The nerve $N(C)$ of a category $C$ is a simplicial set and defines a functor $$ N\colon\operatorname{Categories}\to \operatorname{sSets} $$ from the category of small categories to simplcial sets. It is given by $N(C)_n=\operatorname{Hom_{Categories}}(\Delta[n],C)$ and $\Delta[n]$ is the obvious category.

Does $N$ preserve directed colimits?

This is true if $\Delta[n]$ is a compact object in $\operatorname{Categories}$.


Yes, $\Delta[n]$ is compact: it has finitely many objects and morphisms!

  • 1
    $\begingroup$ (However, not every compact category is finite.) $\endgroup$ – Zhen Lin Jun 24 '13 at 17:02
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    $\begingroup$ (That is correct, @ZhenLin, for example, the one-object category $B\mathbb{N}$ is compact. The parenthesis around these comments make it feel like we're whispering...) $\endgroup$ – Omar Antolín-Camarena Jun 24 '13 at 17:05

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