A survey of higher dimensional algebras

Emre Özel
Anadolu Üniversitesi, Türkiye

The purpose of this talk is to define higher categorical algebras related to 2- and 3-categories and to associate them with crossed modules and 2-crossed modules, which are algebraic models for homotopy 2- and 3-types, respectively. First, we will construct, in a detailed manner, a commutative, strict 2-algebra from the crossed module of commutative K-algebras. Then we will show that the 1-category of crossed modules of commutative K-algebras is equivalent to the 1-category of commutative, strict 2-algebras. Second, we will examine a 2-crossed module of commutative K-algebras from a higher-dimensional categorical perspective and discuss why a commutative strict 3-algebra cannot be constructed. As a result of this discussion, we will construct a special 3-categorical structure (called sestertius algebra) from a 2-crossed module. These constructions will be on commutative structures. Finally, non-commutative, that is, only associative cases will be discussed.