That Logic Blog

April 02, 2009


I graduated from my PhD in December and have finally gotten around to placing my thesis on the arXiv. Because of the space limitations of arXiv metadata, the abstract on the webpage has been truncated somewhat, so I will put the real abstract below.

Coherence for rewriting 2-theories: General theorems with applications to presentations of Higman-Thompson groups and iterated monoidal categories; Jonathan Asher Cohen


The problems of the identity of proofs, equivalences of reductions in term rewriting systems and coherence in categories all share the common goal of describing the notion of equivalence generated by a two-dimensional congruence. This thesis provides a unifying setting for studying such structures, develops general tools for determining when a congruence identifies all reasonable parallel pairs of reductions and examines specific applications of these results within combinatorial algebra. The problems investigated fall under the umbrella of ``coherence'' problems, which deal with the commutativity of diagrams in free categorical structures --- essentially a two-dimensional word problem. It is categorical structures equipped with a congruence that collapses the free algebra into a preorder that are termed ``coherent''.

The first main result links coherence problems with algebraic invariants of equational theories. It is shown that a coherent categorification of an equational theory yields a presentation of the associated structure monoid. It is subsequently shown that the higher Thompson groups $F_{n,1}$ and the Higman-Thompson groups $G_{n,1}$ arise as structure groups of equational theories, setting up the problem of obtaining coherent categorifications for these theories.

Two general approaches to obtaining coherence theorems are presented. The first applies in the case where the underlying rewriting system is confluent and terminating. A general theorem is developed, which applies to many coherence problems arising in the literature. As a specific application of the result, coherent categorifications for the theories of higher order associativity and of higher order associativity and commutativity are constructed, yielding presentations for $F_{n,1}$ and $G_{n,1}$, respectively.

The second approach does not rely on the confluence of the underlying rewriting system and requires only a weak form of termination. General results are obtained in this setting for the decidability of the two-dimensional word problem and for determining when a structure satisfying the weakened properties is coherent. A specific application of the general theorem is made to obtain a conceptually straightforward proof of the coherence theorem for iterated monoidal categories, which form a categorical model of iterated loop spaces and fail to be confluent.