# Acset Taxonomy > Taxonomy of ACSet skills with morphisms to semantically similar categorical/relational skills Use when this capability is needed. - Skill: `tomevault-io/acset-taxonomy` (Agent Skill, multi-file: 2 files) - Install (CLI): `npx skillmds@latest add tomevault-io/acset-taxonomy` - Raw SKILL.md: https://api.skillmd.com/api/skills/tomevault-io/acset-taxonomy/raw - Safety review: pending (external: skill-scanner PASS, skillspector CAUTION) - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - Author: tomevault-io (https://skillmd.com/u/tomevault-io) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/tomevault-io/acset-taxonomy --- # ACSet Skill Taxonomy **Trit**: 0 (ERGODIC - coordinates between schema validation and data generation) ## Overview This skill maps the ecosystem of ACSet-based skills and their morphisms to semantically similar categorical/relational skills. ``` ┌─────────────────────────────────────────┐ │ ACSET SKILL UNIVERSE │ └─────────────────────────────────────────┘ │ ┌──────────────────────────┼──────────────────────────┐ │ │ │ ┌──────▼──────┐ ┌───────▼───────┐ ┌───────▼───────┐ │ EXPLICIT │ │ DOMAIN- │ │ SEMANTICALLY │ │ ACSET │ │ SPECIFIC │ │ SIMILAR │ │ CORE │ │ ACSET │ │ CATEGORICAL │ └─────────────┘ └───────────────┘ └───────────────┘ (3) (12) (12) ``` --- ## I. EXPLICIT ACSET CORE (3 skills) Foundation skills that define ACSet theory and operations: | Skill | Description | Schema Pattern | Trit | |-------|-------------|----------------|------| | [acsets](file:///Users/bob/.claude/skills/acsets/SKILL.md) | Algebraic databases with Specter navigation | `Ob::, Hom::, Attr::` | 0 | | [acsets-relational-thinking](file:///Users/bob/.claude/skills/acsets-relational-thinking/SKILL.md) | Functors X: C → Set, DPO rewriting | Category → Set | -1 | | [specter-acset](file:///Users/bob/.claude/skills/specter-acset/SKILL.md) | Bidirectional navigation, inline caching | Lenses/Prisms | +1 | **GF(3) Triad**: `acsets-relational-thinking (-1) ⊗ acsets (0) ⊗ specter-acset (+1) = 0 ✓` --- ## II. DOMAIN-SPECIFIC ACSET (12 skills) ACSet schemas instantiated for specific domains: ### A. Google Workspace ACSet Family | Skill | Domain | Schema Objects | MCP Equivalence | |-------|--------|----------------|-----------------| | [calendar-acset](file:///Users/bob/.claude/skills/calendar-acset/SKILL.md) | Google Calendar | Event, Attendee, Recurrence | calendar-mcp | | [docs-acset](file:///Users/bob/.claude/skills/docs-acset/SKILL.md) | Google Docs/Sheets | Document, Comment, Cell | docs-mcp | | [drive-acset](file:///Users/bob/.claude/skills/drive-acset/SKILL.md) | Google Drive | File, Folder, Permission | drive-mcp | | [tasks-acset](file:///Users/bob/.claude/skills/tasks-acset/SKILL.md) | Google Tasks | Task, TaskList, Due | tasks-mcp | **Common Pattern**: Transform API operations → GF(3)-typed Interactions → triadic queues ### B. Data Import/Export ACSet Family | Skill | Domain | Schema Objects | Source | |-------|--------|----------------|--------| | [chatgpt-export-acset](file:///Users/bob/.claude/skills/chatgpt-export-acset/SKILL.md) | ChatGPT exports | Conversation, Message, Author | JSON export | | [openai-acset](file:///Users/bob/.claude/skills/openai-acset/SKILL.md) | OpenAI API | Request, Response, Token | API calls | | [browser-history-acset](file:///Users/bob/.claude/skills/browser-history-acset/SKILL.md) | Browser history | Visit, Page, Domain | SQLite | ### C. Infrastructure ACSet