Yoneda Linear Algebra

A programme to read linear algebra not as the theory of a field acting on tuples but as what remains after the Yoneda embedding is applied to a categorified semiring and the higher morphisms are made to collapse. The wager is that scalars, vectors, and co-vectors are shadows of hom-sets, and that the familiar bilinearity is a coherence theorem in disguise.

Start from a semiring and its categorification, a monoidal category whose isomorphism classes of objects recover the elements of , with and lifting the two operations. Modules become -enriched presheaves; the Yoneda embedding sends each object to the hom-functor it represents. The claim under test is that linear-algebraic identities are naturality squares for , so that “linear” means “representable”.

Coherence as the collapse of freedom

The free monoidal structure on carries too much data. Reassociations and unit isomorphisms are genuine 3-morphisms, and a formal expression remembers its whole bracketing history. Mac Lane coherence is exactly the statement that this history is irrelevant, since every diagram of associators and unitors commutes.

Proposition (coherence thins the free structure)

Read one level up, coherence collapses the higher (3-)morphisms of the free monoidal -category to identities, so the free structure thins. Between any two parallel -morphisms there is at most one -morphism. A thin category is a preorder, and the arithmetic of reappears as that preorder’s composition. Bilinearity is thus not an axiom imposed on but the residue of freedom after coherence has quotiented the bracketings.

The monad line

The tensor unit and the representable functors organise into a standard monadic story, which is where the programme expects the computational content to sit.

Remark (Yoneda tensor, monad, Kleisli)

The Yoneda tensor is Day convolution on , extending from ; the Yoneda monad is the resulting free-module monad , whose algebras are the -modules; and vectors-as-computations live in its Kleisli category, where a morphism is a matrix and Kleisli composition is matrix multiplication. This is the linear-algebra face of the same monad line pursued elsewhere in the wiki for algorithms.

Points and co-points

Definition (co-points)

For an object write for the set of monoidal co-points of , the morphisms out of into the unit that respect the monoidal structure on one side only. A vector is then a point and a co-vector an element of . Each is a hom-set with one endpoint fixed. Fixing the other endpoint exchanges the two, and the pairing is the duality bracket, recovered as composition rather than posited.

The three pieces interlock. Coherence supplies the arithmetic, the monad supplies the modules, and the co-point construction supplies duality, all as instances of representability. What is still missing is a theorem that the enriched Yoneda lemma forces these to agree with ordinary linear algebra over a field; at present the correspondence is checked case by case and is best regarded as a conjectural dictionary. See category.

References

• S. Mac Lane, Categories for the Working Mathematician (coherence theorem)
• B. Day, On closed categories of functors (1970) (Day convolution)