulip sarahzrf (Apr 16 2020 at 00:02):

if f ⊣ g then g* ⊣ f*

view this post on Zulip sarahzrf (Apr 16 2020 at 00:02):

so g* ≅ Lan_f and f* ≅ Ran_g

view this post on Zulip sarahzrf (Apr 16 2020 at 00:02):

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

view this post on Zulip sarahzrf (Apr 16 2020 at 00:03):

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

view this post on Zulip sarahzrf (Apr 16 2020 at 00:03):

for any functor categories, even, not just presheaves

view this post on Zulip 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.

view this post on Zulip sarahzrf (Apr 16 2020 at 00:03):

heh

view this post on Zulip sarahzrf (Apr 16 2020 at 00:04):

good timing

view this post on Zulip 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!

view this post on Zulip 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.

view this post on Zulip 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.

view this post on Zulip 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.

view this post on Zulip 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 S3pred3S2pred2S1predS\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 NN\mathbb{N} \to \mathbb{N}, where S\operatorname{S} is successor

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

view this post on Zulip Matteo Capucci (he/him) (Apr 16 2020 at 09:50):

Does the same thing work for other functions?

view this post on Zulip Morgan Rogers (he/him) (Apr 16 2020 at 09:56):

sarahzrf said:

i guess we have a whole adjoint string on PSh(N) then, S3oppred3opS2oppred2opS1oppredopSopRanSop\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 (iNprediopiNSiop)(\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?

view this post on Zulip 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...

view this post on Zulip 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

view this post on Zulip sarahzrf (Apr 16 2020 at 15:43):

elaboration:

view this post on Zulip sarahzrf (Apr 16 2020 at 15:47):

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

  1. P:NN+P : \mathbb{N} \to \mathbb{N}^+ where P(n)=n+1P(n) = n + 1. this is an order isomorphism, so we have P1PP^{-1} \dashv P and PP1P \dashv P^{-1}. on its domain, P1P^{-1} coincides with "pred".

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

view this post on Zulip sarahzrf (Apr 16 2020 at 15:50):

then pred=P1rιP=S\mathrm{pred} = P^{-1} \circ r \dashv \iota \circ P = \mathrm{S}

view this post on Zulip sarahzrf (Apr 16 2020 at 16:01):

hmm, so i think rιr \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

view this post on Zulip 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⁺

view this post on Zulip sarahzrf (Apr 16 2020 at 16:01):

:thinking: