Category Theory
Concept
I like to think about Category Theory as mathematics done by someone who likes talking about transformations and has a dealthy fear of ever talking about elements. This means they are happy to talk about sets and functions, but never elements of a set; groups and group homomorphisms, but never elements of a group; topologies and their homomorphisms, but, again, never elements; and so on.
Our hypothetical mathematician enjoys talking about all the same subjects as the rest of us (meaning, of course, algebra, topology, and some logic), but utilizes a radically different vocabulary than the usual mathematician.
For instance, while most of us would construct a product of sets as a set of pairs:
our hypothetical friend would instead classify the product as the set paired with functions 1 such that for any other object paired with functions there exists a unique function so that and .
These are the projections
but it is not necessary to know or say this in order to define or use the set product.
If this seems far more complex than the explicit construction, that’s because it is.
This complexity is not without payoff. On the contrary, something sort of magical happens. If we ask our hypothetical friend for the group product construction, they need only point to the definition they already gave.
Antics of their wordlessness aside, it’s true that if we take their definition of set product and replace “set” with “group” and “function” with “group homomorphism”, we somewhat amazingly obtain a valid classification of group products.
And if we use “topological space” and “continuous function”, we obtain a classification of a product of topological spaces. And if we use “ring” and “ring homomorphism”, we obtain a classification of products of rings. And ...
The language of Category Theory, exactly because it neglects to talk about elements directly, ends up incredibly general. Its constructions and theorems apply, essentially without modification, accross fields of mathematics.