Catlab.jl x ASI Interleave

Bridge connecting AlgebraicJulia/Catlab.jl (categorical algebra in Julia) to the ASI skill graph (GF(3)-colored capability system).

Catlab Core Concepts (from DeepWiki deep-mine)

ACSets (Attributed C-Sets)

The universal data structure. A schema defines objects, homomorphisms (morphisms between objects), and attribute types. An ACSet instance is a functor from that schema category to Set.

# SchGraph: objects V, E; morphisms src: E->V, tgt: E->V
=======
description: >
  Bridge connecting AlgebraicJulia/Catlab.jl to skill graphs.
  Triggers: ACSets, attributed C-sets, wiring diagrams, decorated cospans,
  DPO rewriting on skill graphs, AlgebraicDynamics, AlgebraicPetri,
  AlgebraicRewriting, Decapodes, categorical algebra in Julia.
---

# Catlab.jl Interleave

Bridge connecting AlgebraicJulia/Catlab.jl (categorical algebra in Julia) to skill graphs.

## Catlab Core Concepts

### ACSets (Attributed C-Sets)

The universal data structure. A schema defines objects, homomorphisms, and attribute types. An ACSet instance is a functor from that schema category to Set.

```julia
>>>>>>> origin/main
@present SchGraph(FreeSchema) begin
  V::Ob; E::Ob
  src::Hom(E,V); tgt::Hom(E,V)
end

<<<<<<< HEAD
# Attributed: add attribute types
=======
>>>>>>> origin/main
@present SchWeightedGraph <: SchGraph begin
  T::AttrType
  weight::Attr(E,T)
end

const WeightedGraph = ACSetType(SchWeightedGraph, index=[:src,:tgt])

<<<<<<< HEAD Every diagram, network, and model in the ecosystem is an ACSet.

Wiring Diagrams as ACSets

SchAttributedWiringDiagram with Box/InPort/OutPort/Wire. Boxes = operations/processes, wires = data flow. Used to compose dynamical systems, Petri nets, and more.

Decorated Cospans

Functor L: A -> X gives "open" ACSets. OpenGraph with hypergraph category structure. Operations: compose, otimes (monoidal product), mcopy, mmerge, delete, create. Enable compositional modeling of open systems.

Wiring Diagrams as ACSets

SchAttributedWiringDiagram with Box/InPort/OutPort/Wire. Boxes = operations, wires = data flow. Used to compose dynamical systems, Petri nets, and more.

Decorated Cospans

Functor L: A -> X gives "open" ACSets. Operations: compose, otimes (monoidal product), mcopy, mmerge, delete, create. Enable compositional modeling of open systems.

origin/main

Downstream Ecosystem

AlgebraicJulia/Catlab.jl (foundation)
<<<<<<< HEAD
  |- AlgebraicDynamics.jl     <- dynamical systems via decorated cospans
  |- AlgebraicPetri.jl        <- Petri nets with reaction network semantics
  |- AlgebraicRewriting.jl    <- DPO/SPO graph rewriting on ACSets
  |- CategoricalTensorNetworks.jl <- tensor contractions as string diagrams
  |- CombinatorialSpaces.jl   <- simplicial sets, discrete exterior calculus
  |- DataMigrations.jl        <- functorial data migration between schemas
  |- DiagrammaticEquations.jl <- physics equations as decorated cospans
  |- Decapodes.jl             <- multiphysics simulation via DEC

No probabilistic/inference capabilities -- must come from downstream packages or bridges. This is where monad-bayes-asi-interleave fills the gap.

GF(3) Tripartite Tag

algebraic-dynamics(-1) otimes catlab-asi-interleave(0) otimes algebraic-rewriting(+1) = 0

Dynamics (-1) x Foundation (0) x Rewriting (+1) = balanced categorical stack.

---

ASI Integration Points

1. acsets / acsets-algebraic-databases / acsets-relational-thinking <-> ACSets Core

Model the ASI skill graph as an ACSet for relational querying:

@present SchASISkills(FreeSchema) begin
=======
  |- AlgebraicDynamics.jl     -- dynamical systems via decorated cospans
  |- AlgebraicPetri.jl        -- Petri nets with reaction network semantics
  |- AlgebraicRewriting.jl    -- DPO/SPO graph rewriting on ACSets
  |- CategoricalTensorNetworks.jl -- tensor contractions as string diagrams
  |- CombinatorialSpaces.jl   -- simplicial sets, discrete exterior calculus
  |- DataMigrations.jl        -- functorial data migration between schemas
  |- DiagrammaticEquations.jl -- physics equations as decorated cospans
  |- Decapodes.jl             -- multiphysics simulation via DEC

Integration Points

Skill Graph as ACSet

@present SchSkills(FreeSchema) begin
>>>>>>> origin/main
  Skill::Ob; Edge::Ob; Hub::Ob
  src::Hom(Edge,Skill); tgt::Hom(Edge,Skill)
  hub_ref::Hom(Hub,Skill)
  SkillName::AttrType; TritVal::AttrType; Category::AttrType
  name::Attr(Skill,SkillName)
<<<<<<< HEAD
  trit::Attr(Skill,TritVal)        # -1, 0, +1
  category::Attr(Skill,Category)   # development | meta | ai-agents | ...
end

const ASISkills = ACSetType(SchASISkills, index=[:src,:tgt,:hub_ref])
skills = ASISkills()

# Conjunctive query: all GF(3)-balanced triads
gf3_triads(s) = filter(parts(s,:Edge)) do e
  t_src = s[s[e,:src], :trit]
  t_tgt = s[s[e,:tgt], :trit]
  (t_src + t_tgt) % 3 == 0
end

2. algebraic-rewriting / topos-adhesive-rewriting <-> DPO/SPO Rewriting

Double-pushout rewriting for safe skill graph mutation (MONOTONIC_SKILL_INVARIANT):

trit::Attr(Skill,TritVal) category::Attr(Skill,Category) end

const Skills = ACSetType(SchSkills, index=[:src,:tgt,:hub_ref])

### DPO Rewriting for Safe Skill Graph Mutation
>>>>>>> origin/main

