I've been thinking a lot about functorial semantics lately - it's probably one of the best ideas in category theory. The idea is relatively simple just that a functor

C -> D represents the structure of C being imposed in D. 1/n

C -> D represents the structure of C being imposed in D. 1/n

Lawvere used this idea in his thesis to revolutionize universal algebra. Universal algebra aims to provide a common framework for all the sorts of gadgets that algebraists study e.g. groups, rings, fields, modules, etc. Lawvere identified a class of categories 2/n

called Lawvere theories representing algebraic gadgets. For a Lawvere theory Q, a finite product preserving functor Q -> C is the same as an instance of the gadget Q represents in C. For example if Q is the Lawvere theory of groups, a finite product preserving functor 3/n