Coherence in Substructural Categories

It is proved that MacLane’s coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in terms of natural transformations equipped with “graphs” (g-natural transformations) and corresponding morphism theorems are given as consequences. Using these results, some basic relations between the free categories of these classes are obtained. In [8] MacLane has shown that monoidal and symmetric monoidal categories are coherent, although the complete definition of the notion was given for the first time in [6]. Strictly keeping to that definition, we show that relevant, affine and cartesian categories are coherent. All the categories above we call substructural because they correspond to the minimal fragments of Associative Lambek’s Calculus, linear, relevant, BCK and intuitionistic logic that are sufficient to describe the underlying structural rules (see [10] to find about different aspects of substructural logics). We use equational axiomatizations of these categories, which originate from [1], rather than postulating the commutativity of certain diagrams. Of course, one who prefers diagram-chasing can easily convert these equations into commutative diagrams. 1 Substructural categories By category with multiplication we mean a category A together with a bifunctor · : A×A → A and a special object I. Categories with multiplication can be axiomatized by postulating the following equations between arrows categorial equations (cat1) f1A = f = 1Bf for all f : A → B (cat2) h(gf) = (hg)f for all f, g, h ∈ Mor(A) functorial equations (·) (g1f1)·(g2f2) = (g1·g2)(f1·f2) (·1) 1A·1B = 1A·B A category with multiplication is monoidal if there are special arrows for all objects A, B and C σA : I·A → A δA : A·I → A σiA : A → I·A δ i A : A → A·I −→ bA,B,C : A·(B·C) → (A·B)·C ←− bA,B,C : (A·B)·C → A·(B·C) and if it satisfies