Tue, 24 Nov 2020, 1 pm MST
Many categorical conditions can be translated into the language of universal algebra as Mal'cev conditions. For example, much of the theory of Mal'cev varieties can be reproduced in the purely categorical context of Mal'cev categories. Then, the famous result that a variety is congruence-permutable if and only if it has a Mal'cev term t(x,y,y)=x and t(x,x,y)=y can be seen as a syntactical characterisation of this categorical condition.
In this talk, we present such a characterisation for the categorical condition of co-extensivity, where a category C is said to be co-extensive if for each pair of objects X,Y in C, the canonical functor ×:X/C×Y/C→(X×Y)/C is an equivalence. As a motivating example, we begin by considering the property of commutative semirings that to present a commutative semiring S as a product is exactly to find two elements of S which sum to 1 and whose product is 0. This very naturally implies that the variety of commutative semirings is co-extensive, and it then becomes interesting to ask exactly which other varieties are co-extensive. In order to answer this question, we first characterise the weaker condition of left co-extensivity. We then show that any co-extensive variety must have what we call a 'diagonalising term'. Lastly, we complete the characterisation by finding a sufficient and necessary set of identities this term must satisfy in order for the variety to be co-extensive.