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

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.

Figures (31)

• Figure 1: Visualizing a 2-Cell in the $\mathbb{HS}$ Double Category.
• Figure 2: Visualization of the (compact) 2-simplex $\Delta^2$ as a subset of the (non-compact) space $\mathbb{R}^3$. Any valid portfolio $p=(x_0, x_1, x_2)$ lies on the triangular plane segment where weights sum to 1 and are non-negative.
• Figure 3: Visualization of the permissible portfolio space $K \subseteq \Delta^2$ from Example \ref{['ex:permissible']}. The ambient simplex (gray) represents all possible portfolios. The permissible region $K$ (blue polygon) is defined by the intersection of the constraints $x_0 \le 0.5$ (Stock Cap) and $x_2 \ge 0.1$ (Cash Floor).
• Figure 4: Visualization of an alignment relation $R \subseteq K_1 \times K_3$ as a closed subset (purple) of the product space. The closedness guarantees that limits of aligned portfolios remain aligned.
• Figure 5: 3D visualization of vertical composition. The relations $R$ (red) and $S$ (orange) are irregular polygons strictly contained within the product of the permissible sets (indicated by the dotted blue frames). The relation $S\circ R$ (blue) is a polygonal subset of the $K_1$--$K_3$ plane.

Theorems & Definitions (255)

• Remark 1.1: Relaxing the long-only restriction
• Remark 1.2: Asset universes
• Definition 2.1: Asset Set
• Definition 2.2: Ambient Portfolio Space
• Proposition 2.3: Simplex Properties
• Definition 2.4: Permissible Portfolio Space (Object)
• Remark 2.5
• Example 2.6: Permissible Region in $\Delta^2$
• Definition 3.1: Re-implementation (Horizontal Morphism)
• Example 3.2: Sector Aggregation