During the academic year 1966/67 a seminar on various aspects of category theory and its
applications was held at the Forschungsinstitut f¨ur Mathemtik, ETH, Z¨urich. This volume
is a report on those lectures and discussions which concentrated on two closely related
topics of special interest: namely a) on the concept of “triple” or standard construction
with special reference to the associated “algebras”, and b) on homology theories in general
categories, based upon triples and simplicial methods. In some respects this report is
unfinished and to be continued in later volumes; thus in particular the interpretation of
the general homology concept on the functor level (as satellites of Kan extensions), is only
sketched in a short survey.
I wish to thank all those who have contributed to the seminar; the authors for their
lectures and papers, and the many participants for their active part in the discussions.
Special thanks are due to Myles Tierney and Jon Beck for their efforts in collecting the
material for this volume