Dear all, I have recently been wondering about the notion of "quality type" introduced by Lawvere in [_Axiomatic cohesion_]: given an extensive category S, a quality type over S is an extensive category F equipped with a fully faithful functor S -> F that has a simultaneous left and right adjoint. There are two classes of examples of quality types over Set that I know of: (a) Hurewicz-style homotopy categories obtained from categories of cohesion [loc. cit.], including e.g. the classical homotopy category of CW or Kan complexes (b) The toposes of presheaves on a small category with a zero object (which, by splitting idempotents, includes the toposes of M-sets where M is a monoid with a two-sided absorbing element) These seem to be the examples Lawvere was thinking about: (a) are "extensive qualities" and (b) are "intensive qualities". What is not clear to me is what these two examples have in common, or more generally, what is the significance of the functor S -> F having a simultaneous left and right adjoint. I would be very grateful for any insights people may have. Sincerely, -- Zhen Lin Low [For admin and other information see: http://www.mta.ca/~cat-dist/ ]