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Adittya Chaudhuri (Jun 25 2024 at 12:31):The composition of Open Petri nets (as structured cospans (horizontal 1-cells)) in the double category of open Petri nets is defined via pushout construction (Theorem 13 of https://arxiv.org/pdf/1808.05415). Are we making a "choice of pushout" in every such composition? Otherwise, how the composition is well defined here? This point was discussed for constructing tensor products via choices of coproducts (Proof of the Lemma 14 of https://arxiv.org/pdf/1808.05415), but I could not find any such argument for the case of composition. Am I missing anything here?
Evan Patterson (Jun 25 2024 at 16:49):Yes, in the order to obtain the structure of a double category, you have to make a choice of pushouts.
Adittya Chaudhuri (Jun 25 2024 at 18:45):Evan Patterson said:
Yes, in the order to obtain the structure of a double category, you have to make a choice of pushouts.
Thanks!!
John Baez (Jun 25 2024 at 19:54):Jade and I should have mentioned this choice of pushout issue since we mentioned the coproduct issue. Often people don't bother mentioning such issues, but if you're going to bring it up you might as well be consistent about it.
I was more explicit about these issues in my paper Structured cospans - see Lemma 2.1 and the remark directly after that - than in this paper on Petri nets where Jade and I were just applying the theory of structured cospans.
The remark shows that there is something left to check, which I'm sure is true but have never bothered to check. So, it would be a good thing for you to try!
Adittya Chaudhuri (Jun 25 2024 at 21:28):John Baez said:
Jade and I should have mentioned this choice of pushout issue since we mentioned the coproduct issue. Often people don't bother mentioning such issues, but if you're going to bring it up you might as well be consistent about it.
I was more explicit about these issues in my paper Structured cospans - see Lemma 2.1 and the remark directly after that - than in this paper on Petri nets where Jade and I were just applying the theory of structured cospans.
The remark shows that there is something left to check, which I'm sure is true but have never bothered to check. So, it would be a good thing for you to try!
Thank you!! I got your point!! I just checked the remark. Yes, I will try !!