# All Concepts are Essentially Algebraic

Abstract

Lawvere’s categorical formulation of algebraic theories enables one to study some of the most common structures found in mathematics – e.g. groups, rings, etc. – at a high level of precision and generality. However, many significant mathematical concepts, including categories, topological spaces, etc., turn out not to be algebraic, in this sense. Notably, the very framework used by Lawvere to describe algebraic theories and their models – categories with finite products and product-preserving functors between them – cannot be described as an algebraic theory, and so it seems that algebra alone cannot encompass the whole of mathematics (nor even itself). There is, however, a deeper sense in which all of mathematics is essentially algebraic. What is needed to reveal this fact is to adapt the classical notion of algebraic theories, which are fundamentally simply typed, to an appropriate notion of dependently typed algebraic theories. At this level of generality, one is capable of defining not only individual mathematical structures, but structures that themselves encompass whole universes of mathematics, including topoi, models of type theory, etc. In particular, the theory of dependently-typed algebraic theories is itself describable as a dependently-typed algebraic theory. This fact has many profound consequences, of which I shall highlight just one: using the framework of dependently-typed algebraic theories, one can construct a type theory whose types themselves correspond to type theories, with functions between these types corresponding to translations between the corresponding type theories.