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International Journal of Mathematics and Mathematical Sciences
Volume 2003, Issue 22, Pages 1383-1395

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 Hindawi Publishing Corporation. 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.


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.