Online seminář z algebry - 14.5.2020 PDF Tisk

Další seminář z algebry se koná 14.5.2020 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Soichiro Fujii

A unified framework for notions of algebraic theory

Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called notions of algebraic theory. In this talk, we present a  unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that algebraic theories correspond to monoid objects therein. To incorporate semantics, we introduce a categorical structure called metamodel, which formalises a  definition of models of algebraic theories. We also define morphisms between notions of algebraic theory, which are a  monoidal version of profunctors. Every strong monoidal functor gives rise to an adjoint pair of such morphisms, and provides a uniform method to establish isomorphisms between categories of models in different notions of algebraic theory. A general structuresemantics adjointness result and a double categorical universal property of categories of models are also shown.

Aktualizováno Úterý, 12 Květen 2020 09:02