Family | Skill | Domain | Schema Objects | Use Case | |-------|--------|----------------|----------| | [nix-acset-worlding](file:///Users/bob/.claude/skills/nix-acset-worlding/SKILL.md) | Nix store | Derivation, StorePath, Dep | GC analysis | | [protocol-acset](file:///Users/bob/.claude/skills/protocol-acset/SKILL.md) | P2P protocols | Message, Peer, Channel | Interop design | | [rg-flow-acset](file:///Users/bob/.claude/skills/rg-flow-acset/SKILL.md) | Ripgrep flows | Match, File, Pattern | Code search | ### D. Language Bridge ACSet Family | Skill | Domain | Schema Objects | Bridge | |-------|--------|----------------|--------| | [lispsyntax-acset](file:///Users/bob/.claude/skills/lispsyntax-acset/SKILL.md) | S-expressions | Sexp, Atom, List | Julia ↔ Lisp | | [compositional-acset-comparison](file:///Users/bob/.claude/skills/compositional-acset-comparison/SKILL.md) | Algorithm analysis | Query, Result, Schema | DuckDB ↔ LanceDB | --- ## III. SEMANTICALLY SIMILAR CATEGORICAL (12 skills) Skills that share categorical/relational structure but don't explicitly use ACSets: ### A. Sheaf/Cohomology Family | Skill | Structure | Relation to ACSet | Morphism | |-------|-----------|-------------------|----------| | [sheaf-cohomology](file:///Users/bob/.claude/skills/sheaf-cohomology/SKILL.md) | Čech cohomology | Sheaves on schemas | H^n(C, F) | | [structured-decomp](file:///Users/bob/.claude/skills/structured-decomp/SKILL.md) | Tree decompositions | Sheaves on trees | FPT algorithms | | [persistent-homology](file:///Users/bob/.claude/skills/persistent-homology/SKILL.md) | Persistence modules | Filtered ACSets | Barcodes | **Morphism**: ACSet schema C → Sheaf F: C^op → Set ### B. Fibration/Extension Family | Skill | Structure | Relation to ACSet | Morphism | |-------|-----------|-------------------|----------| | [covariant-fibrations](file:///Users/bob/.claude/skills/covariant-fibrations/SKILL.md) | Dependent types over 2 | Fibered ACSets | π: E → B | | [kan-extensions](file:///Users/bob/.claude/skills/kan-extensions/SKILL.md) | (L ⊣ R) adjunctions | Schema change functors | Lan_F, Ran_F | | [ordered-locale](file:///Users/bob/.claude/skills/ordered-locale/SKILL.md) | Frames with preorder | Directed ACSets | ≪ relation | **Morphism**: Schema morphism F: C → D induces Lan_F: Set^C → Set^D ### C. Computational/Rewriting Family | Skill | Structure | Relation to ACSet | Morphism | |-------|-----------|-------------------|----------| | [datalog-fixpoint](file:///Users/bob/.claude/skills/datalog-fixpoint/SKILL.md) | Bottom-up iteration | Relational queries | IDB/EDB | | [interaction-nets](file:///Users/bob/.claude/skills/interaction-nets/SKILL.md) | Lafont nets | Graph rewriting | DPO rules | | [open-games](file:///Users/bob/.claude/skills/open-games/SKILL.md) | Compositional games | Wiring diagrams | Play/Coplay | **Morphism**: ACSet rewriting via DPO (Double Pushout) ### D. Distributed/Temporal Family | Skill | Structure | Relation to ACSet | Morphism | |-------|-----------|-------------------|----------| | [topos-catcolab](file:///Users/bob/.claude/skills/topos-catcolab/SKILL.md) | Double theories | Collaborative ACSets | Automerge CRDT | | [crdt-vterm](file:///Users/bob/.claude/skills/crdt-vterm/SKILL.md) | LWW/G-Counter | Mergeable ACSets | ⊔ lattice | | [temporal-coalgebra](file:///Users/bob/.claude/skills/temporal-coalgebra/SKILL.md) | Final coalgebras | Stream of ACSets | νF bisimulation | **Morphism**: ACSet × Time → Temporal ACSet (presheaf on ℕ^op) --- ## IV. MORPHISM DIAGRAM ``` ┌────────────────┐ │ acsets (0) │ │ Core Functor │ │ X: C → Set │ └───────┬────────┘ │ ┌────────────────────┼────────────────────┐ │ │ │ ┌───────▼────────┐ ┌───────▼────────┐ ┌───────▼────────┐ │ Domain-Specific│ │ Sheaf/Cohom │ │ Computation │ │ (Workspace, │ │ (structure on │ │ (rewriting, │ │ Protocol) │ │ schema) │ │ queries) │ └───────┬────────┘ └───────┬────────┘ └───────┬────────┘ │ │ │ │ forgetful │ forgetful │ │ functor U │ functor U │ │ │ │ └────────────────────┼────────────────────┘ │ ┌───────▼────────┐ │ specter-acset │ │ Navigation │ │ Lenses/Prisms │ └────────────────┘ ``` --- ## V. GF(3) TRIADIC ORGANIZATION ### Core Triad ``` acsets-relational-thinking (-1) ⊗ acsets (0) ⊗ specter-acset (+1) = 0 ✓ ``` ### Sheaf Triad ``` sheaf-cohomology (-1) ⊗ structured-decomp (0) ⊗ persistent-homology (+1) = 0 ✓ ``` ### Workspace Triad ``` calendar-acset (-1) ⊗ docs-acset (0) ⊗ drive-acset (+1) = 0 ✓ ``` ### Computation Triad ``` datalog-fixpoint (-1) ⊗ interaction-nets (0) ⊗ open-games (+1) = 0 ✓ ``` ### Temporal Triad ``` temporal-coalgebra (-1) ⊗ crdt-vterm (0) ⊗ topos-catcolab (+1) = 0 ✓ ``` --- ## VI. SCHEMA PATTERNS ### Generic ACSet Schema Template ```julia @present SchDomainACSet(FreeSchema) begin # Objects (Ob) Entity::Ob Relation::Ob Attribute::Ob # Morphisms (Hom) source::Hom(Relation, Entity) target::Hom(Relation, Entity) # Attributes (Attr) Name::AttrType Value::AttrType entity_name::Attr(Entity, Name) attr_value::Attr(Attribute, Value) end ``` ### Common Schema Transformations | Transform | From | To | Functor | |-----------|------|----|---------| | Forget | WorkspaceACSet | ACSet | Forgetful U | | Embed | ACSet | SheafACSet | Yoneda y | | Temporal | ACSet | StreamACSet | Δ^* (nerve) | | Distributed | ACSet | CRDTACSet | Free(Lattice) | --- ## VII. USAGE PATTERNS ### Cross-ACSet Query ```julia # Query across multiple domain ACSets using ACSets, Catlab # Define morphism between schemas F = SchemaMap(CalendarSchema, TasksSchema, Event => Task, due_date => due_date ) # Pull back tasks to calendar view calendar_tasks = Δ(F, my_tasks) ``` ### Sheaf Consistency Check ```julia # Verify local-to-global consistency using StructuredDecompositions decomp = tree_decomposition(my_acset) consistent = verify_sheaf_condition(decomp) ``` ### Temporal Stream ```julia # Create temporal ACSet stream using TemporalCoalgebra stream = unfold(initial_acset) do x (observe(x), step(x)) end ``` --- ## VIII. INVARIANTS ```yaml invariants: - name: functor_preservation predicate: "F(id_A) = id_{F(A)} and F(g∘f) = F(g)∘F(f)" scope: per_morphism - name: schema_pullback predicate: "Δ(F, X) is right adjoint to Σ(F, X)" scope: per_schema_map - name: gf3_triad_conservation predicate: "Σ trit = 0 for each triad" scope: per_skill_group - name: crdt_convergence predicate: "a ⊔ b = b ⊔ a (commutativity)" scope: per_distributed_acset ``` --- ## IX. REFERENCES - [Catlab.jl](https://github.com/AlgebraicJulia/Catlab.jl) - [ACSets.jl](https://github.com/AlgebraicJulia/ACSets.jl) - [StructuredDecompositions.jl](https://github.com/AlgebraicJulia/StructuredDecompositions.jl) - [Relational Thinking Book](https://www.relationalthinking.dev/) - [CatColab](https://catcolab.org/) --- ## 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.* --- > Converted and distributed by [TomeVault](https://tomevault.io/claim/plurigrid) — claim your Tome and manage your conversions.