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Stream: learning: questions

Topic: Adjunctions Internal to Prof / Dualizable Profunctors


view this post on Zulip Joshua Meyers (Oct 03 2023 at 18:18):

Hi,

I'm wondering what sort of profunctors M:Cop×DSetM:C^{\text{op}}\times D \to \text{Set} and N:Dop×CSetN:D^{\text{op}}\times C \to \text{Set} form an adjunction internal to Prof\text{Prof}. By this I mean that there are natural transformations η:HomCMN\eta:\text{Hom}_C\Rightarrow M\otimes N and ϵ:NMHomD\epsilon: N\otimes M \Rightarrow \text{Hom}_D satisfying the triangle identities.

nLab has a enigmatic remark about "semifunctors" in this situation but I'm not sure how a semifunctor would specify a profunctor.

In my own thinking, so far I have concluded that for all cCc\in C, M(c,)M(c,-) and N(,c)N(-,c) must be quotients of representables, but that's all I have so far.

view this post on Zulip Joshua Meyers (Oct 03 2023 at 18:26):

Here is the argument for the last claim:

For f:ccf:c\to c', denote ηc,c(f):=(η1(f),η2(f))\eta_{c,c'}(f):=\overline{(\eta_1(f),\eta_2(f))}.

Consider one of the triangle identities: (MHomCMηMMNMMϵMHomDM)=idM(M\cong \text{Hom}_C\otimes M\xRightarrow{\eta\otimes M} M \otimes N\otimes M \xRightarrow{M\otimes \epsilon} M\otimes\text{Hom}_D\cong M) = id_M.

Take any m:M(c,d)m:M(c,d) and run it through this identity:

m(idc,m)(η1(idc),η2(idc),m)(η1(idc),ϵ(η2(idc),m))M(idc,ϵ(η2(idc),m))(η1(idc))m\mapsto \overline{(\text{id}_c,m)} \mapsto \overline{(\eta_1(\text{id}_c), \eta_2(\text{id}_c), m)} \mapsto \overline{(\eta_1(\text{id}_c), \epsilon(\overline{\eta_2(\text{id}_c),m}))}\mapsto M(\text{id}_c, \epsilon(\overline{\eta_2(\text{id}_c),m}))(\eta_1(\text{id}_c))

By the identity, we must have m=M(idc,ϵ(η2(idc),m))(η1(idc))m = M(\text{id}_c, \epsilon(\overline{\eta_2(\text{id}_c),m}))(\eta_1(\text{id}_c)). So then every m:M(c,d)m:M(c,d) is the result of a DD-action of the single element η1(idc)\eta_1(\text{id}_c), which only depends on cc. Thus there is a surjective transformation D(d,)M(c,)D(d,-)\Rightarrow M(c,-), where dd is the codomain of η1(idc)\eta_1(\text{id}_c), sending h:ddh:d\to d' to M(idc,h)(η1(idc))M(\text{id}_c, h)(\eta_1(\text{id}_c)).

A similar argument can be made for N(,c)N(-, c).

view this post on Zulip Mike Shulman (Oct 03 2023 at 19:11):

Talking about semifunctors is just a cute trick for talking about functors to the Cauchy completion. If F:CDF:C\to D is a semifunctor, then the image of each identity F(idc)F(\mathrm{id}_c) is not necessarily an identity, but it is idempotent, and thus FF induces an actual functor into the Cauchy completion of DD. And oppositely.

view this post on Zulip Joshua Meyers (Oct 03 2023 at 19:24):

So the Cauchy completion DDD\to\overline{D} sends all idempotents to identity? I thought it just makes all idempotents split.

But how does any of this give a profunctor?

view this post on Zulip Mike Shulman (Oct 03 2023 at 20:15):

I didn't mean that the composite CFDDC \xrightarrow{F} D \to \overline{D} is a functor, only that from FF you can define a functor CDC\to \overline{D}, by sending an object cCc\in C to the splitting of the idempotent F(idc)F(\mathrm{id}_c).

view this post on Zulip Mike Shulman (Oct 03 2023 at 20:17):

Once you have such a functor, you get a profunctor C×DopSetC\times \overline{D}^{\mathrm{op}}\to\mathrm{Set} in the usual way, and then restrict to C×DopC\times D^{\mathrm{op}}.

view this post on Zulip Mike Shulman (Oct 03 2023 at 20:18):

Since Set\mathrm{Set} has split idempotents, functors C×DopSetC\times \overline{D}^{\mathrm{op}}\to\mathrm{Set} are equivalent to functors C×DopSetC\times {D}^{\mathrm{op}}\to\mathrm{Set}, so the restriction doesn't lose any information.