Category Coherence Research Articles

A category with group structure is a monoidal category with a contravariant functor *, natural isomorphisms I+ A * @ A, I -+ A @A * and coherence axioms for the commutativity of some diagrams. The structure is abelian if the category is symmetric monoidal and additional coherence axioms are satisfied. In a recent paper K.-H. Ulbrich [3] has studied the coherence of these categories and proved that any arrow composed of elementary arrows of the structure depends only upon the formal expression of its domain and codomain: any diagram commutes if the vertices and edges are such expressions and composites, respectively. This result is similar to that of S. Mac Lane for monoidal categories in [2]. We present here an alternative way for the study of the coherence of these categories. This paper was written after we read [3] and became convinced that the interest and applicability of the results could justify a more conceptual treatment of the topic which could make it more accessible. For the case of the categories with group structure we use essentially a “diamond lemma” argument after some necessary transformations: a similar argument was used in [2]. For the symmetric case the coherence result is a corollary of [ 11: we provide only the convenient background. This approach gives some insight into the relation between categories with group structure and compact categories which can be explored more deeply when, and if, the applications require it. We also detail some elementary results and consequences which are related to our treatment. In the last part of the paper we show that a group structure on a monoidal category can be given only by the assignment for any object A of an object 305 0021-8693/83 $3.00