Is there an example that in some abelian category $A$, direct limit always exist, but filtered colimit does not always exist?
2026-03-28 04:33:18.1774672398
Filtered colimit and directed colimit
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No. Every filtered category admits a cofinal functor from a directed category, so the existence of directed colimits implies that of filtered colimits.
The construction of this directed category is slightly technical. It can be read in the first part of Adamek and Rosicky's monograph on locally presentable and accessible categories.
EDIT: An error in the A-R proof has been pointed out on MSE before and was recalled by @user12580 in the comments, together with a link to a correct proof, which interestingly precedes the A-R book by quite some time.