All Concepts are Essentially Algebraic - A Study of Dependently-Typed Algebraic Theories

Explore a Berkeley seminar lecture that delves into the profound relationship between algebraic theories and mathematical concepts. Learn how Lawvere's categorical formulation of algebraic theories provides a framework for studying fundamental mathematical structures like groups and rings with precision and generality. Discover why certain mathematical concepts, including categories and topological spaces, transcend traditional algebraic theory limitations. Examine the evolution from simply typed to dependently typed algebraic theories, revealing how all mathematics can be understood as essentially algebraic. Understand how this expanded framework enables the definition of mathematical structures that encompass entire universes of mathematics, including topoi and models of type theory. Investigate the self-referential nature of dependently-typed algebraic theories and their implications, particularly in constructing type theories where types themselves correspond to other type theories, with functions representing translations between them.