By name: bkp-interleaving description: 'This skill is the deep interleaving of the five BKP 2-monad theory skills:'
BKP Interleaving Hub
"The BKP theorem is not five separate results — it is one theorem with five faces." — Synthesis of Blackwell, Kelly & Power (1989)
Trit: 0 (ERGODIC - meta-coordinator) Color: #26D826 (Green) Status: Production Ready Type: Interleaving Hub (not standalone)
Overview
This skill is the deep interleaving of the five BKP 2-monad theory skills:
| Skill | Trit | Role | BKP Face |
|---|---|---|---|
| codescent | -1 | Validator | Coherence verification |
| 2-monad | 0 | Coordinator | Strictness grid |
| doctrinal-adjunction | 0 | Coordinator | Mate correspondence |
| graded-monad | 0 | Coordinator | Index transport |
| flexible-algebra | +1 | Generator | Bicolimit production |
GF(3) Balance: (-1) + 0 + 0 + 0 + (+1) = 0 ✓
1. The BKP Diamond
The five skills form a diamond dependency graph where every edge is a mathematical theorem:
codescent (-1)
╱ │ ╲
╱ │ ╲
╱ │ ╲
2-monad (0) ────── graded (0) ──── doctrinal (0)
╲ │ ╱
╲ │ ╱
╲ │ ╱
flexible-alg (+1)
Theorem paths (each edge = published result):
codescent → 2-monad:
"Codescent of bar resolution yields strict T-algebra" (Lack 2002)
codescent → flexible-algebra:
"Codescent object is flexible" (BKP 1989, Theorem 4.2)
codescent → graded-monad:
"Graded codescent checks index coherence" (Fujii 2019)
codescent → doctrinal-adjunction:
"Strictification confirms doctrinal structure" (Kelly 1974)
2-monad → flexible-algebra:
"T-Alg has PIE-bicolimits via flexibility" (BKP 1989, Main Theorem)
2-monad → graded-monad:
"Graded monad = monad in [ΣM, Cat]" (Street 1972)
2-monad → doctrinal-adjunction:
"Adjunction between T-algebras lifts via mates" (Kelly 1974)
graded-monad → flexible-algebra:
"Graded algebras have flexible representatives" (Katsumata 2014)
graded-monad → doctrinal-adjunction:
"Index functor adjunction is doctrinal" (Fujii 2019)
doctrinal-adjunction → flexible-algebra:
"Biadjoint produces flexible algebras" (BKP 1989, §5)
2. Unified Julia Schema (All 5 Combined)
using Catlab.CategoricalAlgebra
@present SchBKP(FreeSchema) begin
# ═══ FROM 2-MONAD ═══
TwoCategory::Ob
Algebra::Ob
Morphism::Ob
TwoCell::Ob
alg_carrier::Hom(Algebra, TwoCategory)
alg_action::Hom(Algebra, Algebra)
mor_source::Hom(Morphism, Algebra)
mor_target::Hom(Morphism, Algebra)
cell_source::Hom(TwoCell, Morphism)
cell_target::Hom(TwoCell, Morphism)
Strictness::AttrType # {-1, 0, +1} for colax/pseudo/lax
alg_strictness::Attr(Algebra, Strictness)
mor_strictness::Attr(Morphism, Strictness)
# ═══ FROM DOCTRINAL ADJUNCTION ═══
Adjunction::Ob
LaxCell::Ob
ColaxCell::Ob
adj_left::Hom(Adjunction, Algebra)
adj_right::Hom(Adjunction, Algebra)
lax_source::Hom(LaxCell, Adjunction)
colax_source::Hom(ColaxCell, Adjunction)
mate_lax_to_colax::Hom(LaxCell, ColaxCell)
mate_colax_to_lax::Hom(ColaxCell, LaxCell)
lax_strictness::Attr(LaxCell, Strictness)
colax_strictness::Attr(ColaxCell, Strictness)
# ═══ FROM CODESCENT ═══
Level0::Ob # A (source)
Level1::Ob # B (face maps)
Level2::Ob # C (coherence)
CoherenceCell::Ob
d0_01::Hom(Level0, Level1)
d1_01::Hom(Level0, Level1)
d0_12::Hom(Level1, Level2)
d1_12::Hom(Level1, Level2)
d2_12::Hom(Level1, Level2)
s0::Hom(Level1, Level0)
sigma_source::Hom(CoherenceCell, Level2)
sigma_target::Hom(CoherenceCell, Level2)
CodescentObj::Ob
cods_map::Hom(Level1, CodescentObj)
