Categories, functors, and natural transformations

A category has objects, arrows between specified pairs of objects, identity arrows, and associative composition whenever targets and sources match. In the category of sets, arrows are functions; in vector spaces over a fixed field, they are linear maps. Composition g∘f means first f and then g. A picture of connected nodes is not automatically a category: composites and identities must exist and obey the axioms. Conversely an object may carry rich internal data that the category deliberately emphasizes only through a chosen class of structure-preserving maps.

For vector spaces, using arbitrary functions as arrows would produce a different category and destroy statements that rely on linearity. The choice of morphisms is therefore part of the mathematical question. An isomorphism is an arrow with a two-sided inverse inside the category. Even a continuous bijection between topological spaces need not be an isomorphism unless its inverse is also continuous. Keeping the category explicit prevents a familiar word such as equivalent from quietly changing meaning when one moves from algebra to geometry or analysis.

A covariant functor assigns an object F(A) to each object A and an arrow F(f):F(A)→F(B) to each f:A→B. It must preserve identities and composition. Forgetting the vector-space structure gives a functor from vector spaces to sets because a linear map is a function and its composition is the same underlying composition. This construction does not manufacture a preferred basis. Any proposal that chooses arbitrary bases and then claims canonical coordinates must explain how those choices interact with every map rather than only with one convenient example.

Dualization reverses direction: a linear map f:V→W gives a pullback f*:W*→V* by precomposition. It is therefore contravariant, or equivalently a covariant functor into an opposite category. Applying dualization twice restores the original direction. In finite dimension the double dual is isomorphic to the original space, but that does not make every arbitrary isomorphism natural. Naturality identifies a specific compatibility with changing V. Infinite-dimensional algebraic vector spaces still have a canonical injection into their double dual, but it need not be surjective.

A natural transformation η:F⇒G supplies a component η_A:F(A)→G(A) for every object. For every arrow f:A→B, taking the F route and then comparing at B must agree with comparing at A and taking the G route. One successful diagram is insufficient: the statement quantifies over all arrows. Components need not be invertible; when every component is an isomorphism, the transformation is a natural isomorphism. The square thus describes a family of compatible comparisons, rather than an additional operation on one isolated object.

For a vector v, evaluation defines j_V(v) in V** by j_V(v)(φ)=φ(v). No basis appears. The identity f**∘j_V=j_W∘f follows by applying both sides to an arbitrary covector on W, making this an excellent first naturality proof. In finite dimension each j_V is invertible. In the lab, a single sigmoid computation graph makes order of operations inspectable; it neither represents an entire category nor checks naturality. The lesson's proof comes from following typed linear maps and testing their evaluations, not from the graph's visual arrows.