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element in an abelian category

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Additive and abelian categories

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

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Lemmas

diagram chasing

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Contents

Idea

Not every abelian category is a concrete category, such as Ab or RRMod, hence its objects do not necessrily have underlying sets whose elements one can reason about. To circumvent this and reason by “diagram chases” of elements in general abelian categories, one can instead use generalized elements in a suitable way.

This is closely related to embedding the abelian category fully-faithfully and exactly into a concrete abelian category. See at abelian category the section Embedding theorems for more on this.

Definition

One definition goes likes this (MacLane, VIII.4 and also for instance Gelfand-Manin, II.5):

An element of an object WW in a given abelian category 𝒜\mathcal{A} is an equivalence class [X,h][X,h] of pairs (X,h)(X,h) where XX is an object of AA and h:XXh:X\to X' a morphism (hence a generalized element) and the equivalence is defined as follows: [X,h]=[Y,k][X,h] = [Y,k] iff there exists an object ZZ in AA and epimorphisms u:ZXu:Z\to X, v:ZYv:Z\to Y such that hu=kv:ZXh\circ u = k\circ v : Z\to X'.

However, beware that the passage to equivalence classes does not respect the abelian group structure and hence generalized elements in this sense cannot be added or subtracted. A more natural approach is discussed in (Bergman) where the actual generalized elements are remembered but a refinement of their domain is allowed, much as familiar from topos theory.

References

Equivalence classes of generalized elements are considered for instance in

  • Sergei Gelfand, Yuri Manin, Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original. Springer 1996. xviii+372 pp.; 2nd corrected ed. 2002.

Genuine generalized elements are considered in

  • George Bergman, A note on abelian categories – translating element-chasing proofs, and exact embedding in abelian groups (1974) (pdf)

Revised on August 28, 2012 15:15:46 by Ingo Blechschmidt (46.244.198.196)