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Totality of Product Completions Jir'i Ad'amek \Lambda , Lurdes Sousa y and Walter Tholen z
 

Summary: Totality of Product Completions
JiŸr'i Ad'amek \Lambda , Lurdes Sousa y and Walter Tholen z
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 A by asking the Yoneda
embedding A ! [A op
; Set] to be right multiadjoint and prove that this
property is equivalent to totality of the formal product completion \PiA of
A. We also characterize multitotal categories with various types of gener­
ators; in particular, the existence of dense generators is inherited by the
formal product completion iff measurable cardinals cannot be arbitrarily
large.
AMS Subj. Class.: 18A05, 18A22, 18A40.
Key words: multitotal category, multisolid functor, formal product com­
pletion.
1 Introduction
The concept of totality, introduced by Street and Walters [15], is a strong
property of categories (implying completeness and cocompleteness -- and more,
see [14], [11]) which, nevertheless, most ``current'' categories enjoy. Recall that

  

Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig

 

Collections: Computer Technologies and Information Sciences