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

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 ?

• 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. – Denis Nardin Mar 24 at 7:08
• 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. – Simon Henry Mar 24 at 7:45