<div class="gmail_extra">The general idea of category theory is to come up with formalizations of common abstract "patterns" found in mathematical constructs. For example, there are homomorphisms of groups, vector spaces (under linear transformations), topological spaces (under continuous functions), etc. Category theory abstracts these into the notions of a "morphism" or an "arrow" (which correspond to the homomorphisms) and "objects" (which correspond to particular spaces of a particular kind). Objects and morphisms are (globally) taken to be undefined primitives, much how sets are undefined primitives in set theory.</div>
<div class="gmail_extra"><br></div><div class="gmail_extra">With the notion of a category in place, we can begin constructing arbitrary categories, or proving that algebras satisfy the category axioms. It is possible to define mappings between categories -- suitably nice mappings are called "functors", and are morphisms in the (small) Category of Categories.</div>
<div class="gmail_extra"><br></div><div class="gmail_extra">The interesting thing about category theory is how rich it can be. It had its origins in algebraic topology, but algebraic topology has rich connections with the theory of lattices and order, set theory, logic, group theory, computation, etc. For example, topological nets and filters can be recast as categories, and limits in these categories generalize to the categorical notion of a limit. Similarly, a topology's "closure operator" can be recast in terms of monads -- 'join' witnesses that a monad is idempotent, in the sense that an (m a) and an (m (m a)) are the same "kind" of structure.</div>
<div class="gmail_extra"><br></div><div class="gmail_extra">Check out Goldblatt's "Topoi - A Categorical Analysis of Logic". It focuses specifically on modeling logics using categorical methods, and the preliminary chapters are a fine introduction to the basics.</div>
<div class="gmail_extra"><br></div><div class="gmail_extra">I'm sure you can turn that into a semester course. </div>