In this talk, we introduce the notion of cocylic morphisms with respect to a
map in the category of pairs proposed by P. Hilton.
The cocylic morphism with respect to a map is the dual concept of
a cyclic morphism of a map and a generalization of a coyclic
morphism in the category of pairs. We study its basic properties
such as preservation of cocyclicity by morphisms and give some
conditions of the set of all cocyclic morphisms with respect to a
map to be a group. We show that there is a relation between
cocyclic morphism with respect to a map and the A-category length
or A-cone length. Finally we introduce a generalized dual
G-sequence and study the condition of the sequence to be exact.
(Joint work with Kee Young Lee)