Given a size pair {\displaystyle \ } 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 {\displaystyle Fun\ }, 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. Wikipedia