Re: horizontal composition

[Marco Grandis <grandis@dima.unige.it>]

Dear Jean (and dear colleagues),

Everyone knows you defined bicategories (and Charles Ehresmann 2-
categories). The point of those two messages is that one can
equivalently define a 2-category or a bicategory using the following
primitive operations (and suitable axioms):

- the vertical composition of cells,
- the whisker composition of cells with maps (instead of the
horizontal composition of cells).

  (Which is precisely what we concretely do in Cat, when we define
horizontal composition of natural transformations: we use vertical
composition and whiskering, showing that the two possible ways of
defining horizontal composition give the same result, by the relevant
part of the middle-four interchange axiom - which I was calling
"reduced interchange".)

All this has some importance in homotopy, which is why I was
interested in it. For instance, take chain complexes with their
homotopies: then the vertical composition of homotopies is (strictly)
associative, whiskering is also associative (in the appropriate
sense), but reduced interchange fails and you do not have a
horizontal composition of homotopies. Such a structure is a sesqui-
category in Ross Street's sense - actually one might say "sesqui-
groupoid".

With best regards

Marco