[Paper Review] Decorated Corelations

Brendan Fong|arXiv (Cornell University)|Mar 29, 2017

Homotopy and Cohomology in Algebraic Topology8 citations

TL;DR

This paper introduces decorated corelations as a method to construct hypergraph categories—symmetric monoidal categories where each object has a special commutative Frobenius monoid structure—by combining corelations (cospans factored via an (E,M)-factorisation system) with a symmetric lax monoidal functor into Set. The key contribution is a general construction that enables efficient, semantics-preserving composition of network-like systems, such as electrical circuits, by discarding irrelevant internal structure during composition, and provides a systematic way to define hypergraph functors via natural transformations between decorating functors.

ABSTRACT

Let $\mathcal C$ be a category with finite colimits, and let $(\mathcal E,\mathcal M)$ be a factorisation system on $\mathcal C$ with $\mathcal M$ stable under pushouts. Writing $\mathcal C;\mathcal M^{\mathrm{op}}$ for the symmetric monoidal category with morphisms cospans of the form $\stackrel{c} o \stackrel{m}\leftarrow$, where $c \in \mathcal C$ and $m \in \mathcal M$, we give method for constructing a category from a symmetric lax monoidal functor $F\colon (\mathcal C; \mathcal M^{\mathrm{op}},+) o (\mathrm{Set}, imes)$. A morphism in this category, termed a \emph{decorated corelation}, comprises (i) a cospan $X o N \leftarrow Y$ in $\mathcal C$ such that the canonical copairing $X+Y o N$ lies in $\mathcal E$, together with (ii) an element of $FN$. Functors between decorated corelation categories can be constructed from natural transformations between the decorating functors $F$. This provides a general method for constructing hypergraph categories---symmetric monoidal categories in which each object is a special commutative Frobenius monoid in a coherent way---and their functors. Such categories are useful for modelling network languages, for example circuit diagrams, and such functors their semantics.

Motivation & Objective

• To develop a category-theoretic framework for modeling network languages such as circuit diagrams in a way that respects their compositional and semantic structure.
• To address the inefficiency of decorated cospans in retaining irrelevant internal details during composition, especially in semantic models.
• To provide a general construction of hypergraph categories using corelations and symmetric lax monoidal functors, ensuring coherence and compatibility with network-style composition.
• To enable the construction of hypergraph functors between such categories via natural transformations between decorating functors, facilitating semantic mappings.
• To demonstrate the method on concrete examples, including linear relations (LinRel) and electrical circuits, showing its utility in practical modeling.

Proposed method

• Use corelations—cospans X →N ←Y where the copairing X+Y →N lies in E from an (E,M)-factorisation system on a category C with finite colimits.
• Construct a symmetric monoidal category C;M^op whose morphisms are cospans with apex in C and legs in M, with monoidal structure given by disjoint union.
• Apply a symmetric lax monoidal functor F: (C;M^op, +) →(Set, ×) to decorate corelations with additional data, such as subspaces or circuit configurations.
• Define decorated corelations as a pair: a corelation X →N ←Y and an element of FN, with composition defined via pushouts and functoriality of F.
• Ensure that the resulting category inherits a hypergraph category structure when M is stable under pushouts, via the Frobenius monoid structure induced by the corelation composition.
• Construct functors between decorated corelation categories via natural transformations between the decorating functors F and G, preserving the hypergraph structure.

Experimental results

Research questions

• RQ1 How can we construct a hypergraph category that models network composition while discarding irrelevant internal structure?
• RQ2 What conditions on a factorisation system (E,M) ensure that corelations form a hypergraph category?
• RQ3 How can decorated corelations be used to define hypergraph functors between network models, such as from circuit diagrams to linear relations?
• RQ4 Can the construction of LinRel as a decorated corelation category be derived from a general method, and what does this imply for semantics of non-controllable systems?
• RQ5 In what way does the decorated corelation approach improve upon decorated cospans for modeling semantics of open systems?

Key findings

• The paper establishes that if C has finite colimits and (E,M) is a factorisation system with M stable under pushouts, then corelations in C form a hypergraph category.
• Decorated corelations are defined as corelations X →N ←Y (with copairing in E) decorated by an element of FN, where F is a symmetric lax monoidal functor to (Set,×).
• The construction yields a hypergraph category whose morphisms are isomorphic to linear relations when applied to the category of finite sets with the isomorphism-morphism factorisation system and the functor Lin: FinSet →Set mapping to subspaces of k^N.
• The category LinCorel, constructed via decorated corelations, is isomorphic to LinRel—the category of finite-dimensional vector spaces and linear relations—providing a corelational foundation for linear systems.
• A hypergraph functor from the decorated cospan category of circuit diagrams to LinRel can be constructed via a monoidal natural transformation from the circuit functor Circ to the composition Lin∘γ, where γ embeds finite sets into cospans.
• The method generalizes to any hypergraph category, enabling the construction of hypergraph functors through natural transformations between decorating functors, thus unifying syntactic and semantic modeling.