A few conceptual questions about category theory. I have the following questions about the basic of Category Theory:


*

*Say I have a category $C$ with objects $\{A_1,A_2,\dots\}$. What exactly are "points"? Are they the elements present inside each object? Are they the object themselves? 

*Why do "points" have to be mappings $1\to C$? Or are they mappings from $1 \to A_k$? Say I have a point $x$. Why can't I simply determine $f(x)$? Why do I have to write $$f\circ x:1\to\text{ co-domain of $f$}$$

*$1_{A_1}$ is the identity morphism of $A_1$. If $A_1$ is not a set but a single element, it is simple enough to define $1_{A_1}$. But what if it is a set? Is $1_{A_1}$ then the identity mapping on $A_1$? Or is it any bijective endomorphism?

*Let $f\in \text{Hom}(A,B)$. Also, let $g,k\in \text{Hom}(B,A)$ such that both $g$ and $k$ are inverses of $f$. Prove that $g=k$. I know how to go about this when I know $A$ and $B$ are sets. What kinds of arguments are valid when $A$ and $B$ are just objects?
Thanks in advance!
 A: For (3,4), you need to pay attention to the fact that categories are rather algebraic in nature: both of these turn out to be fairly simple questions of arithmetic, without having to invoke any sort of analogy between arrows in categories and functions of sets. For (4), the usual trick from group theory to prove the same thing works: simplify $gfk$ in two different ways.
For (1,2), there is a useful notion of "generalized element": any arrow whose codomain is $X$ can be usefully thought of as some sort of element of $X$. One can develop the internal language of a category in a way that resembles set theory: some examples are:


*

*write $x \in X$ to mean "$x$ is a generalized element of $X$",

*if $f : X \to Y$, then write $f(x)$ to mean $f \circ x$, interpreted as a generalized element

*

*and note $f(x) \in Y$ as you might expect


*if $\mathcal{S} \subseteq X$ is a subobject of $X$, represented by some arrow $s : S \to X$, and if $x \in X$, then we say $x \in S$ if and only if we can find $f$ so that $x = sf$.


The nicer a category is, the more we can make the internal language look like the language of set theory. In a Cartesian category, we can form ordered pairs and solution sets to equations. In a Cartesian closed category, we can do typed lambda calculus. The internal language of a topos turns out to be a kind of intuitionistic set theory! (and a nice topos gives us set theory with classical logic!)
The generalized elements with domain $1$ are often called "global elements" (the terminology relates to a geometric interpretation of toposes). I don't think I've heard "point" used as a general term for them before.
The identity morphism $1_X$, when interpreted as a generalized element, does a very good job of capturing the notion of a "generic element of $X$", or of an "indeterminate element of $X$".
In various specific categories, we can find a notion naturally isomorphic to the set-theoretic notion of set. e.g.


*

*In $\mathbf{Set}$, $X$ is naturally isomorphic to $\hom(1, X)$.

*In $\mathbf{Top}$, the set of points of $X$ is naturally isomorphic to $\hom(1, X)$, where $1$ is the space with one point.

*In $\mathbf{cRing}$, the category of commutative rings, $|R|$ (the set of elements of a ring $R$) is naturally isomorphic to $\hom(\mathbb{Z}[T], R)$. 


There is a useful notion of a "separator". e.g. in $\mathbf{cRing}$, if you have two functions $f,g : X \to Y$ and $f \neq g$, then you can find a map $x: \mathbb{Z}[T] \to X$ such that $f \circ x \neq g \circ x$. We can think of this as saying that we can distinguish unequal functions by considering their "values" on the generalized elements of $X$ coming from $\mathbb{Z}[T]$.
Not every category has a single object that can play this role: sometimes you need sets (or even proper classes) of objects to manage this feat. The class of all objects is always a separator, of course, since the generic element of $X$ will satisfy $f \circ 1_X \neq g \circ 1_X$.
A: Usually the objects are sets with same structure an the arrows are {homo,homeo,diffeo,...}morphisms. But this isn't required. The "points" are maps $1\to A_k$ (with 1 "like" a singleton in some sense) because you only have objects and arrows. In the usual cases, $1_A$ is the identity map, but again isn't required that $A$ be a set.
See Examples of categories where morphisms are not functions.
