---- > [!definition] Definition. ([[chain map]]) > A **chain map** is a morphism $f_{\bullet}: C_{\bullet} \to D_{\bullet}$ of [[chain complex of modules|chain complexes]], that is, a sequence of [[linear map|homomorphisms]] such that the squares > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGEB9YMAagEYAviEGl0mXPkIp+5KrUYs2XQqPHY8BImX7z6zVohAARbnyEixIDBqlFZu6vqVHTq+TCgBzeEVAAzACcIAFskMhAcCCRZBQM2KAA9Lh4BYTUQINCkACZqKKQAZidFQxAk01SLDKywxAiCxDy4l0yzNJBqBjoAIxgGAAUJTWkQLDBsWEsA4LrYxuKWsv9OQi7e-qHbLSNxydZBCkEgA > \begin{tikzcd} > C_{n+1} \arrow[r, "d^C_{n+1}"] \arrow[d, "f_{n+1}" description] & C_n \arrow[d, "f_n" description] \\ > D_{n+1} \arrow[r, "d^D_{n+1}"] & D_n > \end{tikzcd} > \end{document} > ``` > commute. > A **cochain map** is a morphism $f^{\bullet}: C^{\bullet} \to D^{\bullet}$ of [[chain complex of modules|cochain complexes]], that is, a sequence of [[linear map|homomorphisms]] such that the squares > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGEA9QgX1PUy58hFAEZyVWoxZsuwMAGpRPEHwHY8BImVGT6zVohAARbqv4gMG4UXG7q+mUdPylKnpJhQA5vCKgAMwAnCABbJDIQHAgkcSkDNigzNRBgsKQAJmpopABmB2lDECTCagY6ACMYBgAFQU0RECCsbwALHHNAkPDESJzELPinVLMyyuq66y0jLDBsWE7U7tjsmMR8oaKAzldlEDGq2vqbGbmsBY8eIA > \begin{tikzcd} > C^n \arrow[r, "d^n_C"] \arrow[d, "f^n" description] & C^{n+1} \arrow[d, "f^{n+1}" description] \\ > D^n \arrow[r, "d^n_D"'] & D^{n+1} > \end{tikzcd} > \end{document} > ``` > commute. > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```