Is the category of groupoids Grpd a locally presentable category? If the answer is yes, can someone sketch a proof or point a reference out?


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    $\begingroup$ The category of small groupoids is sketchable, right? $\endgroup$ – Martin Brandenburg Jan 24 '14 at 18:47
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    $\begingroup$ The easiest way is to observe that it's an accessibly embedded reflective subcategory of $\mathbf{sSet}$, which is locally finitely presentable. $\endgroup$ – Zhen Lin Jan 24 '14 at 20:27
  • $\begingroup$ @ZhenLin for a reflective subcategory you need a full subcategory and an inclusion functor, however for Grpd and sSet you have the nerve functor to mediate between Grpd and sSet, isn't it ? $\endgroup$ – user123619 Jan 27 '14 at 10:22
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    $\begingroup$ Yes, but the nerve functor is a fully faithful (indeed, injective on objects) accessible functor that has a left adjoint, so it is equivalent (indeed, isomorphic) to such a functor. $\endgroup$ – Zhen Lin Jan 27 '14 at 10:29

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