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Stream: learning: questions

Topic: What is this Chu-like construction?

Dylan Braithwaite (Mar 15 2024 at 19:07):

I have situation where I have a bunch of binary relations, for example $\sim_A, \sim_B, \sim_C$, where $\sim_A$ is a relation between sets $A^+$ and $A^-$, and so on. I also have functions $f_1 : A^+ \to B^+ + C^+$, $f_2 : B^- \to C^+ + A^-$ and $f_3 : C^- \to A^- + B^+$ which satisfy a kind of "partial adjointness" condition:

Abusing notation slightly I'll write $f_1(a) \sim_B b$ to mean "there exists some $b'$ such that $f_1(a) = \mathsf{inj}_L(b')$ and $b' \sim b$" and similarly for the other function/relation pairs.
Then I want the functions to satisfy

$$f_1(a) \sim_B b \iff a \sim_A f_2(b)\\ f_1(a) \sim_C c \iff a \sim_A f_3(c)\\ f_2(b) \sim_C c \iff f_3(c) \sim_B b.$$

Viewing the relations as functions, i.e. $(\sim_A) : A^+ \times A^- \to 2$ I think this is the same as asking that the composites are equal: $(\sim_B + \sim_C) \circ f_1 = (\sim_C + \sim_A) \circ f_2 = (\sim_A + \sim_B) \circ f_3$ as functions $A^+ \times B^- \times C^- \to 2 + 2$.

Viewing this data as describing some notion of map $(A^+, A^-, \sim_A) \to (B^+, B^-, \sim_B), (C^+, C^-, \sim_C)$, in this form there's an obvious generalisation to morphisms between arbitrary-sized lists of relations. In fact, at first glance I thought this would be a morphism in the polycategory $\mathsf{Chu}(\mathsf{Set}^\text{op}, 2)$ with + as the monoidal product, but after at little thought I realised this isn't the case: certain parts of the definition are dualised, and others aren't. Indeed the unary morphisms which would consistant with this definition are just those of (the underlying category) of $\mathsf{Chu}(\mathsf{Set}, 2)$.

So it seems like where this is diverging is in the definition of the multimorphisms, and if $\mathsf{Set}$ had co-exponentials I think I could dualise the definition of the standard monoidal product on the Chu construction to get morphisms of the form I want. But since the case I care about won't be representable, I'm hoping instead I could instead dualise some characterisation of the polycategorical version of $\mathsf{Chu}$ to justify the above definition. The nlab gives one characterisation of the ordinary Chu construction as being adjoint to the forgetful functor from *-polycategories to subunary polycategories, but I can't even just naively try dualising something there because the data of a subunary polycategory has a single product/non-representable connective, but my definition necessarily uses 2: $\times$ in the definition of the objects as well as $+$ in the definition of the morphisms.

So my question is whether I've missed a trick to realise this as a special case of the ordinary Chu construction, or whether anyone recognises this as an instance of some other known construction? I realise there's a few more things I could try like checking for adjunctions between *-polycategories and distributive categories or monoidal multicategories, but I thought I'd check here first, because it feels to me like there ought to be a nicer way to think about it than what I have currently.

Mike Shulman (Mar 15 2024 at 19:46):

Dylan Braithwaite said:

I think this is the same as asking that the composites are equal: $(\sim_B + \sim_C) \circ f_1 = (\sim_C + \sim_A) \circ f_2 = (\sim_A + \sim_B) \circ f_3$ as functions $A^+ \times B^- \times C^- \to 2 + 2$.

I'm suspicious of this. Have you checked this carefully, paying attention to which copy of $2$ each of the components of each of the composites maps into? I feel like there's going to be a mismatch when you get around the cycle.

Mike Shulman (Mar 15 2024 at 19:53):

My first thought is that it might be an instance of what I called the 2-Chu-Dialectica construction, but on second thought I don't see it. Even in that very general setup, the objects appearing in the "secondary components" are obtained by combining a relation and a morphism of the same index. So in your case it would be something defined from $f_1$ and $\sim_A$, and something defined from $f_2$ and $\sim_B$, and something defined from $f_3$ and $\sim_C$. Which is not what you have. So this is probably not helpful, sorry.

Dylan Braithwaite (Mar 16 2024 at 12:45):

Mike Shulman said:

I'm suspicious of this. Have you checked this carefully, paying attention to which copy of $2$ each of the components of each of the composites maps into? I feel like there's going to be a mismatch when you get around the cycle.

Thats a good point, thanks. I'll have another think about that

Dylan Braithwaite (Mar 16 2024 at 12:54):

Mike Shulman said:

My first thought is that it might be an instance of what I called the 2-Chu-Dialectica construction
[...]

Yeah I thought the same initially. I started by thinking it seems dual to a multivariable adjunction, and after looking at your paper realised I probably don't need the 2-dimensional structure, and that I can maybe just view it as an instance of the 1-categorical Chu construction