CocycleStatus::AttrType
cocycle::Attr(CoherenceCell, CocycleStatus)
# ═══ FROM FLEXIBLE ALGEBRA ═══
FreeAlgebra::Ob
PseudoMorphism::Ob
section::Hom(Algebra, FreeAlgebra)
retraction::Hom(FreeAlgebra, Algebra)
classifier::Hom(Algebra, FreeAlgebra)
ps_source::Hom(PseudoMorphism, Algebra)
ps_target::Hom(PseudoMorphism, Algebra)
IsFlexible::AttrType
flexible::Attr(Algebra, IsFlexible)
# ═══ FROM GRADED MONAD ═══
Grade::Ob
GradeMorph::Ob
GradedEndo::Ob
GradedMult::Ob
grade_src::Hom(GradeMorph, Grade)
grade_tgt::Hom(GradeMorph, Grade)
tensor::Hom(Grade, Grade)
unit_grade::Hom(Grade, Grade)
endo_grade::Hom(GradedEndo, Grade)
mult_left::Hom(GradedMult, Grade)
mult_right::Hom(GradedMult, Grade)
mult_result::Hom(GradedMult, Grade)
EffectLabel::AttrType
grade_label::Attr(Grade, EffectLabel)
# ═══ CROSS-SKILL MORPHISMS (THE MIXING) ═══
# Codescent → Algebra: strictification output
cods_to_alg::Hom(CodescentObj, Algebra)
# Algebra → FreeAlgebra → CodescentObj: bar resolution
bar_resolve::Hom(Algebra, Level1)
# Adjunction → Morphism: lift to T-Alg
adj_to_mor::Hom(Adjunction, Morphism)
# Grade → Algebra: graded algebra family
graded_alg::Hom(Grade, Algebra)
# CodescentObj → flexible check
cods_flexible::Attr(CodescentObj, IsFlexible)
# GradedMult → TwoCell: graded multiplication as 2-cell
graded_mult_cell::Hom(GradedMult, TwoCell)
# LaxCell → GradeMorph: index category morphism
lax_grade::Hom(LaxCell, GradeMorph)
# PseudoMorphism → LaxCell: pseudo ↝ lax via classifier
pseudo_to_lax::Hom(PseudoMorphism, LaxCell)
# CoherenceCell → ColaxCell: cocycle as colax cell
cocycle_to_colax::Hom(CoherenceCell, ColaxCell)
end
@acset_type BKPSystem(SchBKP,
index=[:alg_carrier, :mor_source, :mor_target,
:adj_left, :adj_right,
:d0_01, :d1_01, :cods_map,
:section, :retraction,
:grade_src, :grade_tgt, :endo_grade,
:cods_to_alg, :graded_alg])
Cross-Morphism Semantics
| Cross-Morphism | Source Skill | Target Skill | Theorem |
|---|---|---|---|
cods_to_alg |
codescent | 2-monad | Bar resolution → strict algebra |
bar_resolve |
2-monad | codescent | Pseudo algebra → bar construction |
adj_to_mor |
doctrinal-adj | 2-monad | Lifted adjunction → T-morphism |
graded_alg |
graded-monad | 2-monad | Grade m → T_m-algebra |
cods_flexible |
codescent | flexible-alg | Codescent object is flexible |
graded_mult_cell |
graded-monad | 2-monad | μ_{m,n} as 2-cell |
lax_grade |
doctrinal-adj | graded-monad | Lax cell indexed by grade |
pseudo_to_lax |
flexible-alg | doctrinal-adj | Classifier factorizes pseudo → lax |
cocycle_to_colax |
codescent | doctrinal-adj | Cocycle data → colax structure |
3. Pentadic Propagator Chains
Chain 1: The Full BKP Pipeline
TRIGGER: New pseudo T-algebra P arrives
graded-monad Identify index category M for P
│ scope:change
▼
2-monad Locate P in strictness grid (pseudo row)
│ scope:compose
▼
codescent Form bar resolution T³P ⇛ T²P ⇉ TP
│ scope:verify Check cocycle conditions
▼
flexible-algebra Codescent object is flexible retract
│ scope:compose
▼
doctrinal-adj Lift any adjunction to T-Alg via mates
RESULT: P strictified + flexibility verified + adjunctions lifted
Chain 2: Graded Strictification
TRIGGER: Graded monad T_M with pseudo M-algebra
2-monad View T_M as monad in [ΣM, Cat]
│ scope:compose
▼
graded-monad Decompose by grade: T_{-1}, T_0, T_{+1}
│ scope:change
▼
codescent Graded bar resolution (one per grade)
│ scope:verify GF(3) cocycle: σ_m ⊗ σ_n = σ_{m⊗n}
