Category Theory and Business Modelling

1. Category Theory and Business Modelling Modelling the change of business structure

2. Organisations Change • Any concept of business has elements of structure and elements of change • The structure can be as simple as roles and the change is the change of staff • If the role is complex then it cannot be staffed by people who leave after a short time (e.g a production manager versus a supermarket checkout clerk. ) • The change of roles is what will concern us here • Example. • A company starts off making telephone parts. HR just means payroll. • The company moves into making mobile telephone parts. HR is charged with training. • The company expands. The work force has to become more technical. HR’s now works with manufacturing to prepare workers in strategic technology changes. • New management outsources much of the manufacture and HR is back to payroll and hiring and firing.

3. How organisations change • The structure of an organisation starts with groups of people. • These people receive information or materials. • They act on this information or these materials and transform them. • They pass the newly transformed information or materials to other groups. • As organisations flourish or decline the actual groups or people that carry out these actions can multiple and specialise (as the volume of work increases) or coalesce (as the volume decreases). This is a transformation.

4. The structure of operations A formal model of an organisation, denoted A, is • a set A of entities: people or groups or even machinery. • Each entity a of A deals with types of information or materials which we write as Typ(a) • Each a of A applies their expertise to transform inputs to outputs. This gives a set of activities W(a) (W for work) • The result of the work is an “action” (a): W(a) x Typ(a)Typ(a) (that is (a)(w,t) is the type of a that results from doing w on materials or information of type t). • a gives his or her or its output to others in a set S+(a) Awhich is the set of “successors” of a. • This is the operational skeleton for our organisation. The details are in the Typ(a) and W(a) • The same structure applies to small factories or (say) accounting firms to global supply chains.

5. Introduction to Category Theory. 1 • Category theory is built on simple concepts: Objects and their transformations. • An object can be anything you like but has transformations that can be performed on it. • The transformations are called morphisms. • Standard examples: • The category of set: Objects are just sets, morphisms are just functions between sets. • The category of vector spaces. Objects are vector spaces, morphisms are linear maps or matrices. • Faces: Objects are pictures of faces. Morphism are distortions. • Axiom: morphisms compose. One distortion followed by another is still a distortion.

6. Introduction to Category Theory. 2 The Category of interest is the category of organisations. We have already described its objects. What are its morphisms? • Morphisms have few constraints beyond that one followed by another is still a morphism. • In general an organisations can get more complicated or it can simplify. • More complicated W(a) and Typ(a) can get bigger and possible split: W(a), Typ(a) get included in a bigger W’(a) and Typ’(a) or even a itself splits into {a1, a2}. • If a morphism is allowed so is its reverse. This allows organisations to grow in some parts and shrink in others. • This gives us a category of organisations (Omitting a few nasty details).

7. Introduction to Category TheoryExample 2. The category of General Ledgers. • Each object is a chart of accounts (for some organisation) denoted as GL (the general ledger). • There are many morphisms f(x):GLGL. Each f (x) updates a chosen set of accounts with an amount x (which might be a complex allocation of amounts). • If g(y):GLGL then we can define f(x) o g(y) and possibly g(y) o f(x). We might have a rule f(-x) o f(x) = 1 That is a reversal. 1 is a “do nothing” morphism and all objects in any category have them. • (This is a special kind of category called a Groupoid)

8. Functors. • Category theory is also called the theory of functors. • Functors are maps between categories. • Categories themselves form categories. The objects are categories, the morphisms are called functors. Let C1 and C2 be categories, a functor takes • The objects of C1 to the objects of C2. • The morphisms of C1 to the morphisms of C2 and preserves composition. • Example. C1 is our category of organisations, C2 is the category of general ledger systems. • Each organisation O in C1 is mapped to its corresponding GL in C2 and a morphism in C1 that changes an organisations (say introducing specialist areas) will have a corresponding change (morphism) in the category of the general ledgers. For example there could be more morphisms of the object GL or some might become obsolete or the GL itself has more or less or different accounts.

9. Category Theory and Supply Chains • The association of one organisational structure with another is the guts of category theory in organisations. This allows us to represent the relationships among structures. • This is why the project to study the operational and expertise structure of supply chains will use these concepts. • Organisations as sketched above are a simple template for a generalisation call combinatorial modules (because organisations are usually about combining things) which are the start of supply chain modelling. • The core of the research is to define the associate category of expertise and the functor from supply chains to their associated expertise. (The  above is only a small part of the story). • And then, of course, examine its properties and see if it stacks up in the real world.

10. Dynamics Structures: models of organisations Dynamics Structures are categories for which • Each object is associated with a set of values for which its current structure is designed. The set of values is common to all the objects and is called the support space. • The values in the support space can be any or all of : tax rates, market share, transport rates, labour rates, interest rates, foreign exchange rates and so on. These are economic environmental settings of the object. • When the economic environment changes the actual values in the support space change. If they go outside the support space of an object the object has to change (adapt). • Each change (so each morphism) has a measure called the resistance. • Changes of least resistance can provide support values consistent with the current economic environment but these need not be sustainable. • This was originally introduced for ecosystems. It is useful for organisations including supply chains. (ref. Dynamic Structures. An Algebraic Approach to Biological and Social Structures. Bulletin of Mathematical Biology, 43, 5, pp579-591,1981 or http://integratedexpertise.org/web_documents/dynamic_structure_theory_a_structural_approach_to_social_and_biological_systems.pdf

11. Summary • Category theory is a powerful but simple concept that is relevant to modelling organisational structure. • Its power and flexibility come from the association of one type of structure with another via a functor. • It has yet to be exploited in the study of organisations. • It will be a significant part in the study of supply chains • Category theory is now the Linga franca of the most powerful constructions in modern mathematics.