What is the right category of Clifford algebras when we want to compare them, not just map one rigidly into another?
Neither obvious choice of morphism serves the comparison of Clifford algebras over a fixed ground field. Taking morphisms to be algebra homomorphisms is far too rigid, since an algebra map must respect on the nose and so sees only the isometries of the underlying forms, forgetting almost every interesting linear comparison. Taking with all linear maps is too loose, since it throws the algebra structure away entirely. The morphisms wanted here are linear maps yet remember how much they fail to be algebra maps.
Several candidate fixes fall short of this target. The full-image factorization keeps only the linear maps whose image is a subalgebra. Enrichment in super vector spaces or filtered spaces records the grading together with the degree filtration by . The Morita bicategory replaces isomorphism by Morita equivalence, with bimodules as its -cells. Each captures something, yet none is quite the hom of linear-maps-with-memory at issue.
The choice made here is Sweedler enrichment. For algebras a coalgebra measures to when a linear map , , respects multiplication in the Sweedler-diagonal sense and the unit. The universal measuring coalgebra is terminal among all such , and it makes algebras into a category enriched in coalgebras, with the pairing of the measuring datum against an element. Its grouplike elements recover the honest algebra maps , while as a whole sits between and . Each of its elements is a linear map that carries, in its comultiplication, the exact record of how far it is from being multiplicative.
Sweedler enrichment suits Clifford algebras precisely because of their -grading. Measuring is compatible with , so running the whole construction internal to super vector spaces lets the Koszul sign rule thread through the Sweedler diagonal automatically, and the even/odd decomposition of is preserved by with the correct signs. The versor picture then rides along. The grouplikes of contain the algebra automorphisms realizing the versor action, and fattens that discrete group of symmetries into a coalgebra of near-symmetries.
Two readings make the payoff concrete. Sheaf-theoretically, spreading Clifford algebras over a cellular-sheaf’s cells leaves the restriction maps free of the algebra-map constraint. Letting them be measuring data turns each restriction into a piece of a coalgebra, so gluing is controlled by -preserving comultiplications rather than strict homomorphisms. Deformation-theoretically, the primitives of over the identity are exactly the Hochschild -cochains, and their bracket is the Gerstenhaber bracket governing associative deformations of the quadratic form. The measuring coalgebra is therefore the linearized, categorified home of Clifford deformation theory, the enriched category in which “how far from an algebra map” is itself the morphism.