Linearly distributive coherence in the absence of units

TL;DR

This paper proves coherence for linearly distributive categories without units, showing that units can obstruct coherence, and extends the result to Frobenius linearly distributive functors.

Contribution

It provides a self-contained proof that linearly distributive categories without units are coherent, addressing a gap in understanding coherence in such structures.

Findings

Coherence holds for linearly distributive categories without units.

Units can obstruct coherence in these categories.

Results relate to directed paths in associahedra and multiplihedra.

Abstract

Coherence in a monoidal category asserts that all morphisms built from structural isomorphisms with a fixed source and target coincide. These structural isomorphisms include, in particular, the associators. Linearly distributive categories carry two tensor products, with structural morphisms given by associators and distributors relating the two tensor products. In several examples, including Grothendieck--Verdier categories, also known as $\ast$-autonomous categories, these distributors need not be invertible. We give a self-contained proof that linearly distributive categories without units are coherent, while units may obstruct coherence. With the same techniques, we also establish an analogous coherence result for Frobenius linearly distributive functors without units. These results admit a reformulation in terms of directed paths in associahedra and multiplihedra.