Size functor
Given a size pair ( M , f ) {\displaystyle (M,f)\ } where M {\displaystyle M\ } is a manifold of dimension n {\displaystyle n\ } and f {\displaystyle f\ } is an arbitrary real continuous function defined on it, the i {\displaystyle i} -th size functor, with i = 0 , … , n {\displaystyle i=0,\ldots ,n\ } , denoted by F i {\displaystyle F_{i}\ } , is the functor in F u n ( R o r d , A b ) {\displaystyle Fun(\mathrm {Rord} ,\mathrm {Ab} )\ } , where R o r d {\displaystyle \mathrm {Rord} \ } is the category of ordered real numbers, and A b {\displaystyle \mathrm {Ab} \ } is the category of Abelian groups, defined in the following way. For x ≤ y {\displaystyle x\leq y\ } , setting M x = { p ∈ M : f ( p ) ≤ x } {\displaystyle M_{x}=\{p\in M:f(p)\leq x\}\ } , M y = { p ∈ M : f ( p ) ≤ y } {\displaystyle M_{y}=\{p\in M:f(p)\leq y\}\ } , j x y {\displaystyle j_{xy}\ } equal to the inclusion from M x {\displaystyle M_{x}\ } into M y {\displaystyle M_{y}\ } , and k x y {\displaystyle k_{xy}\ } equal to the morphism in R o r d {\displaystyle \mathrm {Rord} \ } from x {\displaystyle x\ } to y {\displaystyle y\ } , for each x ∈ R {\displaystyle x\in \mathbb {R} \ } , F i ( x ) = H i ( M x ) ; {\displaystyle F_{i}(x)=H_{i}(M_{x});\ } F i ( k x y ) = H i ( j x y ) .