Lax functor explained

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 diagrammatic order. A lax functor P from C to D, denoted

P: C\to D , consists of the following data:

• for each object x in C, an object

P _{x\in} D ;

• for each pair of objects x,y ∈ C a functor on morphism-categories,

P _{x,y} : C(x,y)\to D(P _{x,P} _{y)} ;

• for each object x∈C, a 2-morphism

P _{id _{x}} : id _{P _{x}} \to P _{x,x} ( id _{x)} in D;

• for each triple of objects, x,y,z ∈C, a 2-morphism

P _{x,y,z} (f,g): P _{x,y} (f);P _{y,z} (g)\to P _{x,z} (f;g) 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

P _{id _{x}} and P _{x,y,z} above, are invertible is called a pseudofunctor.