Coherent Bicartesian and Sesquicartesian Categories

Coherence is here demonstrated for sesquicartesian categories, which are categories with nonempty finite products and arbitrary finite sums, including the empty sum, where moreover 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, and it yields a very easy decision procedure for equality of arrows. Restricted coherence holds also 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. The printed version of this paper (in: R. Kahle et al. eds, Proof Theory in Computer Science, Lecture Notes in Computer Science, vol. 2183, Springer, Berlin, 2001, pp. 78-92) and versions previously posted here purported to prove unrestricted coherence for the bicartesian categories mentioned above. Lemma 5.1 of these versions, on which the proof of coherence for sesquicartesian categories relied too, is however not correct. The present version of the paper differs from the previous ones also in terminology.