```julia
using AlgebraicRewriting

# DPO rule: add bridge skill to hub (never delete)
<<<<<<< HEAD
# L -> K <- R where |R| >= |L| always
add_bridge_rule = Rule(
  ACSetTransformation(L, K),   # L: match hub pattern
  ACSetTransformation(R, K),   # R: hub + new bridge skill
=======
# L -> K <- R where |R| >= |L| always (monotonic)
add_bridge_rule = Rule(
  ACSetTransformation(L, K),
  ACSetTransformation(R, K),
>>>>>>> origin/main
)
new_skills = rewrite(add_bridge_rule, current_skills)
@assert nparts(new_skills, :Skill) >= nparts(current_skills, :Skill)

<<<<<<< HEAD SPO rewriting available for partial matches (non-adhesive contexts).

3. discopy / discopy-operads <-> Wiring Diagram Composition

Catlab's wiring diagrams and DisCoPy's string diagrams are the same mathematical object:

# ASI skill composition as wiring diagram
=======
### Wiring Diagram Composition

```julia
# Skill composition as wiring diagram
>>>>>>> origin/main
wd = @program SchSkillOp (validator::Val, coordinator::Coord, generator::Gen) begin
  validated = validator(input)
  coordinated = coordinator(validated)
  result = generator(coordinated)
  return result
end
<<<<<<< HEAD
# This WiringDiagram ACSet can be exported to DisCoPy format

Bridge: Julia WiringDiagram ACSet <-> Python DisCoPy Diagram via JSON serialization.

4. topos-catcolab / catcolab-* <-> CatColab Collaborative Modeling

CatColab is the web frontend to Catlab. Skills: catcolab-ologs, catcolab-petri-nets, catcolab-stock-flow, catcolab-causal-loop, catcolab-decapodes. All CatColab models are ACSets underneath.

5. interaction-nets <-> Decorated Cospans / Hypergraph Categories

Decorated cospans give OpenGraph a hypergraph category structure. Operations: compose (sequential), otimes (parallel), mcopy (fan-out), mmerge (fan-in), delete, create. Interaction nets are the computational model; decorated cospans are the categorical semantics.

6. crn-topology <-> AlgebraicPetri Reaction Networks

=======

WiringDiagram ACSet can be exported to DisCoPy format via JSON

### AlgebraicPetri Reaction Networks
>>>>>>> origin/main

```julia
using AlgebraicPetri

<<<<<<< HEAD
# Chemical reaction network as Petri net ACSet
=======
>>>>>>> origin/main
sir_model = LabelledPetriNet([:S,:I,:R],
  :infection => ((:S,:I) => (:I,:I)),
  :recovery  => (:I => :R)
)
<<<<<<< HEAD
# Compose via decorated cospans
open_sir = Open(sir_model, [:S], [:R])

Connects to crn-topology for topological analysis of reaction networks.

7. dynamical-system-functor / coupled-system <-> AlgebraicDynamics

======= open_sir = Open(sir_model, [:S], [:R])

