Abstract

Do there exist the theories between the axiomatic theory of Symmetric Monoidal Closed Categories (SMCC) and “fully coherent” partial order? (As examples of SMC Cs one may take the categories of modules over commutative rings with unit.) It turns out that the answer is positive. In terms of diagrams, it means that there exist certain non-commutative diagrams in free SMCC and certain non-free SMCC K such that some of these diagrams are always commutative in K while others are not. More recently, it was obtained an infinite series of diagrams D_n (n\in N) such that the commutativity of D_{n+1} does not imply the commutativity of D_n. It means that there exist infinitely many intermediate theories. This situation is radically different from the well known case of Cartesian Closed Categories. This fact is a strong motivation for the study of dependency of diagrams. Various methods of verification of dependency of diagrams are discussed. They may be of interest to computer algebra.

(The talk is based on joint work with A. El Khoury, L. Mehats and M. Spivakovsky.)