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Stream: learning: questions

Topic: adjoint to decoration

Matteo Capucci (he/him) (Jun 07 2023 at 09:57):

I have a vague memory of somewhere in the literature on structured cospans considering what happens when the decoration functor $L:\bf A \to X$ has a right adjoint $R$, in which case a structured cospan $La \to x \leftarrow Lb$ in $\bf X$ corresponds to a cospan $a \to Rx \leftarrow b$ in $\bf A$. Am I dreaming up this memory? Anyone has something interesting to point out about this situation?

Morgan Rogers (he/him) (Jun 07 2023 at 11:29):

A relationship between the pullbacks of those cospans might tell you something interesting about the functors involved (I don't have a specific idea for this set-up, but I'm thinking along the lines of the Frobenius condition which makes a left adjoint preserve exponentials)

John Baez (Jun 07 2023 at 19:23):

Matteo Capucci (he/him) said:

I have a vague memory of somewhere in the literature on structured cospans considering what happens when the decoration functor $L:\bf A \to X$ has a right adjoint $R$, in which case a structured cospan $La \to x \leftarrow Lb$ in $\bf X$ corresponds to a cospan $a \to Rx \leftarrow b$ in $\bf A$. Am I dreaming up this memory?

You're not dreaming it up; in my paper with @Kenny on Structured cospans we write:

When $L$ has a right adjoint $R : \mathsf{X} \to \mathsf{A}$ we can also think of this [structured cospan] as a cospan in $\mathsf{A}$, where the apex is equipped with extra structure, namely an object $x \in \mathsf{X}$ that it comes from. However, treating structured cospans as living in $\mathsf{X}$ is technically more convenient, since then we only need $\mathsf{X}$ to have pushouts to compose them.

John Baez (Jun 07 2023 at 19:34):

Anyone has something interesting to point out about this situation?

In practical this situation is the typical one for structured cospans, so @Kenny and @Christina Vasilakopoulou and I proved that a large class of decorated cospan double categories are equivalent to structured cospan double categories of this sort:

Suppose $\mathsf{A}$ has finite colimits and $F : (\mathsf{A} , +) \to (\mathbf{Cat}, \times)$ is a symmetric lax monoidal pseudofunctor. Then each category $F(a)$ for $a \in \mathsf{A}$ becomes symmetric monoidal, and $F$ becomes a pseudofunctor $F : \mathsf{A} \to \mathbf{SymmMonCat}$. Using the Grothendieck construction, $F$ also gives an opfibration $U : \mathsf{X} \to \mathsf{A}$ where $\mathsf{X} = \int F$. Let $\mathbf{Rex}$ be the 2-category of categories with finite colimits, functors preserving finite colimits, and natural transformations. We show that if $F : \mathsf{A} \to \mathbf{SymmMonCat}$ factors through $\mathbf{Rex}$ as a pseudofunctor, the opfibration $U : \mathsf{X} \to \mathsf{A}$ is also a right adjoint. From the accompanying left adjoint $L : \mathsf{A} \to \mathsf{X}$, we construct a symmetric monoidal double category ${}_L \mathbb{C}\mathbf{sp}(\mathsf{X})$ of structured cospans. In Theorem 4.1 we prove that this structured cospan double category ${}_L \mathbb{C}\mathbf{sp}(\mathsf{X})$ is isomorphic to the decorated cospan double category $F \mathbb{C}\mathbf{sp}$. In fact, they are isomorphic as symmetric monoidal double categories.

John Baez (Jun 07 2023 at 19:36):

But you perhaps are interested in results where we start with a structured cospan double category where $L$ is a left adjoint.

John Baez (Jun 07 2023 at 19:39):

Under some conditions this should be equivalent to a decorated cospan double category. In other words, the theorem above should have some sort of converse! @Daniel Cicala and @Christina Vasilakopoulou were writing a paper that would prove this converse, but they haven't finished it and yet and I'm betting they never will. So, right now you can read the conclusions to Structured versus decorated cospans for some hints on how this should go.

John Baez (Jun 07 2023 at 19:40):

More generally, I'd say that in this general sort of situation you should try to use opfibrations and the monoidal Grothendieck construction - our paper illustrates how they become important.

Matteo Capucci (he/him) (Jun 08 2023 at 07:51):

Thanks John! In fact I'm interested in a situation in which L is a fibration and R is a right adjoint right inverse thereof. This looks curiously dual to the setup you describe...

Morgan Rogers (he/him) (Jun 08 2023 at 09:03):

Aha, that's just the sort of situation I was talking about. $R$ should be locally cartesian closed in that situation, if memory serves.

Morgan Rogers (he/him) (Jun 08 2023 at 09:04):

I don't have enough context to know if that's a useful observation though heh

Matteo Capucci (he/him) (Jun 08 2023 at 11:23):

What does being locally cartesian closed mean for a functor?

Morgan Rogers (he/him) (Jun 08 2023 at 11:25):

Preserving that structure from the domain category as far as it exists.

John Baez (Jun 08 2023 at 18:39):

Matteo Capucci (he/him) said:

Thanks John! In fact I'm interested in a situation in which L is a fibration and R is a right adjoint right inverse thereof. This looks curiously dual to the setup you describe...

Wow! What's an example of that, apart from taking what I'd consider a "reasonable" example and taking the opposite of both categories involved?

John Baez (Jun 08 2023 at 18:48):

(By "reasonable" examples I mean those giving most of the examples of structured cospan categories that I've ever written about: these arise when you've got a forgetful functor that's also an opfibration.)

Matteo Capucci (he/him) (Jun 21 2023 at 16:23):

Sorry, I left you with the cliffhanger!

I'm considering fibrations of predicates/subobjects. Then if we call $L: \bf A \to X$ such a fibration, having a right adjoint section $R: \bf X \to A$ means having fiberwise terminal objects. If your notion of predicate is sensible then such an adjoint always exists: it picks out the 'always true' predicate on any given object, $\top$.

Now an L-decorated cospan in this setting would be a cospan $L(a,\kappa) \to x \leftarrow L(b,\upsilon)$ in $\bf X$, which adjointly corresponds to a cospan $(a,\kappa) \to (x, \top) \leftarrow (b,\upsilon)$.

The reason I've found myself looking at these guys is that me (together with @Nathaniel Virgo, @Manuel Baltieri and @Martin Biehl, which I hope don't mind me sharing these ideas here) are interested in bisimulation of systems with goals. When 'systems' means 'coalgebras', a bisimulation is just a cospan of coalgebras $^1$. But over coalgebras we can consider predicates of various kinds, in which case we can look at what happens when you have predicates on the feet (or do cospans have hands instead?) of the bisimulation. In particular we are looking at bisimulations between a 'plant' $a$ and a 'controller' $b$. The plant will have a predicate $\kappa$ associated to it, called its regulation goal, while the controller will have a predicate $\upsilon$ called the viability constraint.
Then an $L$-decorated cospan between $(a, \kappa)$ and $(b, \upsilon)$ is precisely a bisimulation of systems with goals. It seems these capture 'good regulation' conditions in the sense of classical control theory (we've been looking at Wonham and Hepburn's papers specifically).

[1] Bisimulations are traditionally represented as spans of coalgebras, but in many cases one can just as well use cospans. Actually, the latter generalize better! Also it is my opinion bisimulation makes much more sense as a corelation than as a relation.