A Classificatory Topos: Refining Evolving Knowledge in Multi-agent Learning Systems

Abstract

We propose a Classificatory Topos, a mathematical framework to model the dynamic evolution of knowledge within a finite system of interacting learning machines. Following guidelines of category theory, the construction establishes a Grothendieck topos, $\text {Sh}(C_{\text {learn}}, J)$, as a mathematical universe for this problem domain. By defining a base site on a category of epistemic states with causal morphisms, and equipping it with a Grothendieck topology that formalizes a logic of justification, the framework provides a rich, non-linear model of system evolution. The use of sheaves ensures causal consistency, while the internal logic of the topos, governed by a subobject classifier, provides the machinery to trace, verify, and explain the refinement of classifications.