**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.