The Unapologetic Mathematician

Mathematics for the interested outsider

2-functors

Of course along with 2-categories, we must have 2-functors to map from one to another.

So, what’s a 2-functor? Since we defined a 2-category as a category enriched over \mathbf{Cat}, a 2-functor should be a functor enriched over \mathbf{Cat}. That is, it consists of a function on objects and a functor for each hom-category, each of which consists of a function on 1-morphisms (the objects of the hom-category) and a function for each set of 2-morphisms. Then there are a bunch of relations.

Let’s expand this a bit. A 2-category \mathcal{C} has a collection \mathrm{Ob}(\mathcal{C}) of objects, a collection \mathrm{Ob}(\hom_\mathcal{C}(A,B)) of 1-morphisms for each pair (A,B) of objects, and a collection \hom_{\hom_\mathcal{C}(A,B)}(f,g) of 2-morphisms for each pair (f,g) of 1-morphisms between the same pair of objects. And all the same remarks go for another 2-category \mathcal{D}.

So a 2-functor F has

  • a function F:\mathrm{Ob}(\mathcal{C})\rightarrow\mathrm{Ob}(\mathcal{D})
  • for each pair (A,B) of objects of \mathcal{C}, a functor F:\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{D}(F(A),F(B)), each consisting of
    • a function F:\mathrm{Ob}(\hom_\mathcal{C}(A,B))\rightarrow\mathrm{Ob}(\hom_\mathcal{D}(F(A),F(B)))
    • for each pair (f,g) of 1-morphisms from A to B a function F:\hom_{\hom_\mathcal{C}(A,B)}(f,g)\rightarrow\hom_{\hom_\mathcal{D}(F(A),F(B))}(F(f),F(g))

Now the composition functors \circ give us functions for composing 1-morphisms and “horizontally” composing 2-morphisms, and the hom-categories give us functions \cdot for “vertically” composing 2-morphisms. For each object we have an identity 1-morphism, and for each 1-morphism we have an identity 2-morphism. A 2-functor will preserve all these structures. First of all, since there are functors between the hom-categories the vertical composition is preserved, along with the identity 2-morphisms. The diagrams for enriched functors say that the identity 1-morphisms, the composition of 1-morphisms, and the horizontal composition of 2-morphisms are all preserved.

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August 18, 2007 - Posted by | Category theory

1 Comment »

  1. Perhaps it’s worth elaborating on this post? Perhaps differentiating between strict 2-functors and the weaker notion of pseudofunctors which I believe is more commonly used.

    Really appreciate this blog, I learnt the majority of my category theory from here.

    Comment by Zeki Mirza | March 25, 2011 | Reply


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