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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cosmoi of internal categories

Author: Ross Street
Journal: Trans. Amer. Math. Soc. 258 (1980), 271-318
MSC: Primary 18D35; Secondary 18C10, 18F20
MathSciNet review: 558176
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Abstract: An internal full subcategory of a cartesian closed category $\mathcal {A}$, is shown to give rise to a structure on the 2-category $Cat(\mathcal {A})$ of categories in $\mathcal {A}$ which introduces the notion of size into the analysis of categories in $\mathcal {A}$ and allows proofs by transcendental arguments. The relationship to the currently popular study of locally internal categories is examined. Internal full subcategories of locally presentable categories (in the sense of Gabriel-Ulmer) are studied in detail. An algorithm is developed for their construction and this is applied to the categories of double categories, triple categories, and so on.

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Keywords: Internal full subcategory, locally presentable category, locally small, fibred category, site, sketched structures, Gabriel theory, internally complete, cartesian closed, multiple category
Article copyright: © Copyright 1980 American Mathematical Society