**Speaker / Predavač** : Prof. Sergei Soloviev

**Title / Naslov**: Non-maximality of the theory of Symmetric Monoidal Closed Categories and dependency of categorical diagrams

**Abstract / Abstrakt**:

Some time ago we (L. Mehats, M. Spivakovsky, S. Soloviev) had shown that there exists diagram of canonical natural transformations in SMCC such that this diagram is commutative while the classical diagram of "triple dual" is still non-commutative in some model, i.e., there exists a non-trivial extension of the theory of SMCC (the theory is non-maximal, as opposed to the case of Cartesian Closed Categories studied by Dosen and Petric). Now in our recent work with Antoine El Khoury we 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. (In the proof the SMCC of semi-modules over semi-rings are used.) As a consequence the study of dependency of diagrams in SMCCs is much more important than in case of CCC. Various methods of verification of dependency are discussed in the end of the talk.