# Preservation of limits

Is there a functor that preserves all small limits but not a large one?

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I did not know that some limits are small and some are large. –  Wadim Zudilin Jul 26 '10 at 12:48
"small" refers to the index category of the diagram; this should be small (in the sense that its object class is a set). –  Martin Brandenburg Jul 26 '10 at 13:00
By the way, I've never seen so far a large limit which exists and does not restrict somehow to a small limit. –  Martin Brandenburg Jul 26 '10 at 13:02
The class of all ordinals is ordered. Add one more element $\infty$ at the end, bigger than all of them. View this "large ordered set" as a large category $\cal C$. A small diagram in $\cal C$ has colimit $\infty$ if $\infty$ occurs in the diagram, and otherwise it has a colimit less than $\infty$. But the large diagram consisting of everything except $\infty$ has colimit $\infty$. The functor to the ordered set $\lbrace 0<1\rbrace$ that sends $\infty$ to $1$ and everything else to $0$ preserves small colimits but not all colimits.