* connected functors
@ 2001-05-07 12:36 Jiri Adamek
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From: Jiri Adamek @ 2001-05-07 12:36 UTC (permalink / raw)
To: categories
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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