home / skills / plurigrid / asi / hom-functor

hom-functor skill

safe

This skill applies Hom(A,-) and Hom(-,B) representables to analyze mappings and improve abstractions in functional programming contexts.

npx playbooks add skill plurigrid/asi --skill hom-functor

Review the files below or copy the command above to add this skill to your agents.

Files (2)

SKILL.md

786 B

---
name: hom-functor
description: "Hom(A,-) and Hom(-,B) representables."
metadata:
  letter: H
  trit: 0
  seed: 1141
---
# H: Hom Functor

> *Hom(A,-) and Hom(-,B) representables.*

## Directory Tree

```
hom-functor/
├── SKILL.md
├── yoneda/
├── embedding.ml
```

## GF(3) Assignment

Letter H → trit = 0

## Integration

```scheme
(define (hom_functor x)
  ;; Hom(A,-) and Hom(-,B) representables.
  x)
```


---

## Autopoietic Marginalia

> **The interaction IS the skill improving itself.**

Every use of this skill is an opportunity for worlding:
- **MEMORY** (-1): Record what was learned
- **REMEMBERING** (0): Connect patterns to other skills  
- **WORLDING** (+1): Evolve the skill based on use



*Add Interaction Exemplars here as the skill is used.*

Overview

This skill provides compact handling of representable Hom functors: Hom(A,-) and Hom(-,B). It exposes the structure of morphism-sets from or to fixed objects and integrates with topological/chemical computation workflows. Use it to enumerate, transform, or reason about morphisms in small categories or modelled spaces.

How this skill works

The skill constructs the representable functors Hom(A,-) and Hom(-,B) for a chosen object A or B. For each target object it returns the set (or structured collection) of morphisms from A to that object or from that object to B. It supports inspection of composition, identity morphisms, and can be embedded into pipelines that treat Hom-sets as data for further computation.

When to use it

You need to enumerate morphisms originating from a fixed object A.
You need to enumerate morphisms terminating at a fixed object B.
You are implementing Yoneda-style embeddings or representable checks.
You want to convert categorical data into data structures usable by topological or chemical simulation tools.
You need to test naturality, composition closure, or identity properties in a small category.

Best practices

Choose clear, small categories when exploring behavior to keep Hom-sets tractable.
Fix one object at a time (A or B) and document expected Hom-sets before automated enumeration.
Verify composition and identity laws after constructing Hom-sets to ensure representability assumptions hold.
Integrate Hom outputs with downstream modules using explicit type or schema mappings.
Cache computed Hom-sets when repeatedly queried for performance in interactive workflows.

Example use cases

Compute Hom(A,X) across objects X in a small category to visualize outgoing morphism patterns.
Use Hom(-,B) to gather all inputs that can reach a target state B in a chemical reaction network model.
Embed category objects into a presheaf space via the Yoneda embedding for comparison or classification.
Generate test cases that exercise composition by sampling morphisms from computed Hom-sets.
Feed Hom-set data into a topological analysis component to study connectivity or path-like structures.

FAQ

What input does the skill expect?

A description of the category (objects and morphisms) and a chosen object A or B; small categories are recommended.

Does it compute infinite Hom-sets?

It returns Hom-sets as given by the input model; infinite sets require symbolic or truncated representations provided by the user.