It is common practice, in category theory, to prove metatheorems attesting that if a statement of some given kind holds in a prescribed reference category, the statement holds more generally in a wide collection of abstract categories satisfying a given set of axioms. The cases of abelian categories, with the category of modules on a ring as reference, and the toposes, with the category of sets as reference, are the most celebrated examples to this respect. This preliminary chapter introduces — or recalls — two rather elementary such metatheorems which, in this book, will frequently provide the quickest way to prove results.
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