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The chain complex built from a chain map by adjoining a shifted source to the target, encoding the map's failure to be a quasi-isomorphism and supporting associated exact sequences.
hatcher-algebraic-topologystacks-derived-categoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
For a chain map , the mapping cone combines with and uses a differential incorporating both the original differentials and .
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
hatcher-algebraic-topologystacks-derived-categoryEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
The canonical short exact sequence involving a mapping cone induces a long exact sequence in homology, relating the source, target, and cone of the map.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
hatcher-algebraic-topologystacks-derived-category