Functor

def. “Cat” Category is a category of all (simple) categories.

• simple, because there are logical issues with things like “set of all sets”

def. Functor. let $F$ be a functor. Think of it as a map from category $C$ to $D$ where

• morphism $f::a\to b$, maps to $Ff::Fa\to Fb$
• “preserves structure”: $h=g.f$ then $Fh=Fg.Fh$
• “preserves identity”: $F{\text{id}}_a={\text{id}}_{Fa}$ Intuition. key is it “preserves structure” This is also called ”functoriality”
• ¶It may map multiple objects into one, $a,b$ both maps to $Fz$
• ¶It may map multiple morphisms in a hom set $\text{Hom}(a,b)$ to single morphism in $\text{Hom}(Fa,Fb)$.

Compositon of Functors §

Motivation. Types and methods, in haskell is objects and morphisms in the category of Hash. But it doesn’t stop there; the whole new system of Algebraic Type System is, in fact, functors. In this case:

• Type constructors (e.g. Maybe a) is a functor
• Composition of type constructors (e.g. Maybe. List a) is functor composition
• Type constructors which take two parameters (e.g. Either a b) is a bifunctor. These concepts will be introduced below.

1. Functors can also be composed like morphisms: Example. consider functors List and Maybe we discussed above, with their own fmap for morphisms:

Then we can consider the composed functor MaybeList functor which composes these two functors together.

Bifunctors §

Motivation. Bifunctors are interesting because they are a special type of functor that takes in two arguments. We consider them mainly because we want to talk about binary operations, like in any other abstract algebra topic. def. Bifunctor is a functor of two arguments, i.e. maps two categories into one category.

• the objects you can map as just a cartesian product of two. There’s no restrictions on how you map objects.
• the morphisms have to map in a functorial way (=preserves identity, composition.)