Axiomatic Definition of a Category
I never learned much logic or set theory, but I am trying to learn a little bit about them and also about the categorial foundations of mathematics (for instance, as in "Topoi" by Robert Goldblatt). Initially, I had the thought that many people do upon hearing that categories could be used in place of sets: if categories are formed from two SETS, whose members are called objects and arrows, then how could using categories as a foundation not be circular, requiring a formal description of sets before defining categories?
I now understand that"objects" are primitive in each system, so the above is really a non-issue. I also understand vaguely that category theory avoids allowing the "set membership" operation in the set-theoretic foundations as being a primitive concept in the categorial-theoretic foundations. I have two questions:
1) Say you define "monic" arrows in category theory. Is one simply not allowed to take some arrow and ask, "is it monic"? This would seem to be about membership: there is a collection of arrows called "monic", and I want to know if a particular arrow is in it or not. Is the answer simply that one is only allowed to ask of a particular arrow, "does it satisfy the property of being monic", but one is not allowed to consider the notion of "the collection of all monic arrows"? It seems totally artificial to me to be allowed to consider "the collection of arrows" as a primitive concept but not to be able to consider "the collection of all monic arrows" once the definition of monic has been made.
2) On p.24 of Goldblatt, for instance, he gives an axiomatic definition of category, introducing collections of things called objects and arrows. He then goes on to say that we assume there are "operations assigning to each arrow f an object dom f and an object cod f". Are we to assume that "operations" and "assignments" like this are primitive? Is there some more precise way to axiomatize "operation" than this? I think I would feel more comfortable with "set membership" as a primitive notion than with these "operations".
No, an arrow is monic if you can prove it has the property of being monic. That isn't about membership. In set theory, a set is infinite if it has certain properties, but there is no set of all infinite sets.
In set theory, there is only one type of object, the set. There are rules and relationships between these objects that are not, however, representable as sets. Indeed, in set theory, there is no set of all functions.
In category theory, we have two types, objects and arrows. We have primitive relationships between these objects, namely, for each arrow there is exactly one "left side" object and exactly one "right side" object. So the property "$x$ is the left side object of arrow $f$" and "$y$ is the right side object of arrow $f$" are primitive in the same way that in set theory, $x\in y$ is a primitive relationship.
You are sort of correct that you only prove individual arrows are monic, but you usually prove that under some condition, there exists a monic arrow from $x$ to $y$.
For example, you say an object $x$ is an initial object in a category if, for every $y$, there is exactly one arrow from $x$ to $y$. (Exactly one? Uniqueness can be stated without set theory, because you can state it as: If $f$ and $g$ are arrows from $x$ to $y$, then $f=g$.) After that definition, you can easily prove that if $x$ is an initial object, and $f$ is an arrow from $x$, then $f$ is monic. So we aren't really proving it for a specific $f$, only for an $f$ which exists in a wider context.
Or you might add an axiom to your category: If $f$ is an arrow from $x$ to $y$, there exists an epic arrow $p$ from $x$ and a monic arrow $i$ to $y$ such that $i\circ p$ is defined and $f=i\circ p$. (Your terminology might differ depending on how you write composition of arrows.)
As a general rule, in category theory, you don't talk about the collection of all arrows or objects, you just state properties about them and about diagrams.