International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 22, Pages 1383-1395
doi:10.1155/S0161171203202222
The category of long exact sequences and the homotopy exact
sequence of modules
Department of Mathematics and Computer Science, Providence College, Providence 02918, RI, USA
Received 25 February 2002
Copyright © 2003 C. Joanna Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The relative homotopy theory of modules, including the (module)
homotopy exact sequence, was developed by Peter Hilton (1965). Our
thrust is to produce an alternative proof of the existence of the
injective homotopy exact sequence with no reference to elements
of sets, so that one can define the necessary homotopy concepts in
arbitrary abelian categories with enough injectives and
projectives, and obtain, automatically, the projective relative
homotopy theory as the dual. Furthermore, we pursue the relative
(module) homotopy theory analogously to the absolute (module)
homotopy theory. For these purposes, we embed the relative
category into the category of long exact sequences, as a full
subcategory, in our search for suitable notions of monomorphisms
and injectives in the relative category.