Preface to the special issue: commutativity of algebraic diagrams

Extract

The problem of the commutativity of algebraic (categorical) diagrams has attracted the attention of researchers for a long time. For example, the related notion of coherence was discussed in Mac Lane's homology book Mac Lane (1963), see also his AMS presidential address Mac Lane (1976). Researchers in category theory view this problem from a specific angle, and for them it is not just a question of convenient notation, though it is worth mentioning the important role that notation plays in the development of science (take, for example, the progress made after the introduction of symbolic notation in logics or matrix notation in algebra). In 1976, Peter Freyd published the paper ‘Properties Invariant within Equivalence Types of Categories’ (Freyd 1976), where the central role is played by the notion of a ‘diagrammatic property’. We may also recall the process of ‘diagram chasing’, and its applications in topology and algebra. But before we can use diagrams (and the principal property of a diagram is its commutativity), it is vital for us to be able to check whether a diagram is commutative.