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.
https://books.google.co.uk/books?id=6B9MDgAAQBAJ&pg=PA62&lpg=PA62&dq=%22definite+article%22+%22category+theory%22&source=bl&ots=AuNUdpyn3Z&sig=cIkBBe7o4hD4TtX0EaT5Hq758ZE&hl=en&sa=X&ved=0ahUKEwidwb-R-4vXAhVEaVAKHSLZCnUQ6AEIJjAA#v=onepage&q=%22definite%20article%22%20%22category%20theory%22&f=false
There may be many isomorphisms between the objects x and y appearing in the proof of ..., but there is a unique natural isomorphism commuting with the chosen representation. On account of this, one typically refers to /the/ representing object of a representable functor. Category theorists often use the definite article "the" in contexts where the object in question is well-defined up to canonical isomorphism.
https://ncatlab.org/nlab/show/generalized+the
https://byorgey.wordpress.com/2014/05/13/unique-isomorphism-and-generalized-the/