We develop a theory of categories which are simultaneously (1) indexed over a base category $S$ with finite products, and (2) enriched over an $S$-indexed monoidal category $V$. This includes classical enriched categories, indexed and fibered categories, and internal categories as special cases. We then describe the appropriate notion of ``limit'' for such enriched indexed categories, and show that they admit ``free cocompletions'' constructed as usual with a Yoneda embedding.

Keywords: monoidal category, enriched category, indexed category, fibered category

2010 MSC: 18D20,18D30

*Theory and Applications of Categories,*
Vol. 28, 2013,
No. 21, pp 616-695.

Published 2013-08-05.

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