ACSets: Algebraic Databases Skill

"The category of simple graphs does not even have a terminal object!" — AlgebraicJulia Blog, with characteristic ironic detachment

What Are ACSets?

ACSets ("attributed C-sets") are a family of data structures generalizing both graphs and data frames. They are an efficient in-memory implementation of a category-theoretic formalism for relational databases.

C-set = Functor

X: C → Set

┌─────────────────────────────────────────────────────────────┐
│  Schema (Small Category C)                                  │
│  ┌─────┐  src   ┌─────┐                                     │
│  │  E  │───────▶│  V  │                                     │
│  │     │  tgt   │     │                                     │
│  └──┬──┘───────▶└─────┘                                     │
│     │                                                       │
│     │ A C-set X assigns:                                    │
│     │   X(V) = set of vertices                              │
│     │   X(E) = set of edges                                 │
│     │   X(src): X(E) → X(V)                                 │
│     │   X(tgt): X(E) → X(V)                                 │
└─────────────────────────────────────────────────────────────┘

Core Concepts

1. Schema Definition

using Catlab.CategoricalAlgebra

@present SchGraph(FreeSchema) begin
  V::Ob
  E::Ob
  src::Hom(E,V)
  tgt::Hom(E,V)
end

@acset_type Graph(SchGraph, index=[:src,:tgt])

2. Symmetric Graphs (Undirected)

@present SchSymmetricGraph <: SchGraph begin
  inv::Hom(E,E)

  compose(inv,src) == tgt
  compose(inv,tgt) == src
  compose(inv,inv) == id(E)
end

@acset_type SymmetricGraph(SchSymmetricGraph, index=[:src])

3. Attributed ACSets (with Data)

@present SchWeightedGraph <: SchGraph begin
  Weight::AttrType
  weight::Attr(E, Weight)
end

@acset_type WeightedGraph(SchWeightedGraph, index=[:src,:tgt]){Float64}

GF(3) Conservation for ACSets

Integrate with Music Topos 3-coloring:

# Map ACSet parts to trits for GF(3) conservation
function acset_to_trits(g::Graph, seed::UInt64)
    rng = SplitMix64(seed)
    trits = Int[]
    for e in parts(g, :E)
        h = next_u64!(rng)
        hue = (h >> 16 & 0xffff) / 65535.0 * 360
        trit = hue < 60 || hue >= 300 ? 1 :
               hue < 180 ? 0 : -1
        push!(trits, trit)
    end
    trits
end

# Verify conservation: sum(trits) ≡ 0 (mod 3)
function gf3_conserved(trits)
    sum(trits) % 3 == 0
end

Higher-Order Functions on ACSets

From Issue #7, implement functional patterns:

map

map(g, :E) do e; ... end

filter

fold

fold(+, g, :E, :weight)

Open ACSets (Composable Interfaces)

# From Issue #89: Open versions of InterType ACSets
using ACSets.OpenACSetTypes

# Create open ACSet with exposed ports
@open_acset_type OpenGraph(SchGraph, [:V])

# Compose via pushout
g1 = OpenGraph(...)  # ports: v1, v2
g2 = OpenGraph(...)  # ports: v3, v4
g_composed = compose(g1, g2, [:v2 => :v3])

Blog Post Series

1. Graphs and C-sets I: What is a graph?
2. Graphs and C-sets II: Half-edges and rotation systems
3. Graphs and C-sets III: Reflexive graphs and C-set homomorphisms
4. Graphs and C-sets IV: Propositional logic of subgraphs

Citation

@article{patterson2022categorical,
  title={Categorical data structures for technical computing},
  author={Patterson, Evan and Lynch, Owen and Fairbanks, James},
  journal={Compositionality},
  volume={4},
  number={5},
  year={2022},
  doi={10.32408/compositionality-4-5}
}

Related Packages

• Catlab.jl: Full categorical algebra (homomorphisms, limits, colimits)
• AlgebraicRewriting.jl: Declarative rewriting for ACSets
• AlgebraicDynamics.jl: Compositional dynamical systems

Xenomodern Integration

The ironic detachment comes from recognizing that:

1. Category theory isn't about abstraction for its own sake — it's about finding the right abstractions that compose
2. Simple graphs are actually badly behaved — the terminal object problem reveals hidden assumptions
3. Functors are data structures — this reframes databases as applied category theory

         xenomodernity
              │
    ┌─────────┴─────────┐
    │                   │
 ironic              sincere
detachment          engagement
    │                   │
    └─────────┬─────────┘
              │
      C-sets as functors
      (both ironic AND sincere)

Commands

just acset-demo          # Run ACSet demonstration
just acset-graph         # Create and visualize graph
just acset-symmetric     # Symmetric graph example
just acset-gf3           # Check GF(3) conservation