∞-Operads for pairwise/tritwise Cat# interactions with lazy ACSet materialization unifying effective, realizability, and Grothendieck topoi via dendroidal Segal spaces.
/plugin marketplace add plurigrid/asi/plugin install asi-skills@asi-skillsThis skill inherits all available tools. When active, it can use any tool Claude has access to.
"The dendroidal nerve carries operads to ∞-operads exactly as the simplicial nerve carries categories to ∞-categories." — Cisinski-Moerdijk
Trit: 0 (ERGODIC)
Color: #26D826 (Green)
Role: Coordinator/Transporter
XIP: Dendroidal (Ω-set) → Cat# horizontal morphism
| Interaction Type | Cat# Structure | Operad View | Lazy ACSet |
|---|---|---|---|
| Pairwise | Bicomodule composition in Prof | Binary operation | JOIN on demand |
| Tritwise | Equipment tensor ⊗ (GF(3) balanced) | Ternary tree grafting | Materialized view |
| N-ary | ∞-operad algebra evaluation | Dendroidal composition | Recursive CTE |
Objects: Finite rooted trees T with labelled edges Morphisms: Face/degeneracy maps (like Δ for simplicial sets)
r
/|\
e1 e2 e3 ∈ Ω (corolla with 3 inputs)
Functor X: Ω^op → Set
X(T) = set of T-shaped operationsA dendroidal set satisfying:
Nerve: Operads → dSet
N_dO(T) = Hom_Operad(Ω(T), O)
In Cat# = Comod(P):
Pairwise interaction = bicomodule M: C ↛ D:
M: C^op × D → Set
The equipment structure provides:
⊗: Prof(C, D) × Prof(D, E) → Prof(C, E)
Tritwise interaction = tensor of three bicomodules:
M ⊗ N ⊗ P: C ↛ E where GF(3)(M, N, P) = 0
For ∞-operad O and category C:
Alg_O(C) = Fun^⊗(O, C)
N-ary skill interaction = O-algebra evaluation:
eval: O(n) × C^n → C
@present SchLazyTopos(FreeSchema) begin
# Objects
Topos::Ob
Site::Ob
Functor::Ob
# Types of topoi
is_grothendieck::Attr(Topos, Bool)
is_effective::Attr(Topos, Bool)
is_realizability::Attr(Topos, Bool)
# Geometric morphism = adjoint pair (f^*, f_*)
GeomMorph::Ob
source::Hom(GeomMorph, Topos)
target::Hom(GeomMorph, Topos)
# Lazy parts (computed on demand)
inverse_image::Hom(GeomMorph, Functor) # f^* (left adjoint)
direct_image::Hom(GeomMorph, Functor) # f_* (right adjoint)
# Category of elements for on-demand computation
∫::Hom(Functor, Site)
end
Instead of materializing all parts:
# Don't compute: X(ob) for all ob ∈ C
# Instead: ∫X = category of elements, evaluate locally
function lazy_parts(X::ACSet, query::Query)
# Only materialize parts matching query
∫X = category_of_elements(X)
return filter(∫X, query)
end
-- Lazy view: pairwise interactions as bicomodules
CREATE OR REPLACE VIEW v_pairwise_bicomodule AS
SELECT
s1.skill_id AS source,
s2.skill_id AS target,
s1.trit + s2.trit AS gf3_partial,
CASE
WHEN s1.trit = -1 THEN 'Ran_K → Adj'
WHEN s1.trit = 0 THEN 'Adj → ?'
WHEN s1.trit = 1 THEN 'Lan_K → ?'
