Convolution Products on Double Categories and Categorification of Rule Algebras

Authors Nicolas Behr , Paul-André Melliès , Noam Zeilberger

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Author Details

Nicolas Behr
  • CNRS, Université Paris Cité, IRIF, France
Paul-André Melliès
  • CNRS, Université Paris Cité, INRIA, France
Noam Zeilberger
  • École Polytechnique, LIX, Palaiseau, France

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Nicolas Behr, Paul-André Melliès, and Noam Zeilberger. Convolution Products on Double Categories and Categorification of Rule Algebras. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Motivated by compositional categorical rewriting theory, we introduce a convolution product over presheaves of double categories which generalizes the usual Day tensor product of presheaves of monoidal categories. One interesting aspect of the construction is that this convolution product is in general only oplax associative. For that reason, we identify several classes of double categories for which the convolution product is not just oplax associative, but fully associative. This includes in particular framed bicategories on the one hand, and double categories of compositional rewriting theories on the other. For the latter, we establish a formula which justifies the view that the convolution product categorifies the rule algebra product.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Categorical rewriting
  • double pushout
  • sesqui-pushout
  • double categories
  • convolution product
  • presheaf categories
  • framed bicategories
  • opfibrations
  • rule algebra


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