Functor of points and category theory

Question

I am trying to read the section on Functor of points from Eisenbud - Harris (and I also referred to Mumford's book). They all motivate functor of points this way :

In general, for any object $Z$ of a category $\mathcal{X}$, the association $X\mapsto\textrm{Hom}_\mathcal{X}(Z,X)$ defines a functor $\phi$ from the category $\mathcal{X}$ to the category of sets. (We wish to identify $\textrm{Hom}_\mathcal{X}(Z,X)$ with the point set $X$).

But the book says that it is not satisfactory to call the set $\textrm{Hom}_\mathcal{X}(Z,X)$, the set of points of $X$ unless this functor $\phi$ is faithful.

I don't understand this statement. If $\phi$ is not faithful, they have given an example in the case of category of $CW$-complexes, where $\textrm{Hom}_\mathcal{X}(Z,X)$ cannot be identified with $X$. But I am not able to understand why $\textrm{Hom}_\mathcal{X}(Z,X)$ can b identified with $X$ if $\phi$ is faithful.

If $\phi$ is faithful, then there is an injection from $\textrm{Hom}_\mathcal{X}(Z,X)\longrightarrow\textrm{Hom}_{Sets}(\textrm{Hom}_\mathcal{X}(Z,Z), \textrm{Hom}_\mathcal{X}(Z,X))$. But what does this tell us? Any help will be appreciated!

Answer 1

If you're a traditional kind of mathematician, you like your categories to have objects that can be described as sets with extra structure and your morphisms to be functions that preserve that structure. This is codified in the notion of a concrete category, namely a category $C$ equipped with a faithful functor $C \to \text{Set}$ sending each object to its underlying set. The point of faithfulness is that it codifies the idea that your morphisms can be identified with functions on underlying sets.

In the context of algebraic geometry, for example, the Nullstellensatz implies that the functor sending an affine variety over an algebraically closed field $k$ to its set of points over $k$ (that is, $X \mapsto \text{Hom}(\text{Spec } k, X)$) is faithful. Thus we can talk about affine varieties over algebraically closed fields as if they are sets of points, and as if morphisms between them are functions between those sets of points.

This fails badly as soon as $k$ is not algebraically closed; for example, if $k = \mathbb{R}$, then even an innocent variety like $\text{Spec } \mathbb{R}[x]/(x^2 + 1)$ already fails to have any points over $\mathbb{R}$, so the $\mathbb{R}$-points functor can't possibly distinguish any morphisms into or out of this object.

Answer 2

I may as well make this an answer; a functor $\hom_{\mathcal{X}}(Z,-)$ is faithful iff $Z$ is a generator of $\mathcal{X}$. Saying that you can "identify" an object $A$ with $\hom(Z,A)$ is perhaps a bit of an abuse of language; the idea is that the morphisms $Z\to A$ tell you everything about $A$ in $\mathcal{X}$. It also gives you an "underlying functor" that turns your category into a concrete category over $\mathbf{Set}$, so you can identify $\hom(Z,A)$ with $A$ in roughly the same sense you can identify a group with $\hom_{\mathbf{Group}}(\mathbb{Z},G)$. It gives you a way to view the abstract objects of the category as structured sets, and gives a nice canonical choice of set.

I'm not familiar with CW complexes, so I can't say whether there's some consideration more specific to that category. But I think these are general considerations for why you might think of an object as identifiable with its image under a faithful hom hom functor. Hopefully this made sense, let me know if I should expand on anything.