nLab
tensor product of presentable (infinity,1)-categories

Idea

The tensor product of presentable $(\mathrm{\infty },1)$-categories is the product in the symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories. See there for more details.

This is the $(\mathrm{\infty },1)$-category $C\otimes D$ which is ‘the universal recipient of a bilinear functor’ from $C\times D$. Here, we think of coproducts in $C$ and $D$ as addition, and then if a functor $C\times D\to E$ preserves colimits in each variable, in particular it preserves coproducts and so is ‘bilinear’. Such a bilinear functor will factor uniquely (in a homotopic sense) through a universal bilinear functor $C\times D\to C\otimes D$, just like for bilinear maps and tensor products of abelian groups.

Related concepts

• tensor product of abelian categories