Do 2-categorical left adjoints preserve $mathcal{V}$-colimits?
Let $mathcal{C}$ be a category enriched in $mathcal{V}$, and $mathcal{D}$ be a category enriched in $mathcal{W}$. It is well known that $mathcal{V}Cat$, the category of $mathcal{V}$-enriched categories, forms a $2$-category.
If we have some $2$-functor, then, say $F : mathcal{V}Cat to mathcal{W}Cat$, that is (in the 2-categorical sense) left-adjoint to another $2$-functor $G : mathcal{W}Cat to mathcal{V}Cat$, then does $F$ necessarily preserve colimits in the sense that the object map of $F$ takes diagrams of $mathcal{V}$-colimits in $mathcal{C}$ to diagrams of $mathcal{W}$-colimits in $mathcal{D}$?
category-theory higher-category-theory enriched-category-theory
asked Sep 9 '17 at 23:35
Nathan BeDell
1,482815
add a comment |
Let $mathcal{C}$ be a category enriched in $mathcal{V}$, and $mathcal{D}$ be a category enriched in $mathcal{W}$. It is well known that $mathcal{V}Cat$, the category of $mathcal{V}$-enriched categories, forms a $2$-category.
If we have some $2$-functor, then, say $F : mathcal{V}Cat to mathcal{W}Cat$, that is (in the 2-categorical sense) left-adjoint to another $2$-functor $G : mathcal{W}Cat to mathcal{V}Cat$, then does $F$ necessarily preserve colimits in the sense that the object map of $F$ takes diagrams of $mathcal{V}$-colimits in $mathcal{C}$ to diagrams of $mathcal{W}$-colimits in $mathcal{D}$?
category-theory higher-category-theory enriched-category-theory
asked Sep 9 '17 at 23:35
Nathan BeDell
1,482815
What do you mean by "takes diagrams of $cal V$-colimits to diagrams of $cal W$-colimits"?
– Fosco Loregian
Sep 10 '17 at 15:48
@FoscoLoregian To be honest, I'm not quite sure. I assumed there would be an a theory analogous to the usual theory of diagram categories in standard category theory in enriched category theory, but perhaps this is not the case when one considers graded limits? In any case, my question I think could be reasonably interpreted (and generalized) as "Do 2-functors with a left adjoint (in the 2-categorical sense) preserve (weighted) 2-colimits?" I'd be interested to know the result for both strict and weak notions of 2-category (and 2-functor).
– Nathan BeDell
Sep 27 '17 at 22:40
add a comment |
Let $mathcal{C}$ be a category enriched in $mathcal{V}$, and $mathcal{D}$ be a category enriched in $mathcal{W}$. It is well known that $mathcal{V}Cat$, the category of $mathcal{V}$-enriched categories, forms a $2$-category.
If we have some $2$-functor, then, say $F : mathcal{V}Cat to mathcal{W}Cat$, that is (in the 2-categorical sense) left-adjoint to another $2$-functor $G : mathcal{W}Cat to mathcal{V}Cat$, then does $F$ necessarily preserve colimits in the sense that the object map of $F$ takes diagrams of $mathcal{V}$-colimits in $mathcal{C}$ to diagrams of $mathcal{W}$-colimits in $mathcal{D}$?
category-theory higher-category-theory enriched-category-theory
asked Sep 9 '17 at 23:35
Nathan BeDell
1,482815
Let $mathcal{C}$ be a category enriched in $mathcal{V}$, and $mathcal{D}$ be a category enriched in $mathcal{W}$. It is well known that $mathcal{V}Cat$, the category of $mathcal{V}$-enriched categories, forms a $2$-category.
