The pair long exact sequence is natural: local context

Project map · Chapter map · Full theorem DAG · Open textbook statement

This local map shows one dependency hop in either direction. The highlighted item is the current focus. What the colours mean

in mathlib1Upstreamed into Mathlib.fully proved4Proved, and every prerequisite is fully proved too.

graph LR n0["Relative singular chains and homology"]:::fully_proved n1("A short exact sequence gives an exact homology sequence"):::mathlib n2("The long exact sequence of a pair"):::fully_proved n3("The pair long exact sequence is natural"):::fully_proved n4("The reduced pair long exact sequence is natural"):::fully_proved n2 --> n3 n3 -.-> n4 n0 --> n2 n0 --> n3 n1 -.-> n2 n1 -.-> n3 n1 -.-> n4 click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-singular-homology" "Relative singular chains and homology — 1 item · 1 fully proved" click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#short-exact-homology-sequence" "A short exact sequence gives an exact homology sequence — 1 item · 1 in mathlib" click n2 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#pair-long-exact-sequence" "The long exact sequence of a pair — 1 item · 1 fully proved" class n3 focus click n3 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#pair-long-exact-sequence-naturality" "The pair long exact sequence is natural — 1 item · 1 fully proved" click n4 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-pair-long-exact-sequence-naturality" "The reduced pair long exact sequence is natural — 1 item · 1 fully proved"