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Types of Relations: Defining Analogies with Category Theory

Claire Ott, Frank Jäkel

TL;DR

The paper addresses formalizing analogy-making by representing knowledge domains as categories and defining analogies as structure-preserving functors between domain categories $\mathcal{C}$ and $\mathcal{D}$ via $F: \mathcal{C} \rightarrow \mathcal{D}$; it introduces domain categories built from base objects $E$, $B$, and $1_\mathcal{C}$, and uses products $X\times Y$ and exponentials $Y^X$ to model n-ary and higher-order relations. It then defines core and blend through pullbacks $\mathcal{C} \times_F \mathcal{D}$ and pushouts $\mathcal{C} +_F \mathcal{D}$, and extends to partial functors $F'$ to handle incomplete mappings; the solar system–hydrogen atom example concretely demonstrates truth-preserving, product-preserving, and exponential-preserving constraints. The framework connects to classic structure-mapping approaches, provides a precise mathematical theory of analogy, and points to practical future work in automated generation, probabilistic extensions, and richer domain types. Overall, it offers a rigorous, scalable method for constructing, comparing, and blending knowledge domains to support intelligent analogy-based reasoning.

Abstract

In order to behave intelligently both humans and machines have to represent their knowledge adequately for how it is used. Humans often use analogies to transfer their knowledge to new domains, or help others with this transfer via explanations. Hence, an important question is: What representation can be used to construct, find, and evaluate analogies? In this paper, we study features of a domain that are important for constructing analogies. We do so by formalizing knowledge domains as categories. We use the well-known example of the analogy between the solar system and the hydrogen atom to demonstrate how to construct domain categories. We also show how functors, pullbacks, and pushouts can be used to define an analogy, describe its core and a corresponding blend of the underlying domains.

Types of Relations: Defining Analogies with Category Theory

TL;DR

The paper addresses formalizing analogy-making by representing knowledge domains as categories and defining analogies as structure-preserving functors between domain categories and via ; it introduces domain categories built from base objects , , and , and uses products and exponentials to model n-ary and higher-order relations. It then defines core and blend through pullbacks and pushouts , and extends to partial functors to handle incomplete mappings; the solar system–hydrogen atom example concretely demonstrates truth-preserving, product-preserving, and exponential-preserving constraints. The framework connects to classic structure-mapping approaches, provides a precise mathematical theory of analogy, and points to practical future work in automated generation, probabilistic extensions, and richer domain types. Overall, it offers a rigorous, scalable method for constructing, comparing, and blending knowledge domains to support intelligent analogy-based reasoning.

Abstract

In order to behave intelligently both humans and machines have to represent their knowledge adequately for how it is used. Humans often use analogies to transfer their knowledge to new domains, or help others with this transfer via explanations. Hence, an important question is: What representation can be used to construct, find, and evaluate analogies? In this paper, we study features of a domain that are important for constructing analogies. We do so by formalizing knowledge domains as categories. We use the well-known example of the analogy between the solar system and the hydrogen atom to demonstrate how to construct domain categories. We also show how functors, pullbacks, and pushouts can be used to define an analogy, describe its core and a corresponding blend of the underlying domains.

Paper Structure

This paper contains 14 sections, 1 theorem, 1 equation, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Formalizing Domains as Categories
  3. Introduction to Categories
  4. Products: Binary and n-ary Relations
  5. Exponentials: Objectification of Relations
  6. Higher-order Relations
  7. Categories for the solar system and hydrogen atom
  8. Analogies between Domain Categories
  9. Functor: Comparing Domains
  10. Core and Blend
  11. Discussion
  12. Funding.
  13. acknowledgments
  14. Pullback and Pushout Combination

Key Result

Lemma A.1

Let $X$ and $Y$ be objects in a category with pullbacks and pushouts and $f:X\rightarrow Y$ be a morphism. If the left square in the diagram below forms a pullback with $X\times_fY$ and the right square forms a pushout with $X+_fY$, then $X\times_fY$ is a pullback of $X$ and $Y$ over $\phi_1$, $\phi

Figures (15)

  • Figure 1: The basic building blocks of a domain category.
  • Figure 2: (a) Product of two objects $X$ and $Y$: For any object $Z$ and morphisms $f$ and $g$ there is a unique morphism $u$ such that the diagram commutes. Dashed arrows indicate unique morphisms. (b) Example of a morphism describing a binary relation: $\text{attracts}$ is a morphism that maps pairs of entities to true or false.
  • Figure 3: A domain category with the object $E\times E$, equipped with morphisms describing attributes ($a$) and binary relations ($r$).
  • Figure 4: (a) An exponential consists of an object $Y^X$ and an evaluation morphism $\varepsilon_{X,Y}$ such that for all morphisms $f$ the above diagram commutes for a unique $\lambda(f)$. (b) The morphism $g\colon X \rightarrow Y$ must also be represented in $Y^X$ because it can be composed with a projection to a morphism form $Z\times X$ to $Y$ for some $Z\in \mathrm{Ob}(\mathcal{C})$.
  • Figure 5: Parts of the atom-category $\mathcal{A}$ showing $E$ and the product $E\times E$ as well as $B^{(E\times E)}$ and some morphisms.
  • ...and 10 more figures

Theorems & Definitions (14)

  • Definition 1: Category
  • Definition 2: Subcategory
  • Definition 3: Unit object
  • Definition 4: Product
  • Definition 5: Exponential
  • Definition 6: Domain Category
  • Definition 7: Functor
  • Definition 8: Partial functor
  • Definition 9: Analogy
  • Definition 10: Coproduct
  • ...and 4 more