coequalizers

npx skills add majiayu000/claude-skill-registry --skill coequalizers-plurigrid-asi-2 --agent claude-code

Same command for any agent — swap --agent for codex, cursor, copilot.

# Coequalizers Skill > **Quotient redundant skill paths via categorical coequalizers** **Version**: 1.0.0 **Trit**: 0 (ERGODIC - coordinates equivalences) **Domain**: category-theory, skill-composition, colimits, behavioral-equivalence --- ## Overview The **coequalizers** skill provides: 1. **Behavioral equivalence checking** via bisimulation (from temporal-coalgebra) 2. **Parallel morphism quotienting** via coequalizers (colimits) 3. **Skill overlap detection** and gluing (from oapply-colimit pushouts) 4. **GF(3) conservation** in quotient spaces 5. **MCP integration** for cross-agent skill synchronization --- ## Core Concept ### What Are Coequalizers? A **coequalizer** is the colimit of two parallel morphisms: ``` X ──f──→ Y │ g │ └────────→ q ↓ Q (coequalizer) Universal property: q ∘ f = q ∘ g ``` **In Sets**: Q = Y / ~ where ~ is the smallest equivalence relation such that f(x) ~ g(x) for all x ∈ X. **For skills**: If two skill paths produce behaviorally equivalent outputs, the coequalizer gives the canonical quotient. --- ## Key Patterns from asi Repository ### 1. oapply-colimit: Pushout = Coproduct + Coequalizer From `/skills/oapply-colimit/SKILL.md`: ```julia function oapply

just coequalizer-find SKILLS...          # Find equivalences among skills
just coequalizer-quotient SYSTEM         # Compute quotient
just coequalizer-verify GF3              # Verify GF(3) conservation
just coequalizer-disperse AGENTS...      # Sync across agents
just coequalizer-pushout SKILL1 SKILL2   # Compose with overlap