Coherence of Associativity in Categories with Multiplication
Matthew G. Brin
Abstract
The usual coherence theorem of MacLane for categories with multiplication assumes that a certain pentagonal diagram commutes in order to conclude that associativity isomorphisms are well defined in a certain practical sense. The practical aspects include creating associativity isomorphisms from a given one by tensoring with the identity on either the right or the left. We show, by reinspecting MacLane's original arguments, that if tensoring with the identity is restricted to one side, then the well definedness of constructed isomorphisms follows from naturality only, with no need of the commutativity of the pentagonal diagram. This observation was discovered by noting the resemblance of the usual coherence theorems with certain properties of a finitely presented group known as Thompson's group F. This paper is to be taken as an advertisement for this connection.
Create a lesson
Related papers
Higher zeroes in pointed infinity-categories
Thomas Rush Brown
Higher-order many-sorted categorial algebras via generalized hep-derivors
J. Climent Vidal, E. Cosme Llópez, R. Ruiz Mora et al.
Normalizers in the category of topological groups
Maria Manuel Clementino, James Richard Andrew Gray
On the representability of actions of algebras with several bilinear multiplications
Manuel Mancini
On closed symmetric monoidal enriched ∞-categories
Rune Haugseng
Absolute descent and the Bénabou--Roubaud theorem
Fernando Lucatelli Nunes