Obstructions to coherence: Natural noncoherent associativity and tensor functors.
Item
-
Title
-
Obstructions to coherence: Natural noncoherent associativity and tensor functors.
-
Identifier
-
AAI9630528
-
identifier
-
9630528
-
Creator
-
Yanofsky, Noson S.
-
Contributor
-
Adviser: Alex Heller
-
Date
-
1996
-
Language
-
English
-
Publisher
-
City University of New York.
-
Subject
-
Mathematics
-
Abstract
-
In the past few years, categorical coherence questions have arisen in many diverse branches of mathematics like quantum groups, knot theory and proof theory. In this paper, we study what happens when coherence fails. We consider categories with a tensor product and a natural associativity isomorphism that does not necessarily satisfy the pentagon coherence requirements (called Associative Categories). Categorical versions of associahedra, {dollar}\rm{lcub}\bf A{rcub}\sb{lcub}{lcub}\bf n{rcub}{rcub},{dollar} are constructed (called Catalan groupoids). The objects correspond to associations of letters, and morphisms correspond to reassociations. These groupoids are used in the constructions of the free associative category. They are also used in the construction of the theory of associative categories (given as a 2-sketch). Generators and relations are given for the fundamental group, {dollar}\rm\pi({lcub}\bf A{rcub}\sb{lcub}{lcub}\bf n{rcub}{rcub}),{dollar} of the of the Catalan groupoids--thought of as a simplicial complex. These groups are shown to be more then just free groups on the number of pentagons. Each associative category, B, has related fundamental groups {dollar}\rm\pi({lcub}\bf B{rcub}\sb{lcub}{lcub}\bf n{rcub}{rcub}){dollar} and homomorphisms {dollar}\pi(P\sb{lcub}n{rcub}){dollar}: {dollar}\rm\pi({lcub}\bf A{rcub}\sb{lcub}{lcub}\bf n{rcub}{rcub})\longrightarrow\pi({lcub}\bf B{rcub}\sb{lcub}{lcub}\bf n{rcub}{rcub}).{dollar} If the images of the {dollar}\pi(P\sb{lcub}n{rcub}){dollar} are trivial, i.e. there is only one associativity path between any two objects, then the category is coherent. Otherwise the images of {dollar}\pi(P\sb{lcub}n{rcub}){dollar} are obstructions to coherence. Some progress is made to classifying noncoherence of associative categories.;Functors between associative categories that do not necessarily satisfy the hexagon coherence requirement are dealt with. Much of the same constructions are also done for this coherence problem. The fundamental groups of the associative categories are shown to be related to the fundamental groups of the functors between them.
-
Type
-
dissertation
-
Source
-
PQT Legacy CUNY.xlsx
-
degree
-
Ph.D.
