I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So far I've seen Lawvere Theories show up a lot.] What's the best way to learn them both?
[EDIT 2: What books (if any) introduce them both in detail?]
Here's my motivation: I'm fascinated by how different areas of Mathematics (and systems of Logic), particularly in Abstract Algebra, relate to one another and by how crucial certain axioms are in those relationships. I know this is quite vague but I've been chasing this stuff around for about a year; I know what it is I'm after when I see it, but I'm not knowledgeable enough (yet) to pin it down precisely. Category Theory & Model Theory (as well as Non-classical Logic) seem to hit the spot frequently.
EDIT: To make things easier/more precise, think of a kind of "mathematical KerPlunk," where the sticks are axioms and the marbles make up a (system of logic or) mathematical structure (with, say, red marbles for theorems, blue for definitions, etc.). If you remove (or change) certain sticks, what falls and why? Which marbles move? Do any change colour? Compare what you get with what you started with. How does the 'new' structure fit into the bigger picture? What are its 'neighbouring structures'? That's the kind of thing I'm interested in.
EDIT 3: Reverse Mathematics looks highly relevant but it's new to me. Is there any way I could get there via CT & MT?
EDIT 4: Suggestions on how to improve this question are welcome. I think Universal Algebra is relevant but I've replaced its tag with the Topos Theory one to narrow things down.