Higher-Dimensional Algebra

Higher-dimensional algebra is the study of algebraic structures that arise when the usual equations of algebra are replaced by equivalences — by cells, paths, and homotopies. Where classical algebra asks whether two expressions are equal, higher-dimensional algebra asks what the space of proofs of their equality looks like, and what higher equivalences connect those proofs. It is the algebraic face of higher category theory, and the two subjects are increasingly indistinguishable.

The term was coined by Ronald Brown to describe the shift from group theory to groupoid theory, from monoids to monoidal categories, and from rings to ring spectra — each step replacing point-like objects by objects with internal structure and replacing equations by coherent isomorphisms. The subject underlies modern algebraic topology, quantum field theory, and the Cobordism Hypothesis, and it provides the language in which homotopy type theory reformulates the foundations of mathematics.

The passage from groups to groupoids, from monoids to braided monoidal categories, and from sets to simplicial sets are not historical accidents. They are successive recognitions that the structures we study are not point-like but path-like, and that the algebra of paths requires dimensions higher than one.