Arboreal Categories and Resources

Authors Samson Abramsky , Luca Reggio

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Samson Abramsky
  • Department of Computer Science, University of Oxford, UK
Luca Reggio
  • Department of Computer Science, University of Oxford, UK


Feedback from Tomáš Jakl and Dan Marsden is gratefully acknowledged.

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Samson Abramsky and Luca Reggio. Arboreal Categories and Resources. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or "static" structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a very general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fraïssé and modal bisimulation games recently introduced in [Abramsky et al., 2017; S. Abramsky and N. Shah, 2018; Abramsky and Shah, 2021] are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Theory of computation → Finite Model Theory
  • factorisation system
  • embedding
  • comonad
  • coalgebra
  • open maps
  • bisimulation
  • game
  • resources
  • relational structures
  • finite model theory


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