### AlgebraicDynamics
>>>>>>> origin/main

```julia
using AlgebraicDynamics, Catlab

<<<<<<< HEAD
# Open continuous dynamical system
=======
>>>>>>> origin/main
rb_system = ContinuousResourceSharer{Float64}(
  [:temperature, :velocity, :pressure],
  (u, p, t) -> rb_dynamics(u, p, t)
)
<<<<<<< HEAD
# Compose via wiring diagram
=======
>>>>>>> origin/main
full_system = oapply(boundary_diagram, [rb_system, thermal_bc])
solution = solve(ODEProblem(full_system, u0, tspan), Tsit5())

<<<<<<< HEAD

8. julia-scientific <-> Julia Runtime for Catlab

Catlab requires Julia >= 1.10. Entry points: julia-scientific, julia-gay. Enzyme.jl autodiff works with AlgebraicDynamics ODE solvers.

9. topos-unified / topos-generate <-> Topos-Theoretic Foundations

Catlab implements presentable categories, limits/colimits, Kan extensions. The topos-theoretic skills (topos-unified, topos-generate, effective-topos) provide the foundation that Catlab operationalizes in code.

10. monad-bayes-asi-interleave <-> Fills Probabilistic Gap

Catlab has no built-in probabilistic inference. monad-bayes-asi-interleave provides SMC/MCMC/PMMH/RMSMC stacks. The bridge: Catlab composes the model structure, monad-bayes performs inference on it.

11. vertex-ai-protein-interleave <-> Protein Reaction Networks

AlgebraicPetri reaction networks model biochemical pathways. Protein folding/docking pipelines from vertex-ai-protein-interleave can use Catlab ACSets to represent reaction networks, pathway composition, and multi-target drug interaction graphs.

---

ASI Skill Graph as an ACSet (Self-Modeling)

The ASI skill graph itself is naturally an ACSet:

# Skills = objects, edges = morphisms, GF(3) trits = attributes
@present SchASIGraph(FreeSchema) begin
  Skill::Ob
  Edge::Ob
  src::Hom(Edge,Skill)
  tgt::Hom(Edge,Skill)
  # Attribute types
  Name::AttrType; Trit::AttrType; Role::AttrType; Version::AttrType
  # Attributes
  name::Attr(Skill,Name)
  trit::Attr(Skill,Trit)          # GF(3): -1, 0, +1
  role::Attr(Skill,Role)          # BRIDGE | HUB | LEAF | ERGODIC
  version::Attr(Skill,Version)
  edge_type::Attr(Edge,Name)      # :citation | :behavioral | :trit_equiv
end

# Invariants as ACSet constraints:
# 1. MONOTONIC: nparts(g, :Skill) >= 1360 (never decreases)
# 2. GF3_CONSERVATION: for any triad (s1,s2,s3), sum of trits = 0 mod 3
# 3. HUB_REACHABILITY: 17 hub skills; every leaf reachable from >= 1 hub

This enables: conjunctive queries over the skill graph, functorial data migration between schema versions, DPO rewriting for safe skill addition, and wiring diagram visualization of skill pipelines.

---

Connection to ASI Topos Stories

Story (task)	Catlab Connection
