# Grpd as a locally presentable category

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?

Thanks

• The category of small groupoids is sketchable, right? – Martin Brandenburg Jan 24 '14 at 18:47
• The easiest way is to observe that it's an accessibly embedded reflective subcategory of $\mathbf{sSet}$, which is locally finitely presentable. – Zhen Lin Jan 24 '14 at 20:27
• @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 ? – user123619 Jan 27 '14 at 10:22
• 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. – Zhen Lin Jan 27 '14 at 10:29