Fungi as functors: A category-theoretic approach to mycelial organisation

2026-03-01

AI Summary

This study addresses the absence of a rigorous mathematical framework for modeling the organizational structure of fungal mycelial networks, which has hindered systematic analysis of the interplay among environmental perturbations, species variation, and ecological feedback. The work introduces category theory into fungal modeling for the first time, formalizing individual fungi as functors from an environmental state category to a mycelial network state category. By leveraging functorial semantics, natural transformations, adjunctions, and pushout constructions, the framework characterizes environmental interactions, sequences of perturbations, hyphal fusion, and interspecies differences. It reveals testable second-order effects arising from the non-commutativity of perturbation order and quantifies procedural sequence effects via local Lie group structures and the Baker–Campbell–Hausdorff expansion. This approach establishes a falsifiable mathematical foundation for cross-species comparison and provides novel tools for analyzing mixed perturbation responses, robustness limits, and structured ecological dynamics.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We develop a rigorous, equation-free category-theoretic foundation for fungal organisation. A fungal organism is formalised as a functor from a category $\Env$ of structured environmental states and admissible transformations to a category $\Myc$ of mycelial network states and biologically meaningful morphisms. An operational program category $\Prog$ models time-ordered exposure protocols, and a semantics functor $\mathcal{F}_{\mathrm{prog}}:\Prog\to\Myc$ maps experimental perturbations to induced network transformations. Species and strain variability are expressed as natural transformations between fungal functors, and ecological feedback is captured via an adjunction between sensing and environment modification. Network fusion (anastomosis) is identified with pushouts in $\Myc$, and order effects in exposure sequences are quantified by a local Lie structure and a Baker--Campbell--Hausdorff expansion near the identity program. A minimal worked exposure example demonstrates how non-commutativity yields experimentally testable quadratic scaling of order asymmetry. The framework provides a structurally explicit and falsifiable basis for analysing compositional perturbations, mixture coupling, robustness limits, and cross-species comparability in fungal systems.

Problem

• fungal organisation
• mycelial networks
• environmental perturbations
• anastomosis
• order effects

Innovation

• category theory
• functor
• mycelial networks
• natural transformation
• adjunction