Foundations of Algebraic Theories and Higher Dimensional Categories

Abstract

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 symmetric operads, non-symmetric operads, generalised operads, and monads. These variants of universal algebra are called notions of algebraic theory. In the first part of this thesis, we develop a unified framework for notions of algebraic theory which includes all of the above examples. Our key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that theories correspond to monoid objects therein. We introduce a categorical structure called metamodel, which underlies the definition of models of theories. We also consider 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 way to establish isomorphisms between categories of models in different notions of algebraic theory. A general structure-semantics adjointness result and a double categorical universal property of categories of models are also shown. In the second part of this thesis, we shift from the general study of algebraic structures, and focus on a particular algebraic structure: higher dimensional categories. Among several existing definitions of higher dimensional categories, we choose to look at the one proposed by Batanin and later refined by Leinster. We show that the notion of extensive category plays a central role in Batanin and Leinster's definition. Using this, we generalise their definition by allowing enrichment over any locally presentable extensive category.