The foundations of programming language theory

As a long-time functional programming adherent, I'm interested in the
mathematical foundations of programming language theory. I've been
learning Haskell for a while now and it's a great language, but
there's a lot of fundamental mathematical theory underlying the design
of Haskell that is a bit harder to grasp without the background. I'm
currently reading Barendregt's extremely terse book "The Lambda
Calculus" and I also picked up "Types and Programming Languages" by
Pierce which I haven't started yet. Anyway, things like Monads,
arrows, and such make a lot more sense now. I was trying to think of
what the "big picture" here is and I came up with this relation****p:

programming language theory <---> type theory
          |                           |
          +---------------------------+
                                      |
lambda calculus <---------------------+
 |                                    |
 +--> category theory <---------------+
 |
 +--> group theory  |
           |        |
           +--------+---> formal logic? number theory?

The arrows here can be interpreted (for example) as lambda calculus
"derives from" or "is based upon" category theory in some (possibly
partial) sense. How accurate is this to reality? Should set theory be
in there somewhere? I believe type theory derives from naive set
theory, but i believe that set theory is a subset of category theory.