by Philip Boxer

Some papers sent to me recently by James D. Smith, II at Carnegie Mellon’s SEI set me thinking about how demand asymmetry might be expressed in Category Theoretic terms. Work by Nick Rossiter et al uses category theory to establish a natural basis for interoperability, comparing functors representing the constructs under which an enterprise organizes its data. This work argues that the conditions for semantic interoperability between systems depend upon their functors^{[1]} commuting at all levels. Heather et al use this to argue the need for a global enterprise:

Modern information systems operate at every level: from data held in a single purpose ﬁxed device, through common PCs at home or mobile computing with business systems of an SME, to databases, intra-acting locally and at national level, inter-acting between nation states and even then open to wider global systems outside of Europe. This interoperability requires global coherence which in synecdoche correlates with the interoperation of the EU itself.

Semantic interoperability defined in this way cannot accommodate the pragmatics of demand that involve going beyond the perimeter of a single sovereign enterprise. The enterprise may be forced to deal with demand asymmetry.

Let us say that an enterprise is an *enterprise context*, defined by the existence of a natural transformation between the functors representing its *sub-contexts* i.e. its functors commute at all levels. These sub-contexts may be further refined by defining sub-sub-contexts etc in terms of sub-functors. In these terms, the *hierarchy* of the enterprise can be expressed in terms of the ‘is-part-of’ relations between the enterprise context and these subordinate functors representing subordinate enterprise contexts.

Let us now say that a *customer situation* (the experiencing of a need) is also defined as a natural transformation between the functors representing its *sub-situations*. These sub-situations may be further refined by defining sub-sub-situations etc in terms of sub-functors. In these terms, an *effects ladder* for the customer situation will be expressed in terms of the relations between the customer situation and its subordinate functors representing subordinate situations.

For example, a hospital is an enterprise, with one of its sub-contexts providing a hip-replacement service. A patient’s condition would then be a customer situation, the need for a hip replacement being one of this condition’s sub-situations.

Now consider that the patient’s condition demands a number of different treatments provided by different specialist organizations – physiotherapy, home nursing, orthoses, home help. This would be a demand higher up the patient’s effects ladder, addressing more of the need arising from their condition (and thereby reducing the patient’s *value deficit*). Let us say that a *geometry-of-use* is a functor composing some number of sub-functors within the hierarchies of some number of different enterprises. A *stratification* can be expressed in terms of the ‘is-used-by’ relations between the geometry-of-use and these sub-functors, representing how these sub-functors are composed.

Which brings us to demand asymmetry. A demand arising from a customer situation is asymmetric for an enterprise if the functor of the geometry-of-use can not be made to commute at all levels with the functors of the enterprise i.e. there is no natural transformation between the functor for the customer situation and that of the enterprise as a whole.^{[2]} It is this demand asymmetry that forces us onto the ground of collaborative systems of systems and beyond.

[1] A functor is a mapping between categories, a category (*aka* abstract category) being a collection of objects that are linked by arrows. Within the PAN toolset, a category in this sense is a complex object defined in terms of dyadic relations (arrows) between the simple objects appearing in a visual PAN model. Value propositions that relate the capabilities of an enterprise to the situations of clients represent particular ways of using those capabilities (*aka* know-how) which are described in PAN as types of restriction placed on the relations between complex objects (*aka* categories). Value propositions are thus themselves functors, the restrictions representing value propositions corresponding to Peircean thirdnesses, making these value propositions *triadic* in nature (see evaluating architectures).

[2] It is when faced by demand asymmetry that the governance of an enterprise has to become horizontally dominant (*aka* East-West dominance). The vertical axis of accountability (*aka* North-South axis) has to create the conditions in which local commutation is present between the geometry-of-use functor and the functors of the enterprise.