A Double Categorical Framework for Multi-Stage Portfolio Construction and Alignment

TL;DR

This paper introduces a double categorical framework to model multi-stage portfolio construction, ensuring structural coherence, safety guarantees, and compliance, with extensions to stochastic and transport-based safety metrics.

Contribution

It formalizes portfolio re-implementation processes using a novel double category, providing structural theorems and extensions without requiring advanced category theory.

Findings

Establishes compositionality of alignment (functoriality)

Provides a pre-trade safety guarantee (adjunction)

Shows order-independence in compliance checking

Abstract

We construct a thin double category HS (Hub-and-Spoke) whose objects are closed subsets of standard simplices, horizontal morphisms are continuous maps representing portfolio re-implementation processes, and vertical morphisms are closed relations representing alignment constraints. This framework models industrial portfolio construction pipelines -- hierarchical structures in which a single investment strategy is translated through multiple stages into thousands of client portfolios. We establish four structural theorems: compositionality of alignment (functoriality), a pre-trade safety guarantee (adjunction), an order-independence result for compliance checking (lax Beck--Chevalley), and a filter-commutation law (Frobenius reciprocity). The topological requirement that permissible portfolio spaces be closed and compact -- ruling out ``phantom portfolios'' that arise from open constraint specifications -- is shown to be essential for coherence. Extensions to set-valued re-implementations via the Double Operadic Theory of Systems, stochastic re-implementations via Markov kernels on Polish spaces, and transport-based safety metrics via Wasserstein distances are developed. An abstract axiomatic treatment identifies the equipment axioms sufficient for the main results. The mathematical content is elementary -- no novel category theory is required. The contribution is the modelling claim: that these particular objects and morphisms formalise portfolio re-implementation correctly.