Is the 2-сategory of groupoids locally presentable?Are reflective subcategories of complete infinity categories complete?Reference request: colimits of locally presentable categoriesAre all locally compact anisotropic groupoids etale up to equivalence?Locally presentable abelian categories with enough injective objectsIs every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?Does the 2 category of Groupoids Admit the Vector Space Monad?What is the consistency strength of weak Vopenka's principle?Locally presentable categories, universes, and Vopenka's principleClosure of presentable objects under finite limits$mu$-presentable object as $mu$-small colimit of $lambda$-presentable objects
Is the 2-сategory of groupoids locally presentable?
Are reflective subcategories of complete infinity categories complete?Reference request: colimits of locally presentable categoriesAre all locally compact anisotropic groupoids etale up to equivalence?Locally presentable abelian categories with enough injective objectsIs every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?Does the 2 category of Groupoids Admit the Vector Space Monad?What is the consistency strength of weak Vopenka's principle?Locally presentable categories, universes, and Vopenka's principleClosure of presentable objects under finite limits$mu$-presentable object as $mu$-small colimit of $lambda$-presentable objects
$begingroup$
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.
ct.category-theory groupoids 2-categories locally-presentable-categories
edited Jul 26 at 8:47
user64494
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asked Jul 25 at 14:37
Ben SprottBen Sprott
7174 silver badges17 bronze badges
$endgroup$
add a comment |
$begingroup$
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.
ct.category-theory groupoids 2-categories locally-presentable-categories
edited Jul 26 at 8:47
user64494
2,1129 silver badges18 bronze badges
asked Jul 25 at 14:37
Ben SprottBen Sprott
7174 silver badges17 bronze badges
$endgroup$
3
$begingroup$
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
Jul 25 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
Jul 25 at 19:30
6
$begingroup$
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
Jul 25 at 21:47
4
$begingroup$
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
Jul 25 at 21:48
add a comment |
$begingroup$
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.
ct.category-theory groupoids 2-categories locally-presentable-categories
edited Jul 26 at 8:47
user64494
2,1129 silver badges18 bronze badges
asked Jul 25 at 14:37
Ben SprottBen Sprott
7174 silver badges17 bronze badges
$endgroup$
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.
ct.category-theory groupoids 2-categories locally-presentable-categories
ct.category-theory groupoids 2-categories locally-presentable-categories
edited Jul 26 at 8:47
user64494
2,1129 silver badges18 bronze badges
asked Jul 25 at 14:37
Ben SprottBen Sprott
7174 silver badges17 bronze badges
edited Jul 26 at 8:47
user64494
2,1129 silver badges18 bronze badges
asked Jul 25 at 14:37
Ben SprottBen Sprott
7174 silver badges17 bronze badges
edited Jul 26 at 8:47
user64494
2,1129 silver badges18 bronze badges
edited Jul 26 at 8:47
user64494
2,1129 silver badges18 bronze badges
edited Jul 26 at 8:47
user64494
2,1129 silver badges18 bronze badges
2,1129 silver badges18 bronze badges
asked Jul 25 at 14:37
Ben SprottBen Sprott
7174 silver badges17 bronze badges
asked Jul 25 at 14:37
Ben SprottBen Sprott
7174 silver badges17 bronze badges
asked Jul 25 at 14:37
Ben SprottBen Sprott
7174 silver badges17 bronze badges
7174 silver badges17 bronze badges
3
$begingroup$
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
Jul 25 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
Jul 25 at 19:30
6
$begingroup$
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
Jul 25 at 21:47
4
$begingroup$
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
Jul 25 at 21:48
add a comment |
3
$begingroup$
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
Jul 25 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
Jul 25 at 19:30
6
$begingroup$
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
Jul 25 at 21:47
4
$begingroup$
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
Jul 25 at 21:48
3
3
$begingroup$
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
Jul 25 at 14:47
$begingroup$
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
Jul 25 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
Jul 25 at 19:30
$begingroup$
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
Jul 25 at 19:30
6
6
$begingroup$
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
Jul 25 at 21:47
$begingroup$
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
Jul 25 at 21:47
4
4
$begingroup$
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
Jul 25 at 21:48
$begingroup$
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
Jul 25 at 21:48
add a comment |
1 Answer
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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.
answered Jul 25 at 18:24
Piotr PstrągowskiPiotr Pstrągowski
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$begingroup$
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.
answered Jul 25 at 18:24
Piotr PstrągowskiPiotr Pstrągowski
7131 gold badge8 silver badges19 bronze badges
$endgroup$
add a comment |
$begingroup$
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.
answered Jul 25 at 18:24
Piotr PstrągowskiPiotr Pstrągowski
7131 gold badge8 silver badges19 bronze badges
$endgroup$
add a comment |
$begingroup$
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.
answered Jul 25 at 18:24
Piotr PstrągowskiPiotr Pstrągowski
7131 gold badge8 silver badges19 bronze badges
$endgroup$
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.
answered Jul 25 at 18:24
Piotr PstrągowskiPiotr Pstrągowski
7131 gold badge8 silver badges19 bronze badges
answered Jul 25 at 18:24
Piotr PstrągowskiPiotr Pstrągowski
7131 gold badge8 silver badges19 bronze badges
answered Jul 25 at 18:24
Piotr PstrągowskiPiotr Pstrągowski
7131 gold badge8 silver badges19 bronze badges
answered Jul 25 at 18:24
Piotr PstrągowskiPiotr Pstrągowski
7131 gold badge8 silver badges19 bronze badges
7131 gold badge8 silver badges19 bronze badges
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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
Jul 25 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
Jul 25 at 19:30
6
$begingroup$
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
Jul 25 at 21:47
4
$begingroup$
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
Jul 25 at 21:48