http://www.cs.ox.ac.uk/people/bob.coecke/AbrNikos.pdf
Why study categories—what are they good for? We can offer a range of answers for readers coming from different backgrounds:
- For mathematicians: category theory organises your previous mathematical experience in a new and powerful way, revealing new connections and structure, and allows you to “think bigger thoughts”.
- For computer scientists: category theory gives a precise handle on important notions such as compositionality, abstraction, representationindependence, genericity and more. Otherwise put, it provides the fundamental mathematical structures underpinning many key programming concepts.
- For logicians: category theory gives a syntax-independent view of the fundamental structures of logic, and opens up new kinds of models and interpretations.
- For philosophers: category theory opens up a fresh approach to structuralist foundations of mathematics and science; and an alternative to the traditional focus on set theory
- For physicists: category theory offers new ways of formulating physical theories in a structural form. There have inter alia been some striking recent applications to quantum information and computation.
http://www.cs.man.ac.uk/~hsimmons/zCATS.pdf
These simple examples tend to give the impression that in any category
an object is a structured set and an arrow is a function of a certain
kind. This is a false impression, and in Section 1.3 we look at some
examples to illustrate this. In particular, these examples show that an
arrow need not be a function (of the kind you first thought of).
An important messages of category theory is that the more important
part of a category is not its objects but the way these are compared, its
arrows. Given this we might expect that a category is named after its
arrows. For historical reasons this often doesn’t happen