Lax functor - Alchetron, The Free Social Encyclopedia

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 → D , consists of the following data:

for each object x in C, an object P x ∈ D ;

for each pair of objects x,y ∈ C a functor on morphism-categories, P x , y : C ( x , y ) → D ( P x , P y ) ;

for each object x∈C, a 2-morphism P id x : id P x → 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 ) → 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.

References

Lax functor Wikipedia

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