Henson,

just a few summer words
about categories and Set theory.

1) Consider a very subtle
category, may be a topos. Let's forget about composition for a moment - we get
digraph.

Formal theory of digraphs
itself is very interesting as it does not have an axioms at all.

We just need signature
<Sorts ob, ar; Functions dom ar:ob, cod ar:ob>.

Well, we keep in mind that
dom, cod are a full functions, but it's usual in math.

Unfortunately this
important theory does not mentioned in great list

2) Goldblatt has mentioned
somewhere in his book that we may look at ∈ as an arrow from element to set.

What Set theory does study
then? Huge digraph:-)

It's more or less
simple:

- there is no
loops and cycles,

- there is no parallel
arrows,

- there are sources (∅,
pra-elements);

- there are sinks -
classes;

...

But how many outcome arrows
does have not a class? It's greater than any cardinal number.

Alex