If we have some $2$-functor, then, say $F : mathcal{V}Cat to mathcal{W}Cat$, that is (in the 2-categorical sense) left-adjoint to another $2$-functor $G : mathcal{W}Cat to mathcal{V}Cat$, then does $F$ necessarily preserve colimits in the sense that the object map of $F$ takes diagrams of $mathcal{V}$-colimits in $mathcal{C}$ to diagrams of $mathcal{W}$-colimits in $mathcal{D}$?
category-theory higher-category-theory enriched-category-theory
category-theory higher-category-theory enriched-category-theory
asked Sep 9 '17 at 23:35
Nathan BeDell
1,482815
asked Sep 9 '17 at 23:35
Nathan BeDell
1,482815
asked Sep 9 '17 at 23:35
Nathan BeDell
1,482815
asked Sep 9 '17 at 23:35
Nathan BeDell
1,482815
asked Sep 9 '17 at 23:35
Nathan BeDell
1,482815
1,482815
What do you mean by "takes diagrams of $cal V$-colimits to diagrams of $cal W$-colimits"?
– Fosco Loregian
Sep 10 '17 at 15:48
@FoscoLoregian To be honest, I'm not quite sure. I assumed there would be an a theory analogous to the usual theory of diagram categories in standard category theory in enriched category theory, but perhaps this is not the case when one considers graded limits? In any case, my question I think could be reasonably interpreted (and generalized) as "Do 2-functors with a left adjoint (in the 2-categorical sense) preserve (weighted) 2-colimits?" I'd be interested to know the result for both strict and weak notions of 2-category (and 2-functor).
– Nathan BeDell
Sep 27 '17 at 22:40
add a comment |
What do you mean by "takes diagrams of $cal V$-colimits to diagrams of $cal W$-colimits"?
– Fosco Loregian
Sep 10 '17 at 15:48
@FoscoLoregian To be honest, I'm not quite sure. I assumed there would be an a theory analogous to the usual theory of diagram categories in standard category theory in enriched category theory, but perhaps this is not the case when one considers graded limits? In any case, my question I think could be reasonably interpreted (and generalized) as "Do 2-functors with a left adjoint (in the 2-categorical sense) preserve (weighted) 2-colimits?" I'd be interested to know the result for both strict and weak notions of 2-category (and 2-functor).
– Nathan BeDell
Sep 27 '17 at 22:40
What do you mean by "takes diagrams of $cal V$-colimits to diagrams of $cal W$-colimits"?
– Fosco Loregian
Sep 10 '17 at 15:48
What do you mean by "takes diagrams of $cal V$-colimits to diagrams of $cal W$-colimits"?
– Fosco Loregian
Sep 10 '17 at 15:48
@FoscoLoregian To be honest, I'm not quite sure. I assumed there would be an a theory analogous to the usual theory of diagram categories in standard category theory in enriched category theory, but perhaps this is not the case when one considers graded limits? In any case, my question I think could be reasonably interpreted (and generalized) as "Do 2-functors with a left adjoint (in the 2-categorical sense) preserve (weighted) 2-colimits?" I'd be interested to know the result for both strict and weak notions of 2-category (and 2-functor).
– Nathan BeDell
Sep 27 '17 at 22:40
@FoscoLoregian To be honest, I'm not quite sure. I assumed there would be an a theory analogous to the usual theory of diagram categories in standard category theory in enriched category theory, but perhaps this is not the case when one considers graded limits? In any case, my question I think could be reasonably interpreted (and generalized) as "Do 2-functors with a left adjoint (in the 2-categorical sense) preserve (weighted) 2-colimits?" I'd be interested to know the result for both strict and weak notions of 2-category (and 2-functor).
– Nathan BeDell
Sep 27 '17 at 22:40
add a comment |
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What do you mean by "takes diagrams of $cal V$-colimits to diagrams of $cal W$-colimits"?
– Fosco Loregian
Sep 10 '17 at 15:48
@FoscoLoregian To be honest, I'm not quite sure. I assumed there would be an a theory analogous to the usual theory of diagram categories in standard category theory in enriched category theory, but perhaps this is not the case when one considers graded limits? In any case, my question I think could be reasonably interpreted (and generalized) as "Do 2-functors with a left adjoint (in the 2-categorical sense) preserve (weighted) 2-colimits?" I'd be interested to know the result for both strict and weak notions of 2-category (and 2-functor).
– Nathan BeDell
Sep 27 '17 at 22:40