Freely adjoining monoidal duals

Given a monoidal category Formula Presented with an object J, we construct a monoidal category Formula Presented by freely adjoining a right dual Formula Presented to J. We show that the canonical strong monoidal functor Formula Presented provides the unit for a biadjunction with the forgetful 2-functor from the 2-category of monoidal categories with a distinguished dual pair to the 2-category of monoidal categories with a distinguished object. We show that Formula Presented is fully faithful and provide coend formulas for homs of the form Formula Presented and Formula Presented for Formula Presented and Formula Presented. If Formula Presented denotes the free strict monoidal category on a single generating object 1, then Formula Presented is the free monoidal category Dpr containing a dual pair - ˧ + of objects. As we have the monoidal pseudopushout Formula Presented, it is of interest to have an explicit model of Dpr: we provide both geometric and combinatorial models. We show that the (algebraist’s) simplicial category Δ is a monoidal full subcategory of Dpr and explain the relationship with the free 2-category Adj containing an adjunction. We describe a generalization of Dpr which includes, for example, a combinatorial model Dseq for the free monoidal category containing a duality sequence X ₀ ˧ X ₁ ˧ X ₂ ˧ … of objects. Actually, Dpr is a monoidal full subcategory of Dseq.