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Totality of colimit closures

Reinhard BörgerWalter Tholen — 1991

Commentationes Mathematicae Universitatis Carolinae

Adámek, Herrlich, and Reiterman showed that a cocomplete category 𝒜 is cocomplete if there exists a small (full) subcategory such that every 𝒜 -object is a colimit of -objects. The authors of the present paper strengthened the result to totality in the sense of Street and Walters. Here we weaken the hypothesis, assuming only that the colimit closure is attained by transfinite iteration of the colimit closure process up to a fixed ordinal. This requires some investigations on generalized notions...

Natural weak factorization systems

Marco GrandisWalter Tholen — 2006

Archivum Mathematicum

In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category 𝒦 is introduced, as a pair (comonad, monad) over 𝒦 2 . The link with existing notions in terms of morphism classes is given via the respective Eilenberg–Moore categories.

Totality of product completions

Jiří AdámekLurdes SousaWalter Tholen — 2000

Commentationes Mathematicae Universitatis Carolinae

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 o p , 𝒮 e t ] to be right multiadjoint and prove that this property is equivalent to totality of the formal product completion Π 𝒜 of 𝒜 . We also characterize multitotal categories with various types of generators; in particular, the existence of dense generators is inherited by the formal product...

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