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Coherence by Normalization for Linear Multicategorical Structures

Author Federico Olimpieri

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ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Categorical semantics
Keywords
  • Coherence
  • Linear Multicategories
  • Resource Calculi
  • Normalization

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Abstract

We establish a formal correspondence between resource calculi and appropriate linear multicategories. We consider the cases of (symmetric) representable, symmetric closed and autonomous multicategories. For all these structures, we prove that morphisms of the corresponding free constructions can be presented by means of typed resource terms, up to a reduction relation and a structural equivalence. Thanks to the linearity of the calculi, we can prove strong normalization of the reduction by combinatorial methods, defining appropriate decreasing measures. From this, we achieve a general coherence result: morphisms that live in the free multicategorical structures are the same whenever the normal forms of the associated terms are equal. As further application, we obtain syntactic proofs of Mac Lane’s coherence theorems for (symmetric) monoidal categories.

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Federico Olimpieri. Coherence by Normalization for Linear Multicategorical Structures. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 43:1-43:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CSL.2024.43

Author Details

Federico Olimpieri
  • School of Mathematics, University of Leeds, UK

Funding

  • Olimpieri, Federico: This work was supported by the US Air Force Office for Scientific Research under award number FA9550-21-1-0007.

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