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?
$begingroup$
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 yesterday
Simon HenrySimon Henry
15.5k14990
$endgroup$
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 yesterday
Simon HenrySimon Henry
15.5k14990
$endgroup$
1
$begingroup$
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
18 hours ago
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
17 hours ago
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 yesterday
Simon HenrySimon Henry
15.5k14990
$endgroup$
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 yesterday
Simon HenrySimon Henry
15.5k14990
asked yesterday
Simon HenrySimon Henry
15.5k14990
edited yesterday
Simon Henry
asked yesterday
Simon HenrySimon Henry
15.5k14990
asked yesterday
Simon HenrySimon Henry
15.5k14990
asked yesterday
Simon HenrySimon Henry
15.5k14990
15.5k14990
1
$begingroup$
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
18 hours ago
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
17 hours ago
add a comment |
1
$begingroup$
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
18 hours ago
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
17 hours ago
1
1
$begingroup$
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
18 hours ago
$begingroup$
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
18 hours ago
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
17 hours ago
$begingroup$
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
17 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered yesterday
Alexander CampbellAlexander Campbell
2,32111316
$endgroup$
add a comment |
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1 Answer
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$begingroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered yesterday
Alexander CampbellAlexander Campbell
2,32111316
$endgroup$
add a comment |
$begingroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered yesterday
Alexander CampbellAlexander Campbell
2,32111316
$endgroup$
add a comment |
$begingroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered yesterday
Alexander CampbellAlexander Campbell
2,32111316
$endgroup$
The canonical reference is Chapter 5 of Greg Bird's thesis.
answered yesterday
Alexander CampbellAlexander Campbell
2,32111316
answered yesterday
Alexander CampbellAlexander Campbell
2,32111316
answered yesterday
Alexander CampbellAlexander Campbell
2,32111316
answered yesterday
Alexander CampbellAlexander Campbell
2,32111316
2,32111316
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
18 hours ago
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
17 hours ago