Given a monoidal category C with an object J, we construct a monoidal category C [J∨] by freely adjoining a right dual J∨ to J. We show that the canonical strong monoidal functor Ω : C →C [J∨] 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 Ω : C →C [J∨] is fully faithful and provide coend formulas for homs of the form C [J∨](U,ΩA) and C [J∨](ΩA, U) for A ∈ C and U ∈ C [J∨]. If N denotes the free strict monoidal category on a single generating object 1, then N[1∨] is the free monoidal category Dpr containing a dual pair −⊣+ of objects. As we have the monoidal pseudopushout C [J∨]≃Dpr +NC , 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 X0 ⊣ X1 ⊣ X2 . . . of objects. Actually, Dpr is a monoidal full subcategory of Dseq.
- monoidal dual
- string diagram