Definition Infinite tensor products [efr-SIKM]

Let \mathcal {C} be a semiCartesian symmetric monoidal category and let \{X_j\}_{j \in J} be a family of objects. Then if P_f(J) denotes the set of finite subsets of J (ordered by inclusion), we obtain a diagram P_f(J)^\mathrm {op} \to \mathcal {C}, where F \mapsto \bigotimes _{j \in F} X_j, and the inclusion F \to F' is mapped to the morphism which applies the deletion X_j \to I for every j not in F.

An infinite tensor product of the collection \{X_j\} is a limit of the cofiltered diagram F \mapsto \bigotimes _{j \in F} X_j, if it exists and is preserved by the tensor Y \otimes - for every Y \in \mathcal {C}.