Collapse-Selection as Idempotent Structure: A 2-Categorical Formulation of Stability and Invariance

This paper develops a categorical formulation of collapse-selection dynamics, modeling stability as invariance under a class of admissible endomorphisms.

Working in a bicategorical setting, admissible dynamics are represented as a lax natural family of endomorphisms. Under mild structural conditions—locality, non-expansiveness, and compatibility with coarse-graining—stabilization induces a lax idempotent comonad. Stable configurations arise as coalgebras of this comonad, forming a coreflective subcategory.

This formulation reframes a common pattern across dynamical systems—convergence toward stable structure—as a universal categorical phenomenon. Rather than depending on a specific dynamical law, stability emerges from invariance under a class of admissible transformations.

A key result is that collapse behaves analogously to a lax idempotent doctrine (in the sense of Kock–Zöberlein structures), but with a comonadic orientation: collapse selects admissible structure rather than freely generating it. This positions collapse-selection as a potential doctrine of admissibility, identifying maximal invariant substructures under constrained dynamics.

To ground the abstraction, the framework is instantiated in CPM(FHilb) from Categorical Quantum Mechanics, where collapse corresponds to decoherence and measurement, and stable objects correspond to classical subalgebras. This demonstrates that the proposed structure is not purely formal, but already present in standard quantum theory.

Context and Motivation

Many systems—physical, biological, computational—exhibit a shared structural behavior: repeated application of local update rules selects a restricted class of stable configurations. This work proposes that such behavior admits a unified description in terms of categorical structure.

Rather than treating collapse, measurement, or convergence as domain-specific mechanisms, this paper suggests they may be instances of a more general phenomenon:

stability as the fixed-point structure of a collapse-induced comonad.

Positioning

This work is intended as a structural proposal rather than a complete theory. It sits at the intersection of:

• category theory (idempotent comonads, coreflective subcategories, KZ doctrines)
• dynamical systems (stability, attractors, invariant structure)
• quantum theory (decoherence, measurement, classical emergence)

The goal is to identify a shared mathematical framework in which these phenomena can be studied uniformly.

Open Questions

• Can collapse be characterized as a canonical lax idempotent comonad?
• Does admissibility correspond to a factorization or localization structure?
• How do invariant families relate to universal constructions (limits, fibrations)?
• Can this framework be extended to enriched or higher categorical settings?