Introductory example(s) of a functor that is full but not faithful

Question

What is your favourite example to offer real beginners of a functor which is full but not faithful?

Answer 1

The example that comes immediately to my mind is the obvious functor from a category $\mathcal{C}$ to its preorder-reflection $\mathcal{D}$, i.e. the category $\mathcal{D}$ whose objects are the same as in $\mathcal{C}$ and with a unique morphism $x \to y$ in $\mathcal{D}$ if and only if there is at least one morphism $x \to y$ in $\mathcal{C}$. It is full by construction, and it is faithful if and only if $\mathcal{C}$ itself is a preorder (in which case $\mathcal{C} \to \mathcal{D}$ is an isomorphism).

There are other constructions in a similar vein. For example, if $\mathcal{H}$ is the category of topological spaces with morphisms modulo homotopy, then the quotient functor $\mathbf{Top} \to \mathcal{H}$ is full but not faithful.

Answer 2

1) Think of monoids as categories with a single object: any homomorphism $f : M \to N$ that is surjective but not an isomorphism is an example of a functor that is full but not faithful.

2) Pick a prime $p$ and let $\mathcal{P}$ be the category whose objects are the $\mathbb{F}_p^n$ and whose morphisms are polynomials in several variables. Let $\mathcal{F}$ be the category whose objects are also the $\mathbb{F}_p^n$ and whose morphisms are all the polynomial functions. There is an obvious functor $\mathcal{P} \to \mathcal{F}$ which is the identity on objects and sends every polynomial to the corresponding polynomial function. Since different polynomials induce the same function, this is not faithful. As stated, it is full by construction; but one could instead define morphisms in $\mathcal{F}$ to be all functions, and then to show that the functor is full is the fun exercise of showing that all function $\mathbb{F}_p^m \to \mathbb{F}_p^n$ can be produced by a polynomial.