Process Category Theory
Michael Heather and Nick Rossiter
University of Northumbria at Newcastle NE1 8ST
http://www.computing.unn.ac.uk/staff/CGNR1
History amply demonstrates that however profound the idea, however great the genius of of its originator, fuller understanding and the benefits of implementation follow in every branch of science and technology when the idea can be represented formally. First it seems the informal stage then to be followed by the formal is the natural order. There can be no better evidence of the truth of this principle for all in due season than from the work of Alfred North Whitehead. For one of the authors of Principia Mathematica then subsequently to proceed to expound the whole philosophy of process with scarcely a mathematical symbol shows not only great self restraint but wisdom that the tools needed to represent process were not then to hand.
However during the course of the second half of the twentieth century there has arrived on the scene the concept of the arrow emerging from within category theory and out of a confluence of algebra, geometry and topology. The arrow is able to represent formally the full glory of the philosophy of process. From its origins in Pure Mathematics with the emphasis of the times wedded to the axiomatic method, category theory has tended to be heavily biased towards the category of sets which stresses Parmedian invariance but the arrow also contains the full richness of Herecleitean variance (both the covariant and the contravariant).
The great detail of Whitehead's philosophy can be abstracted in existing category theory but with an abstraction that loses none of the detail. His 'category of the ultimate' is to be found in the topos. The 'category of existence' is the cartesian closed category which gives not just mathematical existence in the sense of consistency but also justifies physical existence with exponentials and limits. For the arrow is an 'actual entity' satisfying 'an occasion'. The natural transformation as an arrow provides Whitehead's 'category of explanation' and his 'categorical obligation' is the underlying functor. Perhaps most important of all is the monad over adjointness which not only captures every relation of Whitehead's 'category of prehension' but also connects with the perception of Leibniz' monad.
Category theory itself is today itself under attack as without foundations but in fact it has one of the strongest of all philosophies to support it painstakingly worked out by Alfred North Whitehead.