Localizations of transfors (Q1398054)

There has been an enormous amount of interest in recent years in developing some higher-dimensional category theory that would adequately reflect and model homotopy theory. This development has not only proved important in category theory and homotopy theory, but has involved input from various parts of mathematics and related areas ranging from theoretical physics through to logic and computer science. Taking the model of chain complexes as motivation, the author has elsewhere [\textit{S. E. Crans}, Cah. Topology Géom. Différ. Catégoriques 41, 2-74 (2000; Zbl 0945.18005)] examined why the majority of the weak \(\omega\)-category models proposed so far, do not yield dimension-raising operations, yet in classical invariants for homotopy types, the Whitehead product gives pairings \(\pi_p\times \pi_q \to \pi_{p+q-m-1}\). Such dimension-raising operations also occur in the hypercrossed complex theory of \textit{P. Carrasco} and \textit{A. M. Cegarra} [J. Pure Appl. Algebra 75, 195-235 (1991; Zbl 0742.55003)]. The idea is that small models with dimension-raising pairings may be more useful than large models with dimension-preserving compositions. The models proposed by the author are called teisi, the origins of this word are explained! The current paper continues the chain complex analogy and discusses the analogue of natural transformations/homotopies and the corresponding higher-dimensional homotopies in this context, the transfors of the title. The paper is, as might be expected, technically exacting as it is the first to attempt such a process, but its introduction is excellent in providing motivation and aiding intuition, which is essential if the abstract definitions are to be well understood.