Geodesic
Enriched indexed categories
Theory and applications of categories, Tome 28 (2013), pp. 616-695.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

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.
Classification : 18D20, 18D30
Keywords: monoidal category, enriched category, indexed category, fibered category 
@article{TAC_2013_28_a20,
     author = {Michael Shulman},
     title = {Enriched indexed categories},
     journal = {Theory and applications of categories},
     pages = {616--695},
     publisher = {mathdoc},
     volume = {28},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2013_28_a20/}
}
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Michael Shulman. Enriched indexed categories. Theory and applications of categories, Tome 28 (2013), pp. 616-695. http://geodesic.mathdoc.fr/item/TAC_2013_28_a20/