Northumbria University, UK
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:
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;
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.
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).