Symmetric monoidal Ind-completion

Theorem 22.5.4 (Symmetric monoidal Ind-completion, [Lurie (2017), Corollary 4.8.1.14]). Let \(C\) be a small stable symmetric monoidal \(\infty \)-category, and assume that the tensor product on \(C\) is exact separately in both variables. Then \(\Ind (C)\) admits a unique symmetric monoidal structure with the following properties:

(1)

The Yoneda embedding \(C\hookrightarrow \Ind (C)\) is symmetric monoidal.

(2)

The tensor product on \(\Ind (C)\) preserves colimits separately in both variables.

For filtered-colimit presentations \(X\simeq \colim _iX_i\) and \(Y\simeq \colim _jY_j\) with \(X_i,Y_j\in C\), its underlying tensor product is characterized by \[ X\otimes Y\simeq \colim _{(i,j)}(X_i\otimes Y_j). \]