Lax functor
In category theory, a discipline within mathematics, the notion of lax functor between bicategories generalizes that of functors between categories.
Let C,D be bicategories. We denote composition in . A lax functor P from C to D, denoted, consists of the following data:
- for each object x in C, an object ;
- for each pair of objects x,y ∈ C a functor on morphism-categories, ;
- for each object x∈C, a 2-morphism in D;
- for each triple of objects, x,y,z ∈C, a 2-morphism in D that is natural in f: x→y and g: y→z.
These must satisfy three commutative diagrams, which record the interaction between left unity, right unity, and associativity between C and D. See http://ncatlab.org/nlab/show/pseudofunctor.
A lax functor in which all of the structure 2-morphisms, i.e. the and above, are invertible is called a pseudofunctor.
