Lax functor
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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 [math]\displaystyle{ P: C\to D }[/math], consists of the following data:
- for each object x in C, an object [math]\displaystyle{ P_x\in D }[/math];
- for each pair of objects x,y ∈ C a functor on morphism-categories, [math]\displaystyle{ P_{x,y}: C(x,y)\to D(P_x,P_y) }[/math];
- for each object x∈C, a 2-morphism [math]\displaystyle{ P_{\text{id}_x}:\text{id}_{P_x}\to P_{x,x}(\text{id}_x) }[/math] in D;
- for each triple of objects, x,y,z ∈C, a 2-morphism [math]\displaystyle{ P_{x,y,z}(f,g): P_{x,y}(f);P_{y,z}(g)\to P_{x,z}(f;g) }[/math] 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 [math]\displaystyle{ P_{\text{id}_x} }[/math] and [math]\displaystyle{ P_{x,y,z} }[/math] above, are invertible is called a pseudofunctor.
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![]() | Original source: https://en.wikipedia.org/wiki/Lax functor.
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