Bifunktori

Bifunktori is a term used in category theory to describe a functor between categories of functors. More precisely, if C and D are categories, then a bifunktori F is a functor from the product category C × D to another category E. This means that F takes objects from C × D and maps them to objects in E, and it also takes morphisms from C × D and maps them to morphisms in E, respecting composition and identity.

An object in C × D is a pair of objects (X, Y) where X is an object in C and Y is an object in D. A morphism in C × D from (X, Y) to (X', Y') is a pair of morphisms (f, g) where f is a morphism in C from X to X' and g is a morphism in D from Y to Y'. The product category C × D has a universal property that formalizes this notion of pairing objects and morphisms.

A bifunktori F: C × D → E maps a pair of objects (X, Y) to an object F(X, Y) in E, and a pair of morphisms (f: X → X', g: Y → Y') to a morphism F(f, g): F(X, Y) → F(X', Y') in E. The functoriality requires that F preserves composition and identities. Specifically, for any composable morphisms f: X → X' and f': X' → X'' in C, and g: Y → Y' and g': Y' → Y'' in D, we must have F(f' ∘ f, g' ∘ g) = F(f', g') ∘ F(f, g). Also, F(id_X, id_Y) = id_{F(X, Y)} for all objects X in C and Y in D.

Bifunktori are fundamental in understanding how structures in one category relate to structures in another, especially when considering products of categories. They play a crucial role in defining concepts like tensor products and in various constructions within abstract algebra and theoretical computer science.