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Order-enriched solid functors

Lurdes SousaWalter Tholen — 2019

Commentationes Mathematicae Universitatis Carolinae

Order-enriched solid functors, as presented in this paper in two versions, enjoy many of the strong properties of their ordinary counterparts, including the transfer of the existence of weighted (co)limits from their codomains to their domains. The ordinary version of the notion first appeared in Trnková's work on automata theory of the 1970s and was subsequently studied by others under various names, before being put into a general enriched context by C. Anghel. Our focus in this paper is on differentiating...

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 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...

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