---- (Warning: ad-hoc notation ahead.) > [!definition] Definition. ([[singular map of pairs functor]]) > We have a [[covariant functor|functor]] $\mathsf{TopPairs} \to \mathsf{ChainPairs}$ sending: > - (Objects) A [[topological pair]] $(X,A)=(A \hookrightarrow X)$ to an inclusion of [[singular homology|singular]] [[chain complex of modules|chain complexes]] $C_{\bullet}(A) \hookrightarrow C_{\bullet}(X)$; > - (Morphisms) A [[topological pair|map of pairs]] $f:(X, A) \to (Y,B)$ to the [[diagram|diagrams]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGEB9MACgEEAlCAC+pdJlz5CKAIzkqtRizZdeADSGjx2PASJkZC+s1aIO3HgCFNYkBh1Sicw9WPKzqngE1NCmFABzeCJQADMAJwgAWyQyEBwIJAAmVyVTEB5QgAIAH05BbhBqBjoAIxgGAAUJXWkQcKwAgAscEVsI6KQ5eMTEAGZUkzZQwq0QDpjEOISu6hw6LAY2JogIAGs2sMjJlJ6kAfiFpbMV9ZEKYSA > \begin{tikzcd} > C_n(A) \arrow[d, "(f |_A)_n"'] \arrow[r, hook] & C_n(X) \arrow[d, "f_n"] \\ > C_n(B) \arrow[r, hook] & C_n(Y) > \end{tikzcd} > \end{document} > ``` > > ---- #### ---- #### 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 > ```