6. Mac Lane, Lawvere, and Foundational Mappings
Categorical Semantics: Moving Beyond Elements
To formalize our NF Ontology computationally, we must transition from defining objects by their internal elements (standard set theory) to defining objects by their external structural relationships (morphisms). This is the domain of Category Theory.
- Lawvere, F. W. (1964). The Category of Categories as a Foundation for Mathematics.
- Mac Lane, S. (1998). Categories for the Working Mathematician. Ch. I (§1–§4, §8), Ch. III (§1), Ch. IV (§1, §9).
Saunders Mac Lane: Arrow-Based Definitions
In Categories for the Working Mathematician (1971), Saunders Mac Lane systematically replaces element-based definitions with arrow-based (morphism) definitions.
A category is defined by objects and arrows (morphisms), governed by associativity and identity.
- Functors: Morphisms between categories that strictly preserve identity and composition.
- Natural Transformations: Continuous morphisms between functors, ensuring mapping diagrams commute.
- Universal Constructions: Defining structures via unique factorization mechanics (e.g., limits, pullbacks, tensor products).
- Adjunctions: Formalizing the relationship between free constructions (left adjoint) and underlying constraints (right adjoint).
F. William Lawvere: Category of Categories (ETAC)
In 1965, F. William Lawvere published The Category of Categories as a Foundation for Mathematics, proposing a radical departure from Zermelo-Fraenkel (ZFC). Lawvere eliminated sets and element membership entirely as foundational primitives.
The Elementary Theory of Abstract Categories (ETAC) builds mathematics using only a first-order language with unary functions for domain/codomain and a ternary relation for composition. This framework perfectly supports structural mathematics without the ontological baggage of assuming everything is built from atomic sets inside spatial containers.
- Holmes, M. R. (2012). Elementary Set Theory with a Universal Set. Ch. 4 (Building Finite Structures), Ch. 5 (The Theory of Relations), Ch. 10 (Introducing Functions).
- Forster, T. (1995). Set Theory with a Universal Set. Section 2.1.2 (Boffa’s lemma and automorphism).
- Wilshaw, S. (2025). New Foundations is Consistent. Sections 2.1, 2.3, 2.5 (The structural hierarchy and non-spatial model data).
The Pivot to JSON: Modeling Transclusion
Up until now, you have been writing raw First-Order Logic to understand how syntax (Stratification) acts as the operational authorization for set formation.
Now, we must translate this logic into a machine-readable format to build our Categorical Ontology.
Why JSON? As discussed in Module 1, the internet attempts (and largely fails) to produce a non-well-founded graph of pure transclusion. Physical hardware struggles with infinite recursion; you cannot put a spatial folder inside itself without crashing a hard drive.
JSON (JavaScript Object Notation), however, is uniquely suited to model Quinean ontology. 1. It is Flat and Non-Spatial: A JSON object is not a physical box. It is a dictionary of key-value pairs. 2. Syntactic Authorization: By defining object references via ID strings (e.g., "source_node": "NodeA") instead of nested arrays, JSON creates a flat, self-referential graph. It points to objects operationally rather than nesting them spatially.
This makes JSON the perfect semantic vessel to model the Universal Set (\(V \in V\)) and complex categorical morphisms without triggering a mechanical stack overflow.
The JSON Ontology Blueprint
Before solving the following problems, review the core schema format we will use to model Categorical objects and morphisms. Note the specific keys (id, type, source, target):
{
"id": "Morphism_f",
"type": "morphism",
"source": "NodeA",
"target": "NodeB",
"constraints": ["associative"]
}type: Identifies the mathematical entity (object,morphism,functor).source/target: The domains and codomains (can be single strings or arrays of strings for products).constraints: A list of rules ensuring this entity obeys specific axioms.
Guided Example: JSON Morphism Anatomy
If you want to represent the morphism \(f: A \to B\) in JSON, you need to specify its domain and codomain explicitly.
{
"id": "Morphism_f",
"type": "morphism",
"source": "_____",
"target": "_____"
}(Hint: What are the ID strings for the objects representing the domain and codomain in \(f: A \to B\)?)
Problem Set: JSON Relational Morphisms
Problem 6.1: Morphisms vs Elements
In your NF JSON ontology, you cannot rely on looking “inside” a set to determine its properties. You must define it externally via morphisms. Write the JSON morphism structure that defines an abstract Universal Cone (Limit) connecting a vertex object LimitVertex to a target functor BaseFunctor.
Problem 6.2: The Subobject Classifier
How will you represent the Subobject Classifier (\(\Omega\)) in your JSON ontology? Will it map to standard Boolean values, or will you use a multi-valued logic structure to represent subobjects via key-value syntactic authorization?