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A Remark on Conservative Cocompletions of Ji r Velebil 1

Summary: A Remark on Conservative Cocompletions of
Jir Velebil 1
Faculty of Electrical Engineering, Technical University, Prague, Czech Republic,
Jir Adamek 1
Technical University, Braunschweig, Germany, adamek@iti.cs.tu-bs.de
For a set F of small categories, F-conservative cocompletions of a category are co-
completions preserving all existing colimits of type F. We prove that every category
has a free F-conservative cocompletion. However, unless F is trivial, this cocomple-
tion fails in general to be locally small.
Key words: cocompletion, cocontinuous functor.
AMS Subj. Class.: 18A35, 18A40.
1 Introduction
Our paper is devoted to a classical topic of category theory: free (co)completions.
It has already been observed by J. Lambek [8] that every small category A has
a free cocompletion, viz, the Yoneda embedding YA into the presheaf category
[A op ; Set]. Often one wants to work with cocompletions preserving existing
colimits, or at least existing colimits of a certain type F (where F is a set of


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


Collections: Computer Technologies and Information Sciences