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A graded object with degree +1 differential squaring to zero.
The zero composite requires zero morphisms; preadditive categories suffice. Cohomology objects require additional kernel/image or abelian hypotheses.
wp-chain-complexnlab-cochain-complexstacks-complexesstacks-abelian-categoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Add coboundary maps with .
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
stacks-complexeswp-chain-complexnlab-chain-complexstacks-derived-textEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
For a cochain complex of modules in a compatible symmetric monoidal additive setting, add a degree-zero associative graded multiplication and require the graded Leibniz rule.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
stacks-dgaThis is an authored directed relation from the source endpoint to the target endpoint.
The singular cochain complex with coefficients is a particular cochain complex obtained by applying Hom to singular chains.
The target is a member or subtype of the broader source class.
hatcher-algebraic-topology