Task 21: Categorical worlding kit	Catlab.jl engine + CatColab UI + UnwiringDiagrams.jl
Task 22: Compositional game theory	Wiring diagrams encode open game composition
Task 23: Nonlinear dynamics observatory	AlgebraicDynamics for attractor ODE composition
Task 24: ASI skill federation	ACSet as skill registry; DPO rewriting for safe addition
Task 20: Self-walking proof pipeline	AlgebraicRewriting as proof rewrite system

---

Gap Registry

Capability	Status	Filled By
Probabilistic inference on ACSets	MISSING in Catlab	monad-bayes-asi-interleave
GPU-accelerated ACSet operations	MISSING	Future: CUDA.jl + ACSet kernels
ACSet <-> DuckDB serialization	PARTIAL	duckdb-ies (Parquet round-trip)
ACSet <-> JSON-RPC for MCP	MISSING	Need: syrup/JSON bridge for ACSets
Real-time collaborative ACSets	PARTIAL	CatColab (web only, no MCP)
ACSet diff/merge (CRDT semantics)	MISSING	crdt skill + DPO rewriting
Tensor network contraction at scale	MISSING	CategoricalTensorNetworks exists but no GPU
Discrete exterior calculus on DGX	MISSING	Decapodes + CUDA offload needed

---

Related ASI Skills

• acsets / acsets-relational-thinking / acsets-algebraic-databases -- core ACSet skills
• algebraic-rewriting / topos-adhesive-rewriting -- DPO/SPO rewriting on ACSets
• discopy / discopy-operads -- Python string diagram companion to Catlab
• catcolab-ologs / catcolab-petri-nets / catcolab-stock-flow / catcolab-decapodes -- CatColab frontends
• interaction-nets -- computational model for decorated cospan composition
• crn-topology -- topological analysis of AlgebraicPetri reaction networks
• dynamical-system-functor / coupled-system -- AlgebraicDynamics integration
• julia-scientific / julia-gay -- Julia ecosystem entry points
• topos-unified / topos-generate / effective-topos -- topos-theoretic foundations
• monad-bayes-asi-interleave -- fills probabilistic inference gap
• vertex-ai-protein-interleave -- protein reaction networks via AlgebraicPetri
• structured-decomp -- structured decompositions = open graphs (Catlab cospans)
• enzyme-autodiff -- Enzyme.jl autodiff for AlgebraicDynamics simulation
• wolframite-compass -- Wolfram -> Catlab via Julia bridge
• string-diagram-rewriting-protocol -- rewriting protocol for wiring diagrams =======

Runtime

Catlab requires Julia >= 1.10. Enzyme.jl autodiff works with AlgebraicDynamics ODE solvers.

Gap Registry

Capability	Status	Notes
Probabilistic inference on ACSets	MISSING in Catlab	Use monad-bayes bridge
GPU-accelerated ACSet operations	MISSING	Future: CUDA.jl + ACSet kernels
ACSet <-> DuckDB serialization	PARTIAL	Parquet round-trip
ACSet <-> JSON-RPC for MCP	MISSING	Need syrup/JSON bridge
ACSet diff/merge (CRDT semantics)	MISSING	DPO rewriting approach

origin/main