1 Answers
In mathematics, more specifically algebraic topology, a pair {\displaystyle } is shorthand for an inclusion of topological spaces i : A ↪ X {\displaystyle i\colon A\hookrightarrow X}. Sometimes i {\displaystyle i} is assumed to be a cofibration. A morphism from {\displaystyle } to {\displaystyle } is given by two maps f : X → X ′ {\displaystyle f\colon X\rightarrow X'} and g : A → A ′ {\displaystyle g\colon A\rightarrow A'} such that i ′ ∘ g = f ∘ i {\displaystyle i'\circ g=f\circ i}.
A pair of spaces is an ordered pair where X is a topological space and A a subspace. The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of X by A. Pairs of spaces occur centrally in relative homology, homology theory and cohomology theory, where chains in A {\displaystyle A} are made equivalent to 0, when considered as chains in X {\displaystyle X}.
Heuristically, one often thinks of a pair {\displaystyle } as being akin to the quotient space X / A {\displaystyle X/A}.
There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space X {\displaystyle X} to the pair {\displaystyle }.