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segal-space skill

/skills/segal-space

This skill analyzes Segal spaces and infinity-categories via simplicial spaces to help you model complex categorical structures.

npx playbooks add skill plurigrid/asi --skill segal-space

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

Files (2)
SKILL.md
832 B
---
name: segal-space
description: "Segal spaces: ∞-categories via simplicial spaces."
metadata:
  letter: S
  trit: -1
  seed: 1152
---
# S: Segal Space

> *Segal spaces: ∞-categories via simplicial spaces.*

## Directory Tree

```
segal-space/
├── SKILL.md
├── spine/
├── composition.rzk
```

## GF(3) Assignment

Letter S → trit = -1

## Integration

```scheme
(define (segal_space x)
  ;; Segal spaces: ∞-categories via simplicial spaces.
  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 presents Segal spaces as a model for ∞-categories using simplicial spaces and homotopy-theoretic composition data. It packages conceptual notes, minimal code examples, and a small directory of resources that illustrate the spine and composition perspectives. The aim is to give researchers and advanced students a compact, practical entry to work with Segal-type models in computational settings.

How this skill works

The skill inspects a simplicial-space viewpoint: objects appear in degree 0, morphisms in degree 1, and higher coherences in higher simplices. It highlights the Segal condition (local gluing of composition) and points to a composition module that expresses how simplices assemble into composition data. The provided snippet acts as a placeholder for embedding Segal-space constructions into larger workflows and experiments.

When to use it

  • Modeling ∞-categories where composition is defined up to coherent homotopy.
  • Comparing different models of higher categories (Segal spaces vs. quasi-categories).
  • Implementing or prototyping homotopy-coherent compositional structures.
  • Teaching or demonstrating the spine and Segal condition to students.

Best practices

  • Keep degree-wise data explicit: record 0-, 1-, and 2-simplices separately for clarity.
  • Verify the Segal maps are equivalences (or fibrations) in your chosen model structure.
  • Use small, concrete examples before scaling to complex or infinite diagrams.
  • Treat the provided code snippets as integration hooks, not full implementations.
  • Document how memory of interactions modifies or extends the example corpus.

Example use cases

  • Build a compact example of a homotopy-coherent monoid as a Segal space.
  • Prototype a translation layer between simplicial-space data and other ∞-category models.
  • Demonstrate failure modes when Segal maps are not equivalences using explicit simplices.
  • Embed Segal-space checks into a pipeline that verifies composition coherence for generated diagrams.

FAQ

Is this a complete implementation of Segal spaces?

No. This skill provides conceptual guidance, small examples, and integration hooks rather than a finished library.

What does the spine refer to here?

The spine refers to the sub-simplicial structure capturing composable chains of 1-simplices used to state the Segal condition.