Ingredients
Incarnations
Properties
Universal aspects
Classification
Induced theorems
…
In higher category theory
Given a category , we may construct the free cocompletion of , freely adding some class of colimits. Often, however, will already have some colimits, which we wish to preserve. A conservative cocompletion of a category is a cocompletion that preserves the colimits in .
For a small category with -colimits, there is a simple description of the -conservative cocompletion (for a class of colimits). It is the the full subcategory of the presheaf category on spanned by the functors sending -colimits in to limits in the presheaf category.
For a large category, this description does not suffice in general, nor does it suffices to consider categories of small presheaves: in fact, there are locally small categories that do not admit locally small conservative cocompletions (see AV02) (however, they do admit conservative cocompletions that are large and not locally small).
Joachim Lambek, Completions of categories: Seminar lectures given 1966 in Zürich, Vol. 24. Springer.
Věra Trnková, Limits in categories and limit-preserving functors, Commentationes Mathematicae Universitatis Carolinae 7.1 (1966): 1-73.
J. F Kennison, On limit-preserving functors, Illinois Journal of Mathematics 12.4 (1968): 616-619.
Max Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64 (1982)
Jiřı́ Adámek and Jiřı́ Velebil?. A remark on conservative cocompletions of categories. Journal of Pure and Applied Algebra 168.1 (2002): 107-124.
See also Theorem 11.5 of:
See section 6.4 (and Theorem 6.4.3 in particular) of:
Last revised on April 25, 2024 at 12:32:43. See the history of this page for a list of all contributions to it.