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://emorehouse.web.wesleyan.edu/research/notes/intro_categorical_semantics.pdf
Remark 1.1.0.5 (unbiased presentation) There is an equivalent presentation of
categories in terms of unbiased composition. There, instead of a single binary
composition operation acting on a compatible pair of arrows, we have a lengthindexed
composition operation for paths of arrows (still with unit and associative
laws). In this presentation, an identity morphism is a nullary composition, a
morphism itself is a unary composition, and in general, any length n path of
arrows has a unique composite. Although more cumbersome to axiomatize, an
unbiased presentation of categories makes it easier to appreciate the idea at the
heart of the definition: every composable configuration of things should have a
unique composite.