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 diagrammatic order. A lax functor P from C to D, denoted ${\displaystyle P:C\to D}$, consists of the following data:

• for each object x in C, an object ${\displaystyle P_{x}\in D}$;
• for each pair of objects x,y ∈ C a functor on morphism-categories, ${\displaystyle P_{x,y}:C(x,y)\to D(P_{x},P_{y})}$;
• for each object x∈C, a 2-morphism ${\displaystyle P_{{\text{id}}_{x}}:{\text{id}}_{P_{x}}\to P_{x,x}({\text{id}}_{x})}$ in D;
• for each triple of objects, x,y,z ∈C, a 2-morphism ${\displaystyle P_{x,y,z}(f,g):P_{x,y}(f);P_{y,z}(g)\to P_{x,z}(f;g)}$ 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 ${\displaystyle P_{{\text{id}}_{x}}}$ and ${\displaystyle P_{x,y,z}}$ above, are invertible is called a pseudofunctor.