Homotopy Type Theory

Homotopy Type Theory (HoTT) is a modern framework that combines ideas from algebraic topology—the study of shapes and spaces—with logic and computer science. It treats mathematical objects as shapes, where types correspond to spaces and equalities represent continuous transformations or deformations between objects. This perspective allows for more flexible reasoning about mathematical structures, emphasizing the relationships and transformations rather than static identities. Essentially, HoTT provides a unified language for understanding both logical proofs and geometric intuition, leading to new ways of verifying mathematical correctness and exploring the fundamental nature of mathematical concepts.