Category Theory
Category Theory, as originally developed, is mathematics
applied to the practice of mathematics itself. The concept of
a category
(not to be
confused with the colloquial usage) models a particular
mathematical discourse. Hence we have the category of
sets
, modeling set theory, the category of groups
,
modeling group theory, the category of vector spaces
,
modeling linear algebra, etc. This allows us to reason
precisely about mathematics itself, and in particular about the
relationships between different mathematical discourses, using
the concepts of functor
and natural
transformation
.
We had to discover the notion of a natural transformation. That in turn forced us to look at functors, which in turn made us look at categories.
—Saunders Mac Lane, in The Work of Samuel
Eilenberg in Topology
, 1976.
Applied Category Theory
If category theory is mathematics applied to the practice of
mathematics, then it is already applied, by definition.
So applied category theory
is somewhat of a misnomer.
What it really means is category theory applied outside
of mathematics. The strength of this approach is that
central concepts in mathematics, such as number, function, and
equality are no longer taken for granted — instead, the
meta-mathematics of category theory is utilized to create
bespoke abstractions which perfectly fit the target domain,
insofar as it can be described by thought-patterns that are
stable enough that we can work with them collectively
(see David Spivak's definition of math).