Gardener, Patricia Elizabeth (1981) Generalized derived functors. PhD thesis, University of Warwick.

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Abstract

This thesis is about a construction of derived functors which considerably generalises the original definition of Cartan and Eilenberg. The new derived functors can be constructed In the following circumstances:- Let C be a category with an initial object, I, and let M be a subcategory of C containing I. If F is a functor from M to a category of "based sets with structure" then the derived functors of F can be defined. The derived functors of F have domain C and their actions on objects of C are expressed as the homotopy groups of a simplicial set. This simplicial set is the nerve of a category so the derived functors can be regarded as the (topological) homotopy groups of the classifying space of a category.

The 0-th derived functor constructed in this manner is the Kan extension of F. The derived functors satisfy an "acyclic model theorem". Taking M as the full subcategory of projectives of an abelian category the derived functors agree with those of Cartan and Eilenberg, as is shown using the theory of over categories and Γ-spaces. A previous important generalisation of derived functors are the cotriple derived functors. If the cotriple derived functors of F are defined then, for a suitable choice of M, the new derived functors agree with them. This shows that many well-known sequences of functors arise from the construction, including Eckmann-Hilton homotopy groups, Hochschild's K-relatlve Tor, the homology of groups and the homology of commutative algebras. The singular homology functors occur as the derived functors of the 0-th reduced singular homology functor and for these new derived functors the derived functors of the 0-th homotopy functor behave like the higher homotopy functors when applied to connected spaces.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Functor theory, Homotopy theory, Abelian categories
Official Date:	December 1981
Dates:	Date Event December 1981 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Robinson, Christopher Alan
Sponsors:	Science Research Council (Great Britain)
Format of File:	pdf
Extent:	90 leaves : illustrations
Language:	eng

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