I am confused about the definition of a category given in the Wikipedia article on Category theory:
It seems to me that the structure being described (the "arrows" between objects in some class) is just a binary relation that is both reflexive and transitive. If so, what is meant by the set of all morphisms (arrows) from one object to another? The definition says that every morphism (arrow) has a unique source object and target object. Is there not then at most one arrow from one object to another?
I am looking for formal axioms of category theory expressed only in the notation of first-order logic and set theory -- no words, just variables, functions, logical connectors, quantifiers, predicates, '=' signs, '$\in$', etc.
How about: class $ob$ is a category iff...
$\forall a,b \in ob\exists hom \forall f(f\in hom \leftrightarrow \forall d\in a(f(d)\in b)) $
$\forall a\in ob\exists i \forall b\in a(i(b)=b)$
Note that the required properties of composition are a direct result of functionality of each morphism.
I must be trying the patience of the moderators here. Sorry guys! This has turned into a rather open-ended discussion. To be continued at the sci.logic and sci.math newsgroups:
I'm not sure this is right either, but you might consider the following.
I define the Category and Arrow predicates as follows:
$\forall x (Category(x) \leftrightarrow \forall y\in x \exists f \forall z\in y (f(z)=z))$
This would be redundant if the elements of x were all sets because you can prove the existence of an identity function on every set.
$\forall x\forall a,b\in x (Arrow(x,a,b) \leftrightarrow a\in x \land b\in x \land \exists f \forall c\in a (f(c)\in b) \land \forall c\in b \rightarrow \exists d\in a (f(d)=c)$
It is then easy to prove that the Arrow relation is reflexive:
$\forall x (Category(x) \rightarrow \forall a\in x Arrow(x,a,a))$
And that the Arrow relation is transitive:
$\forall x (Category(x)\rightarrow \forall a,b,c\in x (Arrow(x,a,b) \land Arrow(x,b,c) \rightarrow Arrow(x,a,c)))$
Thanks all for your help.