functor is the mathematical model of topological quantum field theory, which has been recently developed. Let be the category of cobordisms of n-1,n-dimensional manifolds with tensor product given by disjoint union, and unit the empty manifold. A topological quantum field theory in dimension n is a symmetric monoidal functor The homology functor is monoidal as via the map . Alternate notions If and are closed monoidal categories with internal hom-functors (we drop the subscripts for readability), there is an alternative formulation ψAB : F(A ⇒ B) → FA ⇒ FB of φAB commonly used in functional programming. The relation between ψAB and φAB is illustrated in the following commutative diagrams: Properties If is a monoid object in , then is a monoid object in . Monoidal functors and adjunctions Suppose that a functor is left adjoint to a monoidal . Then has a comonoidal structure induced by , defined by and . If the induced structure on is strong, then the unit and counit of the adjunction are monoidal natural transformations, and the adjunction is said to be a monoidal adjunction; conversely, the left adjoint of a monoidal adjunction is always a strong monoidal functor. Similarly, a right adjoint to a comonoidal functor is monoidal, and the right adjoint of a comonoidal adjunction is a strong monoidal functor. See also Monoidal natural transformation References Monoidal categories