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Totality of product completions. (English) Zbl 1034.18004
Summary: 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 \({\mathcal A}\) by asking the Yoneda embedding \({\mathcal A} \to [{\mathcal A}^{\text{op}}, {\mathcal S}et]\) to be the right multiadjoint and prove that this property is equivalent to totality of the formal product completion \(\prod {\mathcal A}\) of \({\mathcal A}\). We also characterize multitotal categories with various types of generators; in particular, the existence of dense generators is inherited by the formal product completion if and only if measurable cardinals cannot be arbitrarily large.
MSC:
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18A22 Special properties of functors (faithful, full, etc.)
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18A05 Definitions and generalizations in theory of categories
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