Is there any references on the tensor product of presentable (1-)categories?












11












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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







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$endgroup$








  • 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): $mathrm{Set}$ is an idempotent algebra in $mathrm{Pr}^L$ 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
















11












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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): $mathrm{Set}$ is an idempotent algebra in $mathrm{Pr}^L$ 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














11












11








11


4



$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 23 at 15:02







Simon Henry

















asked Mar 23 at 11:56









Simon HenrySimon Henry

15.6k14991




15.6k14991








  • 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): $mathrm{Set}$ is an idempotent algebra in $mathrm{Pr}^L$ 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




    $begingroup$
    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): $mathrm{Set}$ is an idempotent algebra in $mathrm{Pr}^L$ 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): $mathrm{Set}$ is an idempotent algebra in $mathrm{Pr}^L$ 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): $mathrm{Set}$ is an idempotent algebra in $mathrm{Pr}^L$ 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










1 Answer
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The canonical reference is Chapter 5 of Greg Bird's thesis.






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    14












    $begingroup$

    The canonical reference is Chapter 5 of Greg Bird's thesis.






    share|cite|improve this answer









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      14












      $begingroup$

      The canonical reference is Chapter 5 of Greg Bird's thesis.






      share|cite|improve this answer









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        14












        14








        14





        $begingroup$

        The canonical reference is Chapter 5 of Greg Bird's thesis.






        share|cite|improve this answer









        $endgroup$



        The canonical reference is Chapter 5 of Greg Bird's thesis.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 23 at 13:51









        Alexander CampbellAlexander Campbell

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