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

TL;DR

This paper introduces a category-theoretic framework to model fungal organization, capturing environmental interactions, network dynamics, and variability in a rigorous, equation-free manner, enabling testable predictions and comparative analyses.

Contribution

It develops a novel, formal category-theoretic foundation for fungal systems, integrating environmental states, network transformations, and ecological feedback without relying on equations.

Findings

Network fusion corresponds to pushouts in the category of mycelial networks.

Order effects are quantified using a local Lie structure and BCH expansion.

Non-commutativity leads to quadratic scaling of order asymmetry in experiments.

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.