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By George F. Simmons

This fabric is meant to give a contribution to a much wider appreciation of the mathematical phrases "continuity and linearity". The book's goal is to light up the meanings of those phrases and their relation to one another.

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Next we shall show that a much stronger result can be obtained: any class-indexed family (fi : Xi - Y)iEI of morphisms in X with common codomain, commonly called a sink, can be simultaneously factorized. 21) tni Y" Z in X with n E M , then there is a uniquely determined morphism w : M --, N and with forall iEI. We refer to (2) as the simultaneous diagonalization property. 11). The following assertions are equivalent: THEOREM (i) X has M-pullbacks and M-intersections; (ii) X has M-unions, and every morphism in X has a right M-factorization; (iii) every sink in X has a right M-factorization.

Prove that a morphism in a category is an isomorphism if and only if it is both epic and extremally monic. Dualize the statement. Prove that every monomorphism is extremal if and only if every morphism which is both epic and monic is actually an isomorphism. Dualize the statement. preliminaries on Subobjects, Images, and Inverse Images (d) 21 Prove that extremal monomorphisms are left cancellable, that is: if a composite n m is extremally monic, then also m is extremally monic. (e) Construct categories in which extremal monomorphisms are not closed under composition or stable under pullback or multiple pullback.

T. the class of embeddings. That K is a hereditary and grounded closure operator of FC is easily checked.

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