Freely adjoining monoidal duals

Abstract

Given a monoidal category C with an object J, we construct a monoidal category C [J(v)] by freely adjoining a right dual J(v) to J. We show that the canonical strong monoidal functor Omega : C -> C [J(v)] 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 Omega : C -> C [J(v)] is fully faithful and provide coend formulas for homs of the form C [J(v)](U, Omega A) and C [J(v)]( Omega A, U) for A is an element of C and U is an element of C [J(v)]. If N denotes the free strict monoidal category on a single generating object 1, then N[1(v)] is the free monoidal category Dpr containing a dual pair - (sic) + of objects. As we have the monoidal pseudopushout C [J(v)] similar or equal to Dpr +(N) C, 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 Delta 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 (sic) X-1 (sic) X-2 ... of objects. Actually, Dpr is a monoidal full subcategory of Dseq.