Categorical Coding: Solving Coding Problems with Category Theory

If you've run across category theory, you've probably come away thinking that it was impossibly abstract and wholly impractical. But the wonderful thing about abstract math is that it also applies to concrete problems. The best way to learn how to use Category Theory is to use it to write code.

Category Theory studies relationships. Thinking about code in terms of relationships requires a paradigm shift, but it's a very powerful step that can help you see coding problems from an entirely new perspective. Rather than viewing code in terms of computation, you'll view code in terms of relationships and transformations. This allows us to solve problems much more flexibly and at an arbitrarily high level.

Category Theory Is About Relationships

Most programming focuses on what functions do. Category theory focuses on how things relate:

  • Can you derive B from A? (Does the source contain enough information?)
  • Is this transformation reversible? (Where is information lost?)
  • How do transformations compose? (Do your pipelines work correctly?)

These questions determine whether your software stays flexible or becomes brittle.

Categorical Coding gives you a systematic framework for answering them—grounded in mathematics, applied to real code.

Category Theory You Can Run

Learning from math textbooks is hard. Learning from executable code is natural.

Forthix makes categorical concepts runnable: 

# Stack: [3, 4]
DUP * SWAP DUP * + SQRT
# Result: [5]

# This is a morphism: [3,4] → [5]
# Each word is a transformation
# Composition is literal string concatenation

Why Forthic is perfect for Categorical Coding:

1. Composition is built-in

In most languages: compose(f, g)(x) or g(f(x))

In Forthic: f g — that's it. Concatenation IS composition.

2. Morphisms are visible

Every word is a morphism. Every program is a composition. You're not simulating category theory—you're doing it.

3. Information flow is explicit

The stack shows exactly what information exists at each step. You can see where it's preserved, transformed, or lost.

4. Definitions extend the category

When you define : SQUARE DUP * ; you're adding a morphism. The category grows with your vocabulary.

This isn't about syntax—it's about ergonomics for categorical thinking.

Built for the AI Era

AI's breakthrough capability is transformation. Categorical Coding provides the framework to reason about it.

An Architectural Alternative to MCP

MCP uses servers to connect LLMs to tools. Forthix takes a different approach.

Server-Based (MCP)

LLM → Server Call
→ External Execution
→ Network Round-trip
→ Hidden Logic

Code Generation (Forthix)

LLM → Generates Forthic
→ You Execute Locally
→ Instant Execution
→ Visible, Auditable Code

The LLM generates Forthic. You execute it. You see exactly what it does.

Wrap. Compose. Abstract.

Forthix doesn't replace your code—it wraps it.

1

Wrap Your Existing Code

Build modules around YOUR codebase—Python, JavaScript, Ruby, whatever you use.

2

Compose Across Runtimes

Because Forthic is multi-runtime, one program can orchestrate multiple systems in different languages.

3

Abstract Higher-Level Operations

Create domain vocabularies that LLMs can use to generate programs specific to your business.

The Result:

LLMs generate a single, readable Forthic program that seamlessly executes across your entire stack—no servers, no API translation layers, no egress complexity.

Your domain logic stays in your language. Just wrap it. The LLM speaks Forthic, which runs everywhere.

The Streaming Advantage

LLMs stream tokens incrementally. Forthic's concatenative composition is a perfect match.

// LLM streaming output:
"users" ← valid Forthic
"users ACTIVE-ONLY" ← valid Forthic
"users ACTIVE-ONLY EMAIL-DOMAIN" ← valid Forthic
"users ACTIVE-ONLY EMAIL-DOMAIN UNIQUE" ← valid Forthic

Each word the LLM outputs is immediately valid and composable. The program grows naturally, word by word.

Why This Matters

No server infrastructure needed
No network egress or latency
Complete transparency—see the code
You control execution completely
Wrap legacy systems in any language
Concise, explainable code output

Frequently Asked Questions

Do I need to know category theory already?

No. We start from zero and build up through practical examples.

Do I need advanced math?

No. If you can code, you can follow along.

Do I have to use Forthic?

Technically, no, but Forthic makes these concepts natural by focusing entirely on transformations and abstractions, leaving the lower level details to other languages.

Is this practical or just theoretical?

Both. It solves real problems: API design, data pipelines, provenance tracking, privacy analysis.