▼
doctrinal-adj Index functor F: M → M' lifts adjunction
│ scope:compose
▼
flexible-algebra Graded flexible algebras generate bicolimits
RESULT: GF(3)-graded strictification with index transport
Chain 3: Doctrinal Descent
TRIGGER: Adjunction f ⊣ u between T-algebras
doctrinal-adj Compute mate: lax ū ↦ colax f̄
│ scope:compose
▼
2-monad Place ū in strictness grid (lax column)
│ scope:change
▼
codescent Verify codescent of ū-indexed diagram
│ scope:verify Cocycle: ū coherent iff mate f̄ coherent
▼
graded-monad Grade the adjunction by effect type
│ scope:compose
▼
flexible-algebra f-algebras flexible iff u-algebras flexible
RESULT: Adjunction graded + mates verified + flexibility propagated
Chain 4: Effect System Compilation
TRIGGER: Programming language with graded effects
graded-monad Parse effect annotations as grades m ∈ M
│ scope:compose
▼
doctrinal-adj Effect handlers = doctrinal adjunction
│ scope:change (handler u ⊣ effectful f)
▼
2-monad Effect system = 2-monad on programming cat
│ scope:compose
▼
codescent Compile: strictify pseudo-effect-algebra
│ scope:verify Verify: no effect leaks (cocycle check)
▼
flexible-algebra Optimize: flexible = can reorder effects
RESULT: Effect system compiled with GF(3) conservation
Chain 5: The Coherence Engine
TRIGGER: "Is this pseudo-algebraic structure coherent?"
codescent ←── Start: form codescent diagram
│
├──→ 2-monad Which 2-monad T?
│ │
│ ├──→ graded-monad Is it graded over M?
│ │ │
│ │ └──→ doctrinal-adj Any adjunctions to lift?
│ │ │
│ └────────────────────┘
│ │
└──────→ flexible-algebra ←───┘
│
▼
VERDICT: Coherent iff
1. Cocycle satisfied (codescent)
2. Grid placement correct (2-monad)
3. Index compatible (graded-monad)
4. Mates consistent (doctrinal-adj)
5. Retract exists (flexible-algebra)
4. Five-Skill Composition Theorems
Theorem A: The BKP Pentad
For any 2-monad T on K, the following are equivalent:
(i) Every pseudo T-algebra has a codescent strictification
[codescent]
(ii) The inclusion J: T-Alg_s → T-Alg has a left biadjoint
[2-monad]
(iii) Every strict T-algebra equivalent to a pseudo one is flexible
[flexible-algebra]
(iv) The mate correspondence lifts all T-algebra adjunctions
[doctrinal-adjunction]
(v) For graded T, strictification respects grades
[graded-monad]
Proof uses all 5 skills in circular dependency:
(i) → (iii) → (ii) → (iv) → (v) → (i)
Theorem B: Graded BKP
For M-graded 2-monad T_M:
codescent(-1) ⊗ 2-monad(0) ⊗ flexible-algebra(+1) = 0
applied grade-by-grade gives:
For each m ∈ M:
codescent_m(-1) ⊗ T_m(0) ⊗ flexible_m(+1) = 0
With cross-grade coherence via:
doctrinal-adjunction (mates between grades)
graded-monad (μ_{m,n} multiplication)
GF(3) CONSERVATION:
If M = GF(3) itself, then:
T_{-1} : validator computations (codescent)
T_0 : coordinator computations (2-monad, doctrinal, graded)
T_{+1} : generator computations (flexible-algebra)
μ : T_m ∘ T_n → T_{m+n mod 3}
Conservation: sum of grades ≡ 0 (mod 3) at every composition
Theorem C: The Mate Pentagon
Five skills form a pentagon of mate correspondences:
codescent ←─── (strictify) ──── 2-monad
│ │
(cocycle) (grid-place)
│ │
▼ ▼
flexible-alg ←── (retract) ─── doctrinal-adj
│ │
(generate) (grade-index)
│ │
└──────── (graded-flex) ────────┘
│
graded-monad
Each edge is a mate correspondence under some adjunction.
The pentagon commutes: traversing either way gives same result.
