Mac Lane coherence can be deduced from the simple connectivity of the associahedron Is it written that way anywhere? jim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
-----Original Message----- From: categories@mta.ca [mailto:categories@mta.ca] On Behalf Of jim stasheff Sent: Wednesday, June 24, 2009 12:18 PM To: Categories list Subject: categories: query
Mac Lane coherence can be deduced from the simple connectivity of the associahedron Is it written that way anywhere?
jim
Hi, Yes. My thesis. "Obstructions to Coherence: Natural Noncoherent Associativity and Tensor Functors", City University of New York, 1996. The part about the associahedron was published in Obstructions to Coherence: Noncoherent Associativity The Journal of Pure & Applied Algebra. 147 no. 2, Pgs 175 - 213. (2000). or http://xxx.lanl.gov/abs/math.QA/9804106 The second part about the tensor functors was never published. I look at the fundamental group of the associahedra thought of as groupoids (called the "Catalan groupoids"). They are all trivial. But then I ask, what if the pentagons do not commute? The fundamental group of the Mac Lane non-commuting pentagon is Z. And I get generators and relations for all the higher non-commuting associahedra. They are not free groups from n=7 on. I do a similar thing for non-coherent tensor functors (monoidal functors). All the best, Noson [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Jim, On Wed, 24 Jun 2009, jim stasheff wrote:
Mac Lane coherence can be deduced from the simple connectivity of the associahedron
Surely that's not true, assuming that by "Mac Lane coherence" you mean Mac Lane's coherence theorem for monoidal categories. The associahedra (and in particular the pentagon) say nothing about the unit coherence isomorphisms, X \otimes I ----> X <---- I \otimes X. To make it true, surely you need to weaken Mac Lane's theorem to a statement about "semigroupal" categories, i.e. monoidal categories without unit...? Best wishes, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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jim stasheff -
Noson S. Yanofsky -
Tom Leinster