Friday, June 27, 2014

Shag on a rock

Well, it turns out that I cannot go to the Cambridge meeting, Category Theory 2014, after all. I will be having some treatment next week. I will try to put up on arXiv an account of the lecture I intended to give.

In an earlier post I said that my talk seemed to have no connection with any other that I could see from the titles. However looking a bit more closely perhaps there are one or two related.


The title of Aleks Kissinger's talk, Finite Matrices are Complete for (Dagger) Multigraph Categories, deceived me until I saw his arXiv paper, arXiv:1406.5942, where I see that multigraph categories are what Aurelio Carboni and I introduced at the Louvain-la-Neuve conference in 1987 under the name  well-supported compact closed (wscc) categories.  We (Rosebrugh, Sabadini, Walters,...) have been working on them in the context of automata and processes since 1997 with the introduction of Span(Graph) and later Cospan(Graph). In fact my talk was to be about recent developments in that combined algebra.

I am not sure if he kows the history which is partly described in our 2005 TAC paper "Generic commutative separable algebras and cospans of graphs" (he calls separable algebras special Frobenius algebras). A more colourful story is told at Joachim Kock's page. In view of what I think of the importance of wscc=multigraph categories I did suggest to Peter Selinger that he include them in his survey of graphical languages but he decided not to. However his suggestion that tangling of circuits might be important lead to our papers on those available here and here.

But Aleks might well be interested in that 2005 paper which proves that the free wscc=multigraph category on a monoidal graph M is the full subategory of Cospan(MonoidalGraphs/M) whose objects are discrete monoidal graphs/M. Actually the result in the 2005 paper is a special case but I pointed out CT2007 and CT2010 that the proof easily extends. His talk is about deciding which equations are true in multigraph categories by looking at matrix categories.

The other talk that I suspect has connections with my lecture is that of Bob Paré. This is a bit of a guess because I don't have any details, but also at  CT2010 I described a model of processes as 3x3 "spans of cospans" or "cospans of spans" with two structures as wscc=multigraph categories, and another level of cells between them, in which however there is an interchange which is not an isomorphic exactly because different amounts of synchronization occur in different composites. I did not have an abstract definition of this type of algebra, but I suspect Bob does since he speaks of "intercategories". I am very interested to see if my guess is correct, and if so what he does with them. My proposed talk was in fact to consider the simplification of cutting the corners, and finding distributive laws.

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2 Comments:

Blogger Aleks Kissinger said...

Some nice pointers! I figured these categories have to pop up everywhere, though it is tricky to find this stuff, party due to uneven use of terminology (special vs. separable, monoidal graph vs. signature etc.). I'd be interested to learn more about this automata/process connection.

This characterisation of the free wscc in terms of cospans of monoidal graphs is really beautiful. For the result, I used a presentation of the free category in terms of a generalisation of Hasegawa's cyclic network graphs, where a certain bijection is allowed to be a function. However, if spend a few minutes staring at it, it is exactly the same data as a cospan of discrete monoidal graphs over Sigma.

This is really nice, because I previously thought composition was this fiddly set-theoretic thing, involving pushouts over restrictions of various stuff, but it's really just the normal composition once you pick the *correct* category of cospans.

4:59 AM  
Blogger Aleks Kissinger said...

Some nice pointers! I figured these categories have to pop up everywhere, though it is tricky to find this stuff, party due to uneven use of terminology (special vs. separable, monoidal graph vs. signature etc.). I'd be interested to learn more about this automata/process connection.

This characterisation of the free wscc in terms of cospans of monoidal graphs is really beautiful. For the result, I used a presentation of the free category in terms of a generalisation of Hasegawa's cyclic network graphs, where a certain bijection is allowed to be a function. However, if spend a few minutes staring at it, it is exactly the same data as a cospan of discrete monoidal graphs over Sigma.

This is really nice, because I previously thought composition was this fiddly set-theoretic thing, involving pushouts over restrictions of various stuff, but it's really just the normal composition once you pick the *correct* category of cospans.

4:59 AM  

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