Coherent bicartesian and sesquicartesian categories

Abstract

Sesquicartesian categories are categories with nonempty finite products and arbitrary finite sums, including the empty sum. Coherence is here demonstrated for sesquicartesian categories in which the first and the second projection from the product of the initial object with itself are the same. (Every bicartesian closed category, and, in particular, the category Set, is such a category.) This coherence amounts to the existence of a faithful functor from categories of this sort freely generated by sets of objects to the category of relations on finite ordinals. Coherence also holds for bicartesian categories where, in addition to this equality for projections, we have that the first and the second injection to the sum of the terminal object with itself are the same. These coherences yield a very easy decision procedure for equality of arrows.