Ji\v r\'{\i } Ad\'amek, Lurdes Sousa, Walter Tholen
Totality of product completions

Comment.Math.Univ.Carolinae 41,1 (2000) 9-24.

Abstract: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.

Keywords: multitotal category, multisolid functor, formal product completion
AMS Subject Classification: 18A05, 18A22, 18A40