TITLE: Elements and arrows in a category.
QUESTION [11 upvotes]: Suppose to have two objects $A$, $B$ in a fixed category and an arrow $\eta : A \to B$.

Has an object "elements"? In the sense does the symbol $a \in A$ have sense? (in the most generic context, without any other assumptions). If "yes", why? What does it mean?
If "yes", can we say that if $a \in A$, then $\eta(a) \in B$?
If "yes" again, does an arrow have the same property of a classical function? ("every element in the domain has only one image")

I am pretty sure that all of my statement are completely wrong (just by the definition of category - we are not supposing anything about "elements"), but I should feel more sure if you could help me anyway.
Thank you very much in advance.
Cheers

REPLY [3 votes]: A category does have elements in a suitable sense. First, let's review some examples of the traditional notion of elements:

There is a natural bijection $S \cong \hom_{\mathbf{Set}}(\mathbf{1}, S)$
If $|X|$ is the set of points of a topological space, there is a natural bijection $|X| \cong \hom_{\mathbf{Top}}(\mathbf{1}, X)$
If $|G|$ is the underlying set of a group, there is a natural bijection $|G| \cong \hom_{\mathbf{Grp}}(\mathbf{Z}, G)$
If $|R|$ is the underlying set of a commutative ring, there is a natural bijection $|R| \cong \hom_{\mathbf{CRng}}(\mathbf{Z}[x], R)$
If $\Gamma : E \to B$ is a bundle, $U \subseteq B$ is an open set, and $\Gamma_U$ is the set of sections of $\Gamma$ over $U$, then there is a natural bijection $\Gamma_U \cong \hom_{\mathbf{Bundle}/B}(U, \Gamma)$
If $X$ is a scheme, $P$ is a scheme, and $X(P)$ is the set of $P$-points of $X$, then there is an equality $X(P) = \hom_{\mathbf{Sch}}(P, X)$.

All of these together suggest a useful notion of generalized element: a generalized element of an object $A$ is simply an arrow with target $A$. You can develop a whole internal logic in a category, and familiar facts often take on familiar forms. This logic becomes more and more familiar the nicer the category is -- Cartesian categories, Cartesian closed categories, and toposes in particular.
e.g. $f(x)$ means the (element represented by the) composite of the function $f$ with the (arrow representing the) element $x$. The arrow $f : X \to Y$ is monic if and only if, for all elements $x,y \in X$, you have $f(x) = f(y) \implies x = y$. (here, evaluation of an element is simply composition)
We can even take this further: if $\mathscr{C}$ is a small category, then there is a useful functor $\mathscr{C} \to \mathbf{Set}$ that sends every object $A$ to the set of all arrows with codomain $A$. So we can even treat the generalized elements like elements in the set-theoretic sense! This is closely related with the Yoneda embedding and with the notion of a category acting on a set.

In my opinion, this notion remains useful even outside of the context of category theory. One (IMO) pretty example has to do with choice. Every time you see an argument that includes a set "choose an $x \in S$" you can instead let $x$ be the generic generalized element of $S$ (i.e. the one corresponding to the identity function).
For example, consider the proof by transfinite induction that every vector space has a basis. Reinterpreting "choose a well ordering on $V$" in this way instantly converts the whole argument into a constructive derivation of a function from the set of well-orderings on $V$ to the set of bases for $V$. And one that works whether or not you assume the axiom of choice! (naturally, it's vacuous should the set of well-orderings on $V$ be empty)