The Coherence Theorem for Ann-Categories

TL;DR

This paper proves the coherence theorem for Ann-categories, showing that morphisms built from isomorphisms depend only on source and target, extending classical coherence results to this algebraic structure.

Contribution

It provides the first proof of the coherence theorem for Ann-categories by embedding them into quite strict Ann-categories, building on prior work and methods from monoidal categories.

Findings

Coherence theorem established for Ann-categories.

Morphisms depend only on source and target in Ann-categories.

Embedding into strict Ann-categories is used for the proof.

Abstract

This paper presents the proof of the coherence theorem for Ann-categories whose set of axioms and original basic properties were given in [9]. Let $$\A=(\A,{\Ah},c,(0,g,d),a,(1,l,r),{\Lh},{\Rh})$$ be an Ann-category. The coherence theorem states that in the category $ \A$, any morphism built from the above isomorphisms and the identification by composition and the two operations $\tx$, $\ts$ only depends on its source and its target. The first coherence theorems were built for monoidal and symmetric monoidal categories by Mac Lane [7]. After that, as shown in the References, there are many results relating to the coherence problem for certain classes of categories. For Ann-categories, applying Hoang Xuan Sinh's ideas used for Gr-categories in [2], the proof of the coherence theorem is constructed by faithfully ``embedding'' each arbitrary Ann-category into a quite strict Ann-category. Here, a {\it quite strict} Ann-categogy is an Ann-category whose all constraints are strict, except for the commutativity and left distributivity ones. This paper is the work continuing from [9]. If there is no explanation, the terminologies and notations in this paper mean as in [9].