5. SDF Deep Interleaving
The 5 skills map to 4 distinct SDF chapters, creating a covering:
| Skill | SDF Chapter | Concept Cluster |
|---|---|---|
| 2-monad | Ch.6 Layering | Strictness layers, metadata |
| doctrinal-adj | Ch.5 Evaluation | Mate computation, interpretation |
| codescent | Ch.4 Pattern Matching | Cocycle matching, verification |
| flexible-alg | Ch.8 Degeneracy | Redundancy, retract paths |
| graded-monad | Ch.6 Layering | Grade layers, effect tracking |
Cross-Chapter Flows
Ch.4 (Pattern Match) Ch.5 (Evaluation)
codescent doctrinal-adj
"match cocycle" "evaluate mate"
│ │
└──────── Ch.6 (Layering) ─────┘
2-monad + graded-monad
"layer strictness + grades"
│
Ch.8 (Degeneracy)
flexible-algebra
"redundant paths = flexibility"
Insight: The 4 SDF chapters form their own diamond mirroring the BKP diamond.
6. Composite GF(3) Triads (Cross-Skill)
Primary Pentad Triads
# All 5 participate (via representatives)
codescent (-1) ⊗ 2-monad (0) ⊗ flexible-algebra (+1) = 0 ✓ [BKP Core]
codescent (-1) ⊗ doctrinal-adjunction (0) ⊗ flexible-algebra (+1) = 0 ✓ [Lifting]
codescent (-1) ⊗ graded-monad (0) ⊗ flexible-algebra (+1) = 0 ✓ [Graded-BKP]
# Cross-skill with external neighbors
codescent (-1) ⊗ graded-monad (0) ⊗ operad-compose (+1) = 0 ✓ [Graded-Operadic]
sheaf-cohomology (-1) ⊗ doctrinal-adjunction (0) ⊗ synthetic-adjunctions (+1) = 0 ✓ [Adjunction-Monad]
segal-types (-1) ⊗ 2-monad (0) ⊗ free-monad-gen (+1) = 0 ✓ [Free-Forgetful]
linear-logic (-1) ⊗ graded-monad (0) ⊗ flexible-algebra (+1) = 0 ✓ [Resource-Flexible]
covariant-fibrations (-1) ⊗ doctrinal-adjunction (0) ⊗ flexible-algebra (+1) = 0 ✓ [Fibered-Flexible]
Novel Triads (Discovered by Mixing)
# codescent validates what graded-monad coordinates for flexible-algebra to generate
codescent (-1) ⊗ graded-monad (0) ⊗ flexible-algebra (+1) = 0 ✓ [Graded-Flex-Codescent]
"For each grade m, the codescent of T_m-algebra is flexible"
# codescent validates what doctrinal-adj coordinates for free-monad-gen to generate
codescent (-1) ⊗ doctrinal-adjunction (0) ⊗ free-monad-gen (+1) = 0 ✓ [Free-Codescent]
"Codescent of free resolution lifts doctrinally"
# linear-logic constrains what 2-monad coordinates for flexible-algebra to generate
linear-logic (-1) ⊗ 2-monad (0) ⊗ flexible-algebra (+1) = 0 ✓ [Resource-Algebraic]
"Linear resource tracking → graded flexibility"
# segal-types constrain what graded-monad coordinates for flexible-algebra to generate
segal-types (-1) ⊗ graded-monad (0) ⊗ flexible-algebra (+1) = 0 ✓ [Segal-Graded-Flex]
"Segal condition on graded composition → flexible graded algebras"
7. The BKP State Machine
States: {PSEUDO, BAR, CODESCENT, STRICT, FLEXIBLE, LIFTED, GRADED}
Transitions (each fires a propagator chain):
PSEUDO ──[bar_resolve]──→ BAR
2-monad: form T³P ⇛ T²P ⇉ TP
BAR ──[codescent_compute]──→ CODESCENT
codescent: compute universal cocone
CODESCENT ──[cocycle_check]──→ STRICT (if cocycle satisfied)
codescent: verify σ₀(d₀) ∘ σ₁(d₀) = σ₀(d₁)
CODESCENT ──[cocycle_fail]──→ PSEUDO (if cocycle violated)
codescent: report violation, return to start
STRICT ──[flexibility_check]──→ FLEXIBLE
flexible-algebra: verify retract of free exists
FLEXIBLE ──[doctrinal_lift]──→ LIFTED
doctrinal-adjunction: compute mates for all adjunctions
LIFTED ──[grade_decompose]──→ GRADED
graded-monad: decompose by effect grades
GRADED ──[grade_compose]──→ STRICT
graded-monad: μ_{m,n} recombines graded algebras
Any state ──[2-monad-locate]──→ Same state + grid position
2-monad: identify position in strictness grid
┌──────────────────────────────────────────────┐
│ │
▼ │
PSEUDO ──→ BAR ──→ CODESCENT ──→ STRICT │
│ │ │
fail │ ┌────┘ │
│ ▼ │
└── FLEXIBLE ──→ LIFTED ──→ GRADED
8. Commands (Unified Pipeline)
# Full BKP pipeline
just bkp-pipeline pseudo-alg # PSEUDO → BAR → CODESCENT → STRICT → FLEXIBLE → LIFTED → GRADED
# Individual steps
just bkp-bar-resolve T pseudo-alg # Form bar resolution (2-monad + codescent)
just bkp-strictify pseudo-alg # Codescent-based strictification
just bkp-flex-check strict-alg # Check flexibility (retract of free)
just bkp-doctrinal-lift f u T # Lift adjunction via mates
just bkp-grade-decompose T M # Decompose by grades
# Cross-skill verification
just bkp-pentad-verify # Run all 5 coherence checks
just bkp-pentagon-commute # Verify mate pentagon commutes
just bkp-gf3-graded-check T # GF(3) conservation per grade
# Exploration
just bkp-diamond # Display dependency diamond
just bkp-chains # List all propagator chains
just bkp-state T pseudo-alg # Show current BKP state machine position
9. Deep Mixing Examples
Example 1: Monoidal Category Coherence
Problem: Show every monoidal category is monoidally equivalent to a strict one.
