Geodesic
Totality of product completions
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 1, pp. 9-24.

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Categories whose Yoneda embedding has a left adjoint are known as total categories and are characterized by a strong cocompleteness property. We introduce the notion of multitotal category $\Cal A$ by asking the Yoneda embedding $\Cal A \rightarrow [\Cal A^{op},\Cal Set]$ to be right multiadjoint and prove that this property is equivalent to totality of the formal product completion $\Pi \Cal A$ of $\Cal A$. We also characterize multitotal categories with various types of generators; in particular, the existence of dense generators is inherited by the formal product completion iff measurable cardinals cannot be arbitrarily large.
Classification : 18A05, 18A22, 18A35, 18A40
Keywords: multitotal category; multisolid functor; formal product completion 
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Adámek, Jiří; Sousa, Lurdes; Tholen, Walter. Totality of product completions. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 1, pp. 9-24. http://geodesic.mathdoc.fr/item/CMUC_2000__41_1_a1/