Is there any references on the tensor product of presentable (1-)categories?Example of a locally presentable $2$-category$infty$-ary tensor product on a categoryWhich categories are the categories of models of a Lawvere theory? Lower Algebra: Modules over the monoidal category of abelian groupsHomotopy theory of acyclic categoriesCan tangent ($infty$,1)-categories be described in terms of the higher Grothendieck construction?Model existence theorem in topos theoryHow to understand the Deligne' tensor product of finite abelian categoryLocally presentable abelian categories with enough injective objectsThe induced functor in the definition of Deligne's tensor product is exact?
Is there any references on the tensor product of presentable (1-)categories?
Example of a locally presentable $2$-category$infty$-ary tensor product on a categoryWhich categories are the categories of models of a Lawvere theory? Lower Algebra: Modules over the monoidal category of abelian groupsHomotopy theory of acyclic categoriesCan tangent ($infty$,1)-categories be described in terms of the higher Grothendieck construction?Model existence theorem in topos theoryHow to understand the Deligne' tensor product of finite abelian categoryLocally presentable abelian categories with enough injective objectsThe induced functor in the definition of Deligne's tensor product is exact?
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Is there any references on the tensor product of (locally) presentable categories ?
All I know about this is Lurie's book that deals with the $infty$-categorical version, and a few references that deals with special cases (Grothendieck abelian categories, toposes etc...)
Is there any references that defines it properly and proves the basic properties ?
reference-request ct.category-theory locally-presentable-categories
asked Mar 23 at 11:56
Simon HenrySimon Henry
16.1k15093
$endgroup$
add a comment |
$begingroup$
Is there any references on the tensor product of (locally) presentable categories ?
All I know about this is Lurie's book that deals with the $infty$-categorical version, and a few references that deals with special cases (Grothendieck abelian categories, toposes etc...)
Is there any references that defines it properly and proves the basic properties ?
reference-request ct.category-theory locally-presentable-categories
asked Mar 23 at 11:56
Simon HenrySimon Henry
16.1k15093
$endgroup$
2
$begingroup$
It's certainly not what you have in mind, but I think it might be worth mentioning that Lurie's work does cover this example too (although I don't think he works out the details in his book): $mathrmSet$ is an idempotent algebra in $mathrmPr^L$ and modules over it are precisely presentable 1-categories.
$endgroup$
– Denis Nardin
Mar 24 at 7:08
1
$begingroup$
Yes of course ! I know. But as you suspected, I was hopping for more elementary references that could also be read by people only familiar with ordinary category theory.
$endgroup$
– Simon Henry
Mar 24 at 7:45
add a comment |
$begingroup$
Is there any references on the tensor product of (locally) presentable categories ?
All I know about this is Lurie's book that deals with the $infty$-categorical version, and a few references that deals with special cases (Grothendieck abelian categories, toposes etc...)
Is there any references that defines it properly and proves the basic properties ?
reference-request ct.category-theory locally-presentable-categories
asked Mar 23 at 11:56
Simon HenrySimon Henry
16.1k15093
$endgroup$
Is there any references on the tensor product of (locally) presentable categories ?
All I know about this is Lurie's book that deals with the $infty$-categorical version, and a few references that deals with special cases (Grothendieck abelian categories, toposes etc...)
Is there any references that defines it properly and proves the basic properties ?
reference-request ct.category-theory locally-presentable-categories
reference-request ct.category-theory locally-presentable-categories
asked Mar 23 at 11:56
Simon HenrySimon Henry
16.1k15093
asked Mar 23 at 11:56
Simon HenrySimon Henry
16.1k15093
edited Mar 23 at 15:02
Simon Henry
asked Mar 23 at 11:56
Simon HenrySimon Henry
16.1k15093
asked Mar 23 at 11:56
Simon HenrySimon Henry
16.1k15093
asked Mar 23 at 11:56
Simon HenrySimon Henry
16.1k15093
16.1k15093
2
$begingroup$
It's certainly not what you have in mind, but I think it might be worth mentioning that Lurie's work does cover this example too (although I don't think he works out the details in his book): $mathrmSet$ is an idempotent algebra in $mathrmPr^L$ and modules over it are precisely presentable 1-categories.
$endgroup$
– Denis Nardin
Mar 24 at 7:08
1
$begingroup$
Yes of course ! I know. But as you suspected, I was hopping for more elementary references that could also be read by people only familiar with ordinary category theory.
$endgroup$
– Simon Henry
Mar 24 at 7:45
add a comment |
2
$begingroup$
It's certainly not what you have in mind, but I think it might be worth mentioning that Lurie's work does cover this example too (although I don't think he works out the details in his book): $mathrmSet$ is an idempotent algebra in $mathrmPr^L$ and modules over it are precisely presentable 1-categories.
$endgroup$
– Denis Nardin
Mar 24 at 7:08
1
$begingroup$
Yes of course ! I know. But as you suspected, I was hopping for more elementary references that could also be read by people only familiar with ordinary category theory.
$endgroup$
– Simon Henry
Mar 24 at 7:45
2
2
$begingroup$
It's certainly not what you have in mind, but I think it might be worth mentioning that Lurie's work does cover this example too (although I don't think he works out the details in his book): $mathrmSet$ is an idempotent algebra in $mathrmPr^L$ and modules over it are precisely presentable 1-categories.
$endgroup$
– Denis Nardin
Mar 24 at 7:08
$begingroup$
It's certainly not what you have in mind, but I think it might be worth mentioning that Lurie's work does cover this example too (although I don't think he works out the details in his book): $mathrmSet$ is an idempotent algebra in $mathrmPr^L$ and modules over it are precisely presentable 1-categories.
$endgroup$
– Denis Nardin
Mar 24 at 7:08
1
1
$begingroup$
Yes of course ! I know. But as you suspected, I was hopping for more elementary references that could also be read by people only familiar with ordinary category theory.
$endgroup$
– Simon Henry
Mar 24 at 7:45
$begingroup$
Yes of course ! I know. But as you suspected, I was hopping for more elementary references that could also be read by people only familiar with ordinary category theory.
$endgroup$
– Simon Henry
Mar 24 at 7:45
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered Mar 23 at 13:51
Alexander CampbellAlexander Campbell
2,44611417
$endgroup$
add a comment |
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$begingroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered Mar 23 at 13:51
Alexander CampbellAlexander Campbell
2,44611417
$endgroup$
add a comment |
$begingroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered Mar 23 at 13:51
Alexander CampbellAlexander Campbell
2,44611417
$endgroup$
add a comment |
$begingroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered Mar 23 at 13:51
Alexander CampbellAlexander Campbell
2,44611417
$endgroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered Mar 23 at 13:51
Alexander CampbellAlexander Campbell
2,44611417
answered Mar 23 at 13:51
Alexander CampbellAlexander Campbell
2,44611417
answered Mar 23 at 13:51
Alexander CampbellAlexander Campbell
2,44611417
answered Mar 23 at 13:51
Alexander CampbellAlexander Campbell
2,44611417
2,44611417
add a comment |
add a comment |
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$begingroup$
It's certainly not what you have in mind, but I think it might be worth mentioning that Lurie's work does cover this example too (although I don't think he works out the details in his book): $mathrmSet$ is an idempotent algebra in $mathrmPr^L$ and modules over it are precisely presentable 1-categories.
$endgroup$
– Denis Nardin
Mar 24 at 7:08
1
$begingroup$
Yes of course ! I know. But as you suspected, I was hopping for more elementary references that could also be read by people only familiar with ordinary category theory.
$endgroup$
– Simon Henry
Mar 24 at 7:45