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i just worked out that i'm pretty sure we have this adjoint string of monotone functions , where is successor
i dont have a real discussion to start on it, i just thought it was kinda fun :upside_down:
is predecessor (except 0 goes to itself), and each is like except that on the first naturals it is the identity
so i guess i'm using that notation kinda backwards even insofar as it's only suggestive, huh
oops
oh, i bet on you could turn it into one that goes off on the right too, each entry holds a lower set of the integers fixed
actually i think this has some relation to cantor-schröder-bernstein, or at least the nice graph-y proof of it
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
Nice! Long chains of adjunctions are pleasant and a bit spooky.
i know, right?
actually i was originally playing with just the pred ⊣ S adjunction b/c Ran_{S^op} : PSh(N) → PSh(N) is important
and it gives a nice concrete expression for Lan_{S^op}
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?"
i guess we have a whole adjoint string on PSh(N) then,
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.
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)
if f ⊣ g then g* ⊣ f*
so g* ≅ Lan_f and f* ≅ Ran_g
well, i guess it was the first of those two lines you were questioning
but yeah precomposition turns adjunctions into adjunctions (altho it flips which way around they go)
for any functor categories, even, not just presheaves
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.
heh
good timing
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!
I'm trying to make "presheaves" seem as irrelevant as they really are here... any 2-functor Cat -> Cat^op would do this.
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.
I might put this on StackExchange if I get around to checking the decidability condition you ask for.
sarahzrf said:
i just worked out that i'm pretty sure we have this adjoint string of monotone functions , where is successor
Hahaha this is a funny way to justify transport rule for inequalities :)
Does the same thing work for other functions?
sarahzrf said:
i guess we have a whole adjoint string on PSh(N) then,
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 (typesetting this made me realise you skipped an index haha). Do these have further adjoints?
Matteo Capucci said:
Does the same thing work for other functions?
not sure what "the same thing" would mean...
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
elaboration:
let ; i.e., the positive natural numbers. then we have
where . this is an order isomorphism, so we have and . on its domain, coincides with "pred".
has a left adjoint defined by and when .
then
hmm, so i think 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
i think that adjoint string comes from the fact that the 2 structures i mentioned keep nesting inside N⁺
:thinking: