\chapter{Functors}




\begin{definition}[Functor]
A \emph{functor} is a function $F : \mathcal{C} \to \mathcal{D}$ between categories such that the following hold.
\[ \forall X \in \mathrm{ob}(\mathcal{C})[ \exists F(X) \in \mathrm{ob} (\mathcal{D}) ] \]
\[ \forall f : X \to Y \in \mathrm{hom}(\mathcal{C}) [ \exists F(f) : F(X) \to F(Y) \]

\[ \forall X \in \mathbf{ob}(\mathcal{C}) [ F(1_X) = 1_{F(X)} ] \]
\[ \forall f : X \to Y [ \forall g : Y \to Z [ F(g \circ f) = F(g) \circ F(f) ] ] \]
\end{definition}


Most spaces and structures are simply sets with some extra properties imposed. For instance, a topological space is a set with a topology, a measurable space is a set with a $\sigma$-algebra. Perhaps more familiar to us, a group is a set with an associative binary operation with an identity element and with all elements invertible.

We can form a functor from $\mathbf{Grp}$ to $\mathbf{Set}$ that 'forgets' the group structure, functors from $\mathbf{Ring}$ to $\mathbf{Ab}$ that 'forgets' the multiplication operation of its rings! Functors like this are called \emph{forgetful functors}; functors which translate objects and morphisms into a category with less structure.



Contravariant functor
Covariant functor
Faithful functor
Presheaf

Opposite functor
Bifunctor
multifunctor

cones
cocones

