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Yoneda splicing is well-defined on equivalence classes
Statement
Splicing - and -fold extensions induces a well-defined map on their generated equivalence classes.
Facts & Assumptions
Given: Equivalent representatives of two composable higher extensions.
Proof
A generating chain map between one pair of representatives can be concatenated with the identity chain map on the other factor; the result is a chain map of the spliced exact sequences that is the identity at both ends.
Such maps generate the equivalence relation of Equivalence of n-fold extensions, so repeated generating replacements do not change the class of the splice defined in The Yoneda splice product.
Depends on
Used by
Dependency tree · two levels
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Sources
- Charles A. Weibel, An Introduction to Homological Algebra, Chapters 3–4 (standard reference, not scraped)