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Stream: theory: applied category theory

Topic: On the category of commutative monoid labelled-graphs

Adittya Chaudhuri (Jan 22 2025 at 16:16):

I am interested in the 2nd approach mentioned in https://johncarlosbaez.wordpress.com/2024/11/14/polarities-part-5/, where a graph (a directed multigraph) is labelled by elements of a fixed commutative monoid $M$ (coming from a rig $R$). They form a category $L$-Gph, whose morphisms are compatible with the underlying commutative monoid structure. Now, I think we can upgrade the notion to a $R$-labelled causal loop diagram (as the notion is defined in https://johncarlosbaez.wordpress.com/2024/11/04/polarities-part-2/) whose morphisms should be compatible with both the underlying monoid structure(under product) and commutative monoid structure (underlying addition). In the first approach of https://johncarlosbaez.wordpress.com/2024/11/14/polarities-part-5/, (when the labels are elements of a set ) the category is same as the slice category ${\rm{Gph}}/G_L$, and hence is cocomplete. But I am not able to see how to represent the category $L$-Gph or the suitable category of $R$-labelled causal loop diagrams as a "slice category of some cocomplete category" . However, to use the essential results of ACT (like the theory of structured cospans) for a composable theory of such structures, we may need to ensure the finite cocompleteness of such categories. So, is there a way to show the category $L$-Gph- or the suitable category of $R$-valued causal loop diagrams is finitely cocomplete? Maybe I am missing something.

Thanks in advance.

John Baez (Jan 22 2025 at 16:33):

Wow, I didn't think anyone actually read that blog article. Thanks for thinking about it!

I don't think $L$-Gph is cocomplete. By the way, the $L$ here is the commutative monoid of labels, so it's the same as what you're calling $M$, the underlying additive monoid of the rig $R$.

In a category with binary coproducts every object $G$ has a 'fold map' or [[codiagonal]]

$G + G \to G$

I don't see how we'd get this if we're working in the category of labeled graphs where when two edges $e_1, e_2$ get mapped to a single edge $e$, we have to add the labels of $e_1$ and $e_2$ to get the label of $e$.

So, I don't think structured cospans will work well for dealing with open $L$-graphs. I was planning to use decorated cospans.

John Baez (Jan 22 2025 at 16:35):

Are you planning to do something with these ideas?

Adittya Chaudhuri (Jan 22 2025 at 17:02):

Thank you so much!! I really loved the idea of "turning quantitative information into qualitative" Yes, by $L$, I meant to say $M$, and wanted to keep the notation the same as in the blog article. I see!! I have not yet thought about in the direction of decorated cospans. I will think about it. Yes, I was thinking of using some of these ideas for developing suitable composable frameworks for biochemical signalling pathways .

John Baez (Jan 22 2025 at 18:14):

Nice! This issue of needing decorated rather than structured cospans already showed up in the case of open dynamical systems, where the morphisms again involve addition. That was discussed on page 30 of Structured versus decorated cospans. We may have talked about that already....

Adittya Chaudhuri (Jan 22 2025 at 23:44):

Thank you so much!! I see the point. Yes, we discussed this issue in the context of open Petri nets with rates #learning: questions > Necessity of decorated cospans in open Petri nets with rate @ 💬