By M.M. Cohen

Cohen M.M. A direction in simple-homotopy idea (Springer, [1973)(ISBN 3540900551)

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Q ! X/. t u These factorizations will be called respectively the (quotient, injection) factorization and the (surjection, insertion) factorization. Lemma. Suppose given a commutative rectangle of functions m ı u D v ı e with e surjective and m injective. Then there exists a function d W X ! Y such that m ı d D v and d ı e D u. e ✲ X W u d ❄ Y v ❄ ✲ Z m Proof. w//. x/. x/ D y we obtain the required diagonal function d. Proposition C. Every function f W X ! Y has an essentially unique factorization e m X !

There exists an inductive set. If Y and Z are inductive sets, then so is Y \ Z; furthermore, the intersection X of all inductive subsets of Y is clearly again inductive. , no proper subset of X can be inductive. In other words, if there exists an inductive set, then there exists a unique smallest inductive set. We def def denote it by N. Its members, called natural numbers, include 0 D ;, 1 D f;g, def def 2 D f0; 1g. . and in general n C 1 D n [ fngg. A set X is called finite if there exists a bijection b W n !

A; B/ is a class whenever A and B are both classes (because every member is then a set). A D C and B D D/ is an immediate consequence of the definition. The definition extends without problem to ordered triples and other ordered multiples. Therefore a function between classes (being an ordered triple of classes) is also a class. 2 The Class S of Functions Between Sets The symbol S will denote the class of all functions for which both domain and codomain are sets. The class S has the following properties—noteworthy for reasons to become clear.