 
Summary: A Remark on Conservative Cocompletions of
Categories
Jir Velebil 1
Faculty of Electrical Engineering, Technical University, Prague, Czech Republic,
velebil@math.feld.cvut.cz
Jir Adamek 1
Technical University, Braunschweig, Germany, adamek@iti.cs.tubs.de
Abstract
For a set F of small categories, Fconservative cocompletions of a category are co
completions preserving all existing colimits of type F. We prove that every category
has a free Fconservative 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
