# Faithful functors vs forgetful functors: exact category-theoretic defs?

In category theory, a functor between two categories $C,D$ is a map $F$ that assigns to each object (resp. morphism) $x$ of $C$ a corresponding object (resp. morphism) $F(x)$ of $D$ by respecting the incidence relations.

For each pair of objects $x,y$ of $C$, we may then define a map $F_{x,y}$ that takes any morphism $m : x \rightarrow y$ to a morphism $F(m) : F(x) \rightarrow F(y)$. I understand that $F$ is called faithful if every such mapping $F_{x,y}$ is injective, which means intuitively that the relational structure of the category is preserved, although the objects may not be.

There is a related notion of forgetful functor for which I couldn't find a precise definition, so is there anyone willing to help? I mean, is it just the opposite of faithful or is it the combination of unfaithfulness with some other implicit property?

• See page 148 of Awodey (second edition) for a discussion that distinguishes functors that are forgetful, faithful, and injective on arrows. – András Salamon Apr 17 '14 at 13:39

## 1 Answer

you should have a look at http://ncatlab.org/nlab/show/forgetful+functor. As the nLab article says, in its common usage there is no precise definition of forgetful functor, though there are some typical examples (such as a functor $U : Alg \to Set$ forgetting some kind of algebraic structure, which will typically have a left adjoint $F : Set \to Alg$ corresponding to the construction of free algebras). And in another (useful) sense, every functor may be regarded as a forgetful functor, and classified according to how much it forgets. There is no requirement that a functor be unfaithful for it to be called "forgetful", and many common examples of forgetful functors are faithful (such as the forgetful functor $Grp \to Set$ from groups to their underlying sets), while others are not (such as in general the functor $cod : C^\to \to C$ from the arrow category of $C$, forgetting everything about an arrow except its codomain).