Summary: Totality of Product Completions
Jir'i Ad'amek \Lambda , Lurdes Sousa y and Walter Tholen z
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
AMS Subj. Class.: 18A05, 18A22, 18A40.
Key words: multitotal category, multisolid functor, formal product com
The concept of totality, introduced by Street and Walters , is a strong
property of categories (implying completeness and cocompleteness -- and more,
see , ) which, nevertheless, most ``current'' categories enjoy. Recall that