On traced monoidal closed categories

The structure theorem of Joyal, Street and Verity says that every traced monoidal category C arises as a monoidal full subcategory of the tortile monoidal category Int C. In this paper we focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into Int C has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixed-point computation. To make the paper more self-contained, we also include various background results for traced monoidal categories.