*Northumbria University, UK*

http://computing.unn.ac.uk/staff/CGNR1/

Category Theory as a creature of its time in the mid-twentieth century was developed within the framework of axiomatic set theory. The majority of most standard texts on the subject of Category Theory concentrate on the category of sets. This may well be the main interest for pure mathematicians but the power of Category Theory for applied mathematics extends well beyond the bounds of set theory to problems of the real world that set theory can only model partially or very inadequately. Database theory is an example where even a very pure implementation of set theory in terms of Codd's Relational Model cannot do justice to real world applications of any complexity [Heather & Rossiter 2007]. For reliability and consistency formal representations of natural phenomena need compositional closure, which the work of Gödel has shown cannot be guaranteed in set theory. Many current scientific problems require higher order solutions where reliance on first order models can be not only misleading but possibly dangerous.

Whitehead and Russell who had such an influence in
establishing the prominence of set theory by publishing *Principia Mathematica *both moved on to categorial
way of thinking. Russell produced his advanced type theory and Whitehead his
philosophy of *process*.
Unfortunately both were too early to see the fulfilment of their work in
formal categories. Russell's types only had the tools of set theory and could
never rise above it: Whitehead's *process *however can be elegantly and
(it is possible) exhaustively captured by the *arrow *of category theory
[Heather & Rossiter 2006] but this is still yet to be fully exploited. As
the fundamental nature of *process *is open and dynamic it requires large cartesian closed categories and corresponds to Topos Theory
without the usual additional assumption of the category of natural number
objects.

Consequently applied categories might be considered under the following heads:

1.
__Categorification__

This method reworks an existing or potential set theoretic model to present it in the language of the category of sets. This can only be cosmetic as there will still be the limitations of axiomatic or possibly naïve set theory. There may be advantages of style that can give better insight than the corresponding set theory model from which it is derived. However any advantage may be illusory. If there is no added value it is often easier just to use set theory in these cases;

2.
__N-categories__

The interest in pure Category Theory is currently in higher order categories but as sets and number are comparable the same qualifications operate as for categorification;

3.
__The Category Theory of Models__

By the use of adjunctions in large categories it is possible to classify any first order model, whether constructed using sets or the category of sets, according to its power of modelling higher order natural phenomena in respect of the creativity of the model (the unit of adjointness) or its quality (co-unit of adjointness).

4.
__Applied Topos Theory__

Whitehead's interpretation of the World as *process*
gives an experimental result for the meaning of the categorial
arrow to replace the mathematical axioms of the category of sets. The Universe
is also experimentally found to have the properties of limits and exponentials
which define a topos. Applied Topos Theory does not
therefore derive from mathematics but from physics.

References:

Heather, Michael, & Rossiter, Nick, Process Category Theory, Salzburg International Whitehead Conference, University of Salzburg 3-6 July (2006).

Heather, Michael, Livingstone, David, & Rossiter, Nick, Higher Order Logic and Interoperability in Global Enterprise, AREIN, I-ESA, Madeira, 26-30 March 12pp (2007).