A category theory approach to HCI

ion. Category Theory for HCI David England 2 2. CATEGORY THEORY Our argument in this paper is that Category theory is a prime candidate for the formal modelling of interaction. At its simplest Category theory is the mapping of one group of mathematical objects to another group. It is a general mathematical theory of structures and of systems of structures [Lawwere 2009]. As such it is said to supersede Set Theory as a general approach to mathematical modeling. Category theory models the mapping of one system to another. The power behind the mapping is the general set of rules that can be used to prove the correctness of the mapping regardless of the type of the Category under consideration. We have already since the value of Category Theory in Computing in the guise of Functional Programming [Bird 1998]. Here category theory is used to establish the correctness of type systems to ensure that programmes behave as expected. However, Functional Programming treats input and output (i.e. all Human Interaction!) as a side effect in the formal system. As such, HCI is said to be outside the formal system being modeled. Given the massive growth in interactive and social computing, leaving the majority of a system outside the formal model is a major restriction of the functional approach. The answer to this problem is to go back to the mapping power of Category Theory. If we consider HCI to be primarily a mapping problem (between and through the artificial system to users), we can begin put these “side effects” to the fore. 3. A TUTORIAL EXAMPLE For our first example let us consider the apparently simple case of a screen layout. We are going to assume that the quality of the screen layout is a function of Fitts’ Law and Signal Detection theory. Optimising for Fitt’s law means the interactive elements are grouped to minimise input device movement and error, whereas Signal Detection theory tells us to optimise the signal elements (foreground) of the screen over noise elements (background and decoration). For simplest of this example let us assume all our elements are rectangle. Our screen design therefore consists of rectangle components whose position conform to Fitts’ law and Signal detection theory. Therefore our hypothesis would be that there exists one or more sets of screen elements positions that optimise user interaction. So we have created a filter on all possible designs to rule out those which are non-optimal. In Category theory terms we have to express categories as nodes in a graph which are connected by arrows which represent mappings or functions. Our first mapping, f, is from non-optimised Fitts’ law positions, NOPFL to optimised positions OPFL