However, a monomorphism need not be left-invertible. Examples Edit Every morphism in a concrete category whose underlying function is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense. In the category of sets the converse also holds, so the monomorphisms are exactly the injective morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator. In particular, it is true in the categories of all groups, of all rings, and in any abelian category. It is not true in general, however, that all monomorphisms must be injective in other categories; that is, there are settings in which the morphisms are functions between sets, but one can have a function that is not injective and yet is a monomorphism in the categorical sense.
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Other textbooks[ edit ] Awodey, Steven Category Theory Oxford Logic Guides Oxford University Press. Borceux, Francis Handbook of categorical algebra Encyclopedia of Mathematics and its Applications Cambridge Univ.
Freyd, Peter J. Categories, allegories North Holland Mathematical Library North Holland. Hatcher, William S. The Logical Foundations of Mathematics, 2nd ed. Sets for mathematics. Cambridge University Press. Conceptual mathematics: a first introduction to categories. McLarty, Colin Elementary Categories, Elementary Toposes. Mac Lane, Saunders Graduate Texts in Mathematics 5.
ISBN An introduction to the subject making judicious use of category theoretic concepts, especially commutative diagrams. May, Peter A Concise Course in Algebraic Topology. Categorical foundations Encyclopedia of Mathematics and its Applications Pierce, Benjamin Basic Category Theory for Computer Scientists. MIT Press. Taylor, Paul Practical Foundations of Mathematics.
An introduction to the connection between category theory and constructive mathematics.
A reasonable estimate of the number of these different items would be somewhere between 50, and , Many of these have been named and many more could and perhaps should have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it is worthwhile to pursue the quest. In addition to the primary information given in the Handbook, there are references to relevant articles, books or lecture notes to help the reader.
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