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Freely adjoining monoidal duals

Published online by Cambridge University Press:  28 October 2020

Kevin Coulembier ,
Ross Street Open the ORCID record for Ross Street [Opens in a new window]  and
Michel van den Bergh
Kevin Coulembier
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, Australia
Ross Street*
Affiliation:
Department of Mathematics and Statistics, Macquarie University, Sydney, Australia
Michel van den Bergh
Affiliation:
Hasselt University, Department of Mathematics Physics Informatics, Diepenbeek, Belgium
*
*Corresponding author. Email: ross.street@mq.edu.au
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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 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.

Keywords

Autonomizationmonoidal dualstring diagramadjunctionbiadjoint
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 7: The Power Festschrift , August 2021 , pp. 748 - 768
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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