A commutative diagram is a collection of maps A_i ⟶^(ϕ_i) B_i in which all map compositions starting from the same set A and ending with the same set B give the same result. In symbols this means that, whenever one can form two sequences A = A_(i_0) ⟶^(ϕ_(i_0)) B_(i_0) = A_(i_1) ⟶^(ϕ_(i_1)) B_(i_1) = A_(i_2) ⟶^(ϕ_(i_2)) ...⟶^(ϕ_(i_(n - 1))) B_(i_(n - 1)) = A_(i_n) ⟶^(ϕ_(i_n)) B_(i_n) = B, and A = A_(j_0) ⟶^(ϕ_(j_0)) B_(j_0) = A_(j_1) ⟶^(ϕ_(j_1)) B_(j_1) = A_(j_2) ⟶^(ϕ_(j_2)) ...⟶^(ϕ_(i_(m - 1))) B_(i_(m - 1)) = A_(i_m) ⟶^(ϕ_(i_m)) B_(i_m) = B, the following equality holds: ϕ_(i_n) °ϕ_(i_(n - 1)) °...°ϕ_(i_1) °ϕ_(i_0) = ϕ_(j_m) °ϕ_(j_(m - 1)) °...°ϕ_(j_1) °ϕ_(j_0).

coequalizer | diagram chasing | diagram lemma | eight lemma | equalizer | five lemma | four lemma | injective module | nine lemma | projective module | pullback map | snake lemma | split exact sequence | zig-zag lemma