Decomposing Comonad Morphisms

Authors Danel Ahman , Tarmo Uustalu

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Author Details

Danel Ahman
  • Faculty of Physics and Mathematics, University of Ljubljana, Slovenia
Tarmo Uustalu
  • Department of Computer Science, Reykjavik University, Iceland
  • Dept. of Software Science, Tallinn University of Technology, Estonia


We thank our anonymous reviewers for very useful remarks.

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Danel Ahman and Tarmo Uustalu. Decomposing Comonad Morphisms. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The analysis of set comonads whose underlying functor is a container functor in terms of directed containers makes it a simple observation that any morphism between two such comonads factors through a third one by two comonad morphisms, whereof the first is identity on shapes and the second is identity on positions in every shape. This observation turns out to generalize into a much more involved result about comonad morphisms to comonads whose underlying functor preserves Cartesian natural transformations to itself on any category with finite limits. The bijection between comonad coalgebras and comonad morphisms from costate comonads thus also yields a decomposition of comonad coalgebras.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Categorical semantics
  • container functors (polynomial functors)
  • container comonads
  • comonad morphisms and comonad coalgebras
  • cofunctors
  • lenses


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