By Robert L. Knighten
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Extra info for Notes on Category Theory
But these notions do not work nearly so well in other categories. ) Now q is a surjection, but it certainly does not have a section - indeed the only homomorphism of Z2 to Z takes both elements of Z2 to 0. 7 of the Catalog of Categories. ] Fortunately there is a weaker property than having a right inverse, that captures the notion of a surjection in a categorical fashion for a great many categories. Moreover it turns out to be important and useful quite generally. Note that in many “algebraic” categories such as the categories of groups, Abelian groups, rings, etc, a surjective homomorphism is usually called an epimorphism, and that is the name used throughout these notes.
2, as a subcategory of the category of sets; the category of Abelian groups, Ab cf. 7, as a subcategory of the category of groups, Group (cf. 5;) the category of lattices, Lattice cf. 3, as a subcategory of the category of partially ordered sets, Poset cf. 2; the category of compact Hausdorff spaces, Comp cf. 4, as a subcategory of the category of topological spaces, Top cf. 3; and the category of Hilbert spaces, Hilbert cf. 4, as a subcategory of the category of Banach spaces, cf. 1. In addition the notion of subcategory generalizes the notion of submonoid, subgroup, etc.
6 CHAPTER I. MATHEMATICS IN CATEGORIES With the arrival of the theory of topoi, that became the most important tool for discussing category theory and foundations. See Joyal and Moerdijk, , Mac Lane and Moerdijk,  and Lawvere and Rosebrugh  . After that digression, we make the following definitions. 2: A small category is one in which the collection of morphisms is a set. Note that as a consequence the collection of objects is a set as well. 3: A large category is one in which the collection of morphisms is not a set.