By Peter J. Hilton, Urs Stammbach

ISBN-10: 1441985662

ISBN-13: 9781441985668

ISBN-10: 1461264383

ISBN-13: 9781461264385

We have inserted, during this variation, an additional bankruptcy (Chapter X) entitled "Some purposes and up to date Developments." the 1st component of this bankruptcy describes how homological algebra arose through abstraction from algebraic topology and the way it has contributed to the data of topology. the opposite 4 sections describe purposes of the tools and result of homological algebra to different components of algebra. lots of the fabric provided in those 4 sections was once no longer on hand while this article used to be first released. evidently, the remedies in those 5 sections are just a little cursory, the purpose being to provide the flavour of the homo logical tools instead of the main points of the arguments and effects. we want to specific our appreciation of support acquired in writing bankruptcy X; specifically, to Ross Geoghegan and Peter Kropholler (Section 3), and to Jacques Thevenaz (Sections four and 5). the one different adjustments include the correction of small error and, after all, the expansion of the Index. Peter Hilton Binghamton, manhattan, united states Urs Stammbach Zurich, Switzerland Contents Preface to the second one version vii advent. . I. Modules.

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L. 1)) F(fg) = F(g) F(f). We give the following examples of contravariant functors. (a) (£:( - , B), for B an object in (£:, is a contravariant functor from (£: to 6 . Similarly, 9Jl~( -, B), 9Jl~( -, B) are contravariant functors from 9Jl~, 9Jl~ respectively to \lIb. We say that these functors are represented by B. h~\l(b). (c) Let A be an object of 9Jl~ and let G be an abelian group. We saw in Section I. 8 how to give Hom;l(A, G) the structure of a left A-module. 2. Functors 47 Hom z ( -, G) thus appears as a contravariant functor from 9Jl A to 9Jl~.

For plainly Abel (G) = G, Abel (Ho) = Ho, Abel( 'lIb. Similarly there are free functors 6->ffi, 6->m F , 6->9R~, 6->9R~, etc. (d) Underlying every topological space there is a set.

In which some structure is "forgotten" or "thrown away". (e) The fundamental group may be regarded as a functor n: ;:to->ffi, where ;:to is the category of spaces-with-base-point (see [21J). It may also be regarded as a functor n: ;:t~ -> ffi, where the subscript h indicates that the morphisms are to be regarded as (based) homotopy classes of (based) continuous functions. Indeed there is an evident classifying functor Q : ;:to ->;:t~ and then n factors as n = nQ. (f) Similarly the (singular) homology groups are functors ;:t->'11b (or ;:th->'11b).

### A Course in Homological Algebra by Peter J. Hilton, Urs Stammbach

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