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Trees in Coalgebra from Generalized Reachability

Thorsten Wißmann, Bálint Kocsis, Jurriaan Rot, Ruben Turkenburg

TL;DR

This work develops a unified coalgebraic framework for generalized reachability and tree structures by combining a universal property with an iterative level construction. Central to the approach is generalized minimality via factorization systems and precise morphisms, which captures both reachability and tree unravellings as instances of a single theory. The authors show that, for analytic set functors, the iterative construction yields the tree unravelling and that universal properties characterize trees as subcoalgebras without proper unravellings. The results recover classical notions for many functors (e.g., bag, automata) while explaining why powerset coalgebras are rarely trees, and pave the way for broader applications to graph-like structures and recursion on infinite trees.

Abstract

An automaton is called reachable if every state is reachable from the initial state. This notion has been generalized coalgebraically in two ways: first, via a universal property on pointed coalgebras, namely, that a reachable coalgebra has no proper subcoalgebras; and second, a coalgebra is reachable if it arises as the union of an iterative computation of successor states, starting from the initial state. In the current paper, we present corresponding universal properties and iterative constructions for trees. The universal property captures when a coalgebra is a tree, namely, when it has no proper tree unravellings. The iterative construction unravels an arbitrary coalgebra to a tree. We show that this yields the expected notion of tree for a variety of standard examples. We obtain our characterization of trees by first generalizing the previous theory of reachable coalgebras and of a minimal object in a category, related to projectivity. Surprisingly, both the universal property and the iterative construction for trees arise as instances of this generalized notion of reachability. Our iterative construction works for all analytic set functors.

Trees in Coalgebra from Generalized Reachability

TL;DR

This work develops a unified coalgebraic framework for generalized reachability and tree structures by combining a universal property with an iterative level construction. Central to the approach is generalized minimality via factorization systems and precise morphisms, which captures both reachability and tree unravellings as instances of a single theory. The authors show that, for analytic set functors, the iterative construction yields the tree unravelling and that universal properties characterize trees as subcoalgebras without proper unravellings. The results recover classical notions for many functors (e.g., bag, automata) while explaining why powerset coalgebras are rarely trees, and pave the way for broader applications to graph-like structures and recursion on infinite trees.

Abstract

An automaton is called reachable if every state is reachable from the initial state. This notion has been generalized coalgebraically in two ways: first, via a universal property on pointed coalgebras, namely, that a reachable coalgebra has no proper subcoalgebras; and second, a coalgebra is reachable if it arises as the union of an iterative computation of successor states, starting from the initial state. In the current paper, we present corresponding universal properties and iterative constructions for trees. The universal property captures when a coalgebra is a tree, namely, when it has no proper tree unravellings. The iterative construction unravels an arbitrary coalgebra to a tree. We show that this yields the expected notion of tree for a variety of standard examples. We obtain our characterization of trees by first generalizing the previous theory of reachable coalgebras and of a minimal object in a category, related to projectivity. Surprisingly, both the universal property and the iterative construction for trees arise as instances of this generalized notion of reachability. Our iterative construction works for all analytic set functors.

Paper Structure

This paper contains 13 sections, 40 theorems, 35 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Precise Morphisms
  4. Precise Factorizations
  5. Comparison to least bounds
  6. Generalized Minimality
  7. Generalized Reachability
  8. Universal Property
  9. Construction
  10. Trees
  11. Universal Property
  12. Construction
  13. Conclusions and Future Work

Key Result

Proposition 2.10

If $F \colon \mathcal{C} \to \mathcal{D}$ preserves $\mathcal{M}$ (that is, $Fm \in \mathcal{M}$ for every $m \in \mathcal{M}$), then everyNote that this includes non-proper factorization systems; earlier proofs, e.g. by Kurz kurzPhd, impose restrictions on $\mathcal{E}$ and $\mathcal{M}$.$(\mathcal

Figures (9)

  • Figure 1: Relations between different minimality concepts in the realm of coalgebra and objects in an abstract category. An edge $X Y$ indicates that concept $X$ generalizes concept $Y$. Previous Version refers to the conference paper WKRT25 that the present paper is extending.
  • Figure 2: A non-precise map $f$ that factors through the $H_B$-precise $p$ for $H_B X = X\times X+ {\emptyset}$WDKH2019.
  • Figure 3: Surjective coalgebra morphisms without splittings.
  • Figure 4: $\mathbb{R}^{(-)}$-coalgebra morphisms $g$ and $h$
  • Figure 5: Unravelling for $H_B X=X\times X + {\emptyset}$.
  • ...and 4 more figures

Theorems & Definitions (125)

  • Definition 2.2
  • Definition 2.4
  • Example 2.5
  • Definition 2.6: joyal81joyal86
  • Definition 2.7: joyofcats
  • Definition 2.9
  • Proposition 2.10: Wissmann22
  • Remark 2.11
  • Definition 2.12
  • Example 2.13
  • ...and 115 more