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I am wondering if the 2-сategory of groupoids is locally presentable. Locally presentable means the category is accessible and co-complete.

It has been pointed out that the category of groupoids is locally presentable. It would be useful if I could find a theorem that says that if your base category, namely groupoids, was LP, then so is the 2-category on that base. I have done a bit of research and I cannot find anything to that effect. I still cannot say if the 2-category of groupoids is LP.

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    $\begingroup$ Locally presentable 2-categories have been studied by Kelly in "Structures defined by finite limits in the enriched context, I", in the Cahiers de topologie et géométrie différentielle catégoriques, tome 23, nr 1, pp. 3-42 (1982). I think you might find the answer there. $\endgroup$
    – godelian
    Commented Jul 25, 2019 at 14:47
  • $\begingroup$ For what is worth, the (2,1)-category of groupoids is locally presentable as an $(\infty,1)$-category (it has as a compact generator the contractible groupoid) $\endgroup$
    – Denis Nardin
    Commented Jul 25, 2019 at 19:30
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    $\begingroup$ Note that these two comments refer to two different ways in which a strict (2,1)-category like $\rm Gpd$ might be called "locally presentable": if it is locally presentable in Kelly's enriched sense over $\rm Cat$ (or $\rm Gpd$), or if it is locally presentable as an $(\infty,1)$-category in Lurie's sense (one might call this being "locally presentable as a bicategory"). It's not entirely clear to me which you had in mind. Note that as an $(\infty,1)$-category the "base" (i.e. underlying) 1-category is not even well-defined, while Piotr's answer refers to this $(\infty,1)$-version. $\endgroup$
    – Mike Shulman
    Commented Jul 25, 2019 at 21:47
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    $\begingroup$ In the particular case of $\rm Gpd$ it happens to be locally presentable in both senses. But note that any 2-category that's biequivalent to $\rm Gpd$ will still be locally presentable as a bicategory, while it will generally not even be complete and cocomplete as a strict 2-category (and hence not locally presentable as such). $\endgroup$
    – Mike Shulman
    Commented Jul 25, 2019 at 21:48

1 Answer 1

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This is true.

Since the $2$-category of groupoids is equivalent to the $2$-category of $1$-truncated spaces, the statement follows immediately from 5.5.1.8 and 5.5.6.21 in Lurie's Higher Topos Theory.

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