Freely adjoining monoidal duals

Kevin Coulembier, Ross Street*, Michel van den Bergh

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
JournalMathematical Structures in Computer Science
Early online date28 Oct 2020
Publication statusE-pub ahead of print - 28 Oct 2020


  • Autonomization
  • monoidal dual
  • string diagram
  • adjunction
  • biadjoint

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