Transfors and their localizations

Sjoerd Crans – 16 April 1997

We all know what natural transformations are. I gave a slightly different viewpoint: the commutativity condition is a higher-dimensional cell which happens to be an identity. In higher dimensions this leads to the notion of transfor, and to the important distinction between naturality and functoriality: the former is a cell relating composites in D, and the latter is an axiom on how composition in C relates to composition in D. Transfors should be composable, leading to an \omega-tas Hom(C,D), and to an \omega-tas \omega-Teisi. There are interesting, unsolved, conceptual problems in this, though. I briefly hinted at localizations of transfors: a q-transfor r: C -> D gives rise to a q-transfor C(c,c') -> something which is almost, but not quite, D(r(c),r(c')). This talk was based on sections 2.5.1 and 2.5.3 of the forthcoming 100+ page paper Central observations on \omega-teisi.