View source for 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|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|algebraic topology]], [[Quantum Field Theory|quantum field theory]], and the [[Cobordism Hypothesis|Cobordism Hypothesis]], and it provides the language in which [[Homotopy Type Theory|homotopy type theory]] reformulates the foundations of mathematics.

[[Category:Mathematics]]
[[Category:Systems]]