## On the size of categories.(English)Zbl 0854.18003

Under what condition(s) is a category equivalent to a small category? This has been an interesting question for a long time in category theory, and has attracted several researchers’ attention. In 1971, the second author conjectured that a category is equivalent to a small category if and only if the category and its presheaf category are locally small (i.e., categories with small homsets). The first published proof to this conjecture was given in 1979 by F. Foltz [“Légitimité des catégories de préfaisceaux”, Diagrammes 1, Exposé No. 3 (1979; Zbl 0517.18008)]. The paper under review gives a shorter proof to the same conjecture. As a matter of fact, the authors completed this shorter proof in 1977, but the result remained unpublished until recently.
Reviewer: H.-P.Yu (Emory)

### MSC:

 18A25 Functor categories, comma categories 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects)

### Keywords:

locally small category; small category; presheaf category

Zbl 0517.18008
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