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Alexander Gietelink Oldenziel (May 04 2023 at 15:54):It is my understanding that *-autonomy and tracial structure are in some vague way 'orthogonal'/ independent'. If you ask for both the par and the tensor collapse to a single monoidal product and you get a compact closed category.
Which leads me to the question: is there a notion of structured category in which both *-autonomous category and trace categories sit snuggly?
feel free to interpret sit snuggly fairly broadly.
I'd love a notion of structured category in which we can somehow (partly) reconstruct both structures without it collapsing to a compactg closed category.
Alexander Gietelink Oldenziel (May 04 2023 at 18:10):for instance - there is a notion of feedback category which slightly weakens the conditions on traced categories. see e.g. http://www.numdam.org/item/ITA_2002__36_2_181_0/
do we still know that a star-autonomous feedback category collapses to a compact closed category?
Matteo Capucci (he/him) (May 04 2023 at 19:08):That's a nice question
Matteo Capucci (he/him) (May 04 2023 at 19:10):Do I understand correctly that you are talking about *-autonomy and traceability on two different monoidal products?
JS PL (he/him) (May 04 2023 at 19:51):Alexander Gietelink Oldenziel said:
Which leads me to the question: is there a notion of structured category in which both *-autonomous category and trace categories sit snuggly?
A star-autononmous category which is traced (on the same tensor product) is compact closed:
http://www.tac.mta.ca/tac/volumes/28/7/28-07.pdf
JS PL (he/him) (May 04 2023 at 19:52):You might be interested in traces on linearly distributive categories:
https://www.math.mcgill.ca/rags/linear/trace.pdf
Alexander Gietelink Oldenziel (May 12 2023 at 10:49):Matteo Capucci (he/him) said:
Do I understand correctly that you are talking about *-autonomy and traceability on two different monoidal products?
*-autonomous categories have two monoidal products. Trace categories have one.
I'm not sure exactly what I'm after. It seems the two concepts are closely related (e.g. they both have close connections with linear logic, for *-autonomous categories this is directly. For trace categories it is through the geometric of interactions) but at the same time they appear orthogonal.
Matteo Capucci (he/him) (May 15 2023 at 08:35):Yeah I was being silly. Which one do you trace though?
Alexander Gietelink Oldenziel (May 15 2023 at 09:15):I don't know. I need to understand the landscape better. I am probably fundamentally confused somewhere