Publication
Freely adjoining monoidal duals
Publisher:
Cambridge University Press (CUP)
Date:
28-10-2020
DOI:
10.1017/S0960129520000274
Abstract: Given a monoidal category $\\mathcal C$ with an object J , we construct a monoidal category $\\mathcal C[{J^ \\vee }]$ by freely adjoining a right dual ${J^ \\vee }$ to J . We show that the canonical strong monoidal functor $\\Omega :\\mathcal C \\to \\mathcal C[{J^ \\vee }]$ 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 $\\Omega :\\mathcal C \\to \\mathcal C[{J^ \\vee }]$ is fully faithful and provide coend formulas for homs of the form $\\mathcal C[{J^ \\vee }](U,\\,\\Omega A)$ and $\\mathcal C[{J^ \\vee }](\\Omega A,U)$ for $A \\in \\mathcal C$ and $U \\in \\mathcal C[{J^ \\vee }]$ . If ${\\rm{N}}$ denotes the free strict monoidal category on a single generating object 1, then ${\\rm{N[}}{{\\rm{1}}^ \\vee }{\\rm{]}}$ is the free monoidal category Dpr containing a dual pair – ˧ + of objects. As we have the monoidal pseudopushout $\\mathcal C[{J^ \\vee }] \\simeq {\\rm{Dpr}}{{\\rm{ + }}_{\\rm{N}}}\\mathcal C$ , 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 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 ex le, 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.