Algebras in tensor categories appear in several interesting research areas, like e.g. VOA extensions or spin topological field theories, but they are usually tricky to find. In this talk, we will explain how to generalize a result by Ostrik and Natale on algebra objects in categories related to lattice VOAs to the case of so-called group-theoretical fusion categories. The algebra objects we find for these also have very good properties that we will describe in detail. We will assume little knowledge of categories. Joint work with the WINART2 team Y. Morales, M. Mueller, J. Plavnik, A. Tabiri and C. Walton
The Promise Constraint Satisfaction Problem is a variant of the Constraint Satisfaction Problem (CSP). In PCSP, we are promised that either there exists a homomorphism from A to B, or there is even no homomrphism from A to a structure C; our goal is to decide between these two cases. An example of PCSP is PCSP(K_3,K_4), where we are deciding if the input graph is 3-colorable or not even 4-colorable.
The natural generalization of polymorphisms to pairs of structure are minions of polymorphisms. Minons have very little structure; compared to algebras we lose the ability to compose operations. Surprisingly, what little structure is left still leads to nontrivial mathematics.
In the talk, we will have a look at polymorphisms in general and polymorphisms from K_3 to K_4 in particular.