categories: Re: Time for functors to grow up; three queries

[John Duskin <duskin@math.buffalo.edu>]

-- 
Tom Leinster's comments on David Benson's post concerning 
"unreasonable(?) strictness" brings up some interesting questions 
which may not be so easily answered. What is "unreasonable" and what 
is "not" seems to very much depend on historical context and the 
corresponding state of our collective sophistication ("mathematical 
maturity"). I vividly remember attending a Bourbaki Seminar in Paris 
in the late sixties where Henri Cartan was giving a lecture and 
asserted (I translate freely) "These two categories are isomorphic". 
Benabou, who was in the audience interjected, "You mean they are 
equivalent.", only to be dismissed slightly sarcastically by Cartan 
with, "Oh we aren't all category theorists (des categoriciens)!".
I can imagine a similar exchange occurring thirty years earlier with 
"These two groups are the same."--"No, you mean they are isomorphic!"

In the case of bicategories and bifunctors preserving composition "on 
the nose" or only "up to isomorphism", if you treat everything 
simplicially, identifying bicategories with their nerves, the 
2-simplices t of a bicategory consist of a triangular 1-cell boundary 
(f,h,g) together with a 2-cell interior Int(t):h==>f \tensor g. 
Morphisms  of bicategories become maps F which preserve faces and 
thus for 2-simplices send t to (F(f),F(h),F(g)) and Int(F(t)): 
F(h)==>F(f)\tensor F(g). The tensor product corresponds to the 
2-simplex which has (f,f\tensor g,g) for boundary and  id: f\tensor 
g==>f\tensor g for interior and is thus sent to (F(f),F(f\tensor 
g),F(g)) with interior F(f\tensor g)==>F(f)\tensor F(g), which is 
exactly Benabou's composition 2-cell c_{f,g}. From this simplicial 
point of view, it is obviously quite special (unreasonable?) to 
require c_{f,g} to be the identity (composition preserved "on the 
nose"), but not so unreasonable to require it to be an isomorphism, 
as is the case for homomorphisms of bicategories.

In contrast, when categories are viewed simplicially, the 2-simplices 
are just the commutative triangles of the category , (f,h,g) with 
h=fg, (viewed as a bicategory, all 2-simplex interiors are identity 
2-cells). Functors between categories are exactly simplicial maps and 
are face preserving, (f,h,g) goes to (F(f),F(h),F(g)), but then 
F(h)=F(f)F(g) and composition must be preserved on the nose if the 
target is a category, but need not be  if the target were a 
bicategory!


Jack