Subject: categories: connected functors

From: Jiri Adamek <[email-masked]>
Date: Mon, 7 May 2001 14:36:35 +0200 (MET DST)

Connected Functors

Peter Johnstone has asked during the last PSSL for a characterization of
functors  p: E -> B which are "connected" in the sense that the functor
Set^B -> Set^E of composition with  p  is fully faithful. We have found
two necesary and sufficient conditions; in the following  E  and  B  are
arbitrary small categories.

Theorem 1. A functor p: E -> B is connected iff every object  X  of  B  is
an absolute limit of the diagram of all arrows X -> p(Z) for  Z  ranging
through  E .

Theorem 2. A functor  p: E -> B is connected iff for every morphism
x: X -> X' of  B  the category of all factorizations of  x  through
objects of  p[E]  is connected.

More precisely, in Thm 1 we form the diagram of all arrows  X -> p(Z)
and all E-morphisms whose p-image forms a commutative triangle in  B. 
Then  X  is equipped with a canonical cone of that diagram; this
cone is requested to be an absolute limit.
In Thm 2 we consider the category of all triples (Z,q,m) where  Z  is an
object of  E  and  m,q  are morphisms of  B  with  x = q.m (and morphisms
between these triples are the E-morphisms whose p-images form two
commutative triangles in  B ). Connectedness of that category has been,
for the case of  x = id , observed as a necessary condition by Peter.

J. Adamek, R. El Bashir, M. Sobral and J. Velebil

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