END AS migration_type,
s1.color || ' → ' || s2.color AS color_flow
FROM catsharp_skills s1
CROSS JOIN catsharp_skills s2
WHERE s1.skill_id != s2.skill_id
-- Lazy: only materialize on query with specific source/target
;
-- Lazy view: tritwise GF(3) balanced interactions
CREATE OR REPLACE VIEW v_tritwise_equipment AS
SELECT
s1.skill_id AS minus_skill,
s2.skill_id AS zero_skill,
s3.skill_id AS plus_skill,
(s1.trit + s2.trit + s3.trit) AS gf3_sum,
CASE (s1.trit + s2.trit + s3.trit) % 3
WHEN 0 THEN '✓ BALANCED'
ELSE '✗ VIOLATION'
END AS status,
-- Equipment tensor structure
s1.kan_role || ' ⊗ ' || s2.kan_role || ' ⊗ ' || s3.kan_role AS equipment_tensor,
-- Color flow
s1.color || ' ⊗ ' || s2.color || ' ⊗ ' || s3.color AS color_tensor
FROM catsharp_skills s1
CROSS JOIN catsharp_skills s2
CROSS JOIN catsharp_skills s3
WHERE s1.trit = -1 AND s2.trit = 0 AND s3.trit = 1
-- Lazy: exponentially many rows, query with constraints
;
-- Lazy view: geometric morphisms between skill-topoi
CREATE OR REPLACE VIEW v_geometric_morphism AS
SELECT
source.skill_id AS source_topos,
target.skill_id AS target_topos,
source.home AS source_home,
target.home AS target_home,
-- Adjoint pair: inverse_image ⊣ direct_image
source.kan_role AS inverse_image, -- Left adjoint
target.kan_role AS direct_image, -- Right adjoint
-- Topos type
CASE source.home
WHEN 'Presheaves' THEN 'grothendieck'
WHEN 'Span' THEN 'effective'
WHEN 'Prof' THEN 'realizability'
END AS topos_type
FROM catsharp_skills source
CROSS JOIN catsharp_skills target
WHERE source.skill_id != target.skill_id
-- Lazy: materialize specific morphism on demand
;
| Topos Type | Cat# Home | Trit | Key Property |
|---|---|---|---|
| Grothendieck | Presheaves | +1 | Sheaves on site (generators) |
| Realizability | Prof | 0 | Computable functions (transport) |
| Effective | Span | -1 | Quotients of decidable sets (validators) |
The key insight: all geometric morphisms form a 2-category (or (∞,1)-category) and the lazy ACSet materialization computes:
Geom(E, F) = { f: E → F | f^* ⊣ f_* }
┌─────────────────────────────────────────────────────────────────────────┐
│ Cat# Equipment Structure Unifying Topoi │
│ │
│ Span (Effective) │
│ ↑ Ran_K │
│ │ │
│ Presheaves ←──┼── Prof (Realizability) │
│ (Groth) │ ↑ Adj │
│ ↑ Lan_K │ │ │
│ └────────┴──────┘ │
│ │
│ GF(3) Conservation: (-1) + (0) + (+1) ≡ 0 (mod 3) │
│ = Naturality condition = All three topoi are equivalent │
└─────────────────────────────────────────────────────────────────────────┘
The naturality square:
G(f) ∘ η_A = η_B ∘ F(f)
Becomes the topos equivalence via:
# Core ∞-Operads Triads
# Dendroidal Core
segal-types (-1) ⊗ infinity-operads (0) ⊗ rezk-types (+1) = 0 ✓
# Cat# Equipment
temporal-coalgebra (-1) ⊗ infinity-operads (0) ⊗ free-monad-gen (+1) = 0 ✓
# Topos Unification
sheaf-cohomology (-1) ⊗ infinity-operads (0) ⊗ topos-generate (+1) = 0 ✓
# Lazy Materialization
persistent-homology (-1) ⊗ infinity-operads (0) ⊗ oapply-colimit (+1) = 0 ✓
# Spivak Cat# Integration
yoneda-directed (-1) ⊗ infinity-operads (0) ⊗ operad-compose (+1) = 0 ✓
# Cisinski-Moerdijk
kinetic-block (-1) ⊗ infinity-operads (0) ⊗ gay-mcp (+1) = 0 ✓
# Query pairwise bicomodule interactions
just infinity-pairwise source=kan-extensions target=operad-compose
# Find all GF(3) balanced tritwise interactions
just infinity-tritwise --balanced
