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 ?

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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): $\mathrm{Set}$ is an idempotent algebra in $\mathrm{Pr}^L$ and modules over it are precisely presentable 1-categories. $\endgroup$ Mar 24, 2019 at 7:08
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    $\begingroup$ 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. $\endgroup$ Mar 24, 2019 at 7:45

1 Answer 1


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


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