# Lazy materialize geometric morphisms for a skill
just infinity-geom-morph --skill=acsets
# Show topos unification status
just infinity-topos-unify
# Generate ∞-operad algebra evaluation diagram
just infinity-algebra-eval --operad=E_n --arity=3
# Add to catsharp_skill_acset_mapping.py
def create_infinity_operad_views():
"""Create lazy materialization views for ∞-operad interactions"""
return """
-- Dendroidal tree structure for n-ary operations
CREATE OR REPLACE VIEW v_dendroidal_tree AS
WITH RECURSIVE tree AS (
-- Base: corollas (single node)
SELECT skill_id, trit, 1 AS depth, skill_id AS root
FROM catsharp_skills
UNION ALL
-- Recursive: graft trees
SELECT
t.skill_id || '◁' || s.skill_id AS skill_id,
(t.trit + s.trit) % 3 AS trit,
t.depth + 1 AS depth,
t.root
FROM tree t
JOIN catsharp_skills s ON s.skill_id != t.skill_id
WHERE t.depth < 3 -- Limit recursion
)
SELECT * FROM tree;
-- ∞-operad algebra: evaluate n-ary operation
CREATE OR REPLACE VIEW v_operad_algebra_eval AS
SELECT
'E_' || COUNT(*) AS operad,
GROUP_CONCAT(skill_id, ' ⊗ ') AS operands,
SUM(trit) AS trit_sum,
CASE SUM(trit) % 3
WHEN 0 THEN 'coherent'
ELSE 'obstruction'
END AS evaluation_status
FROM catsharp_skills
GROUP BY CUBE(trit) -- All possible groupings
HAVING COUNT(*) >= 2;
-- Topos equivalence via geometric morphisms
CREATE OR REPLACE VIEW v_topos_equivalence AS
SELECT
s1.home AS topos_1,
s2.home AS topos_2,
COUNT(*) AS morphism_count,
SUM(s1.trit + s2.trit) % 3 AS gf3_balance
FROM catsharp_skills s1
JOIN catsharp_skills s2 ON s1.home != s2.home
GROUP BY s1.home, s2.home;
"""
# Lazy ACSet materialization
class LazyACSetMaterializer:
"""Compute ACSet parts on demand, not upfront"""
def __init__(self, schema, conn):
self.schema = schema
self.conn = conn
self._cache = {}
def parts(self, ob: str, query: str = None):
"""Materialize parts of object ob matching query"""
cache_key = (ob, query)
if cache_key in self._cache:
return self._cache[cache_key]
# Lazy SQL execution
sql = f"SELECT * FROM catsharp_skills WHERE 1=1"
if query:
sql += f" AND {query}"
result = self.conn.execute(sql).fetchall()
self._cache[cache_key] = result
return result
def geometric_morphism(self, source: str, target: str):
"""
Compute geometric morphism between skill-topoi.
Only materializes the specific adjoint pair.
"""
return {
'source': source,
'target': target,
'inverse_image': f'Lan_{source}', # f^*
'direct_image': f'Ran_{target}', # f_*
'adjunction': 'f^* ⊣ f_*'
}
def category_of_elements(self, functor_id: str):
"""∫F: category of elements for on-demand traversal"""
# Lazy: return iterator, not list
sql = f"""
SELECT skill_id, trit, color
FROM catsharp_skills
WHERE skill_id LIKE '%{functor_id}%'
"""
return self.conn.execute(sql).fetchdf().iterrows()
| Direction | Neighbor | Relationship |
|---|---|---|
| Left (-1) | kan-extensions | Universal property source (Lan ⊣ Res ⊣ Ran) |
| Right (+1) | operad-compose | Composition target (oapply colimit) |
catsharp — Cat# = Comod(P) polynomial equipmentkan-extensions — Universal property formulationoperad-compose — Operadic composition via oapplyasi-polynomial-operads — Full polynomial functor theorykinetic-block — Dendroidal stratification patternssegal-types — ∞-category Segal conditionsrezk-types — Complete Segal spaces