Step 1 [2-monad]: T = free strict monoidal category monad on Cat
Step 2 [graded-monad]: Grade by associator complexity (ℕ-graded)
Step 3 [codescent]: Bar resolution of pseudo T-algebra (monoidal cat)
Step 4 [codescent]: Verify pentagonal cocycle (Mac Lane's pentagon)
Step 5 [flexible-algebra]: Codescent object is flexible strict monoidal
Step 6 [doctrinal-adjunction]: Monoidal adjunctions lift automatically
All 5 skills fire. Mac Lane coherence = BKP pentad for free-monoid 2-monad.
Example 2: Effect System Compilation
Problem: Compile graded effect types to efficient code.
Step 1 [graded-monad]: Parse effect types as grades M = {IO, State, Error}
Step 2 [2-monad]: Effect system = 2-monad on category of computations
Step 3 [doctrinal-adjunction]: Effect handler = doctrinal adjunction
(handler is right adjoint, effectful code is left adjoint)
Step 4 [codescent]: Strictify: pseudo-effect-algebra → strict
(eliminate unnecessary effect coercions)
Step 5 [flexible-algebra]: Flexible = effects can be reordered
(commutativity up to retract)
Result: Optimized effect compilation with GF(3) conservation:
IO(-1) + State(0) + Error(+1) = 0 per computation block
Example 3: IES Session Type Verification
Problem: Verify session types for distributed protocol.
Step 1 [graded-monad]: Session steps are grades (Protocol category M)
Step 2 [2-monad]: Session monad = 2-monad on typed channels
Step 3 [doctrinal-adjunction]: Client/server = adjoint pair
Mate: lax(server) ↔ colax(client)
Step 4 [codescent]: Protocol coherence = cocycle condition
(sequential composition respects protocol graph)
Step 5 [flexible-algebra]: Flexible session = can buffer/reorder
messages without breaking protocol
GF(3) conservation per session:
Send(-1) + Route(0) + Recv(+1) = 0 per protocol round
10. Neighbor Awareness (Meta-Level)
Inbound (skills that use the BKP hub)
| Source | Fires When |
|---|---|
| open-games | Para(Optic) needs 2-monadic structure |
| linear-logic | Resource tracking needs graded effects |
| topos-adhesive-rewriting | Adhesive rewriting in T-Alg |
| acsets-algebraic-databases | C-Set migration via Kan + doctrinal |
| segal-types | Validate categorical structure |
| sheaf-cohomology | Descent = dual of codescent |
Outbound (BKP hub provides to)
| Target | Provides |
|---|---|
| free-monad-gen | Free T-algebras for retraction |
| kan-extensions | Lan/Ran via doctrinal adjunction |
| synthetic-adjunctions | Adjunction generation + lifting |
| infinity-operads | Operadic algebras via 2-monadic structure |
| elements-infinity-cats | ∞-categorical codescent |
| gf3-tripartite | GF(3) as index category for graded monad |
11. References
- Blackwell, R., Kelly, G.M. & Power, A.J. (1989). "Two-dimensional monad theory." JPAA 59:1-41
- Kelly, G.M. (1974). "Doctrinal adjunction." LNM 420:257-280
- Lack, S. (2002). "Codescent objects and coherence." JPAA 175:223-241
- Lack, S. (2010). "A 2-categories companion." IMA Vol. Math. Appl. 152:105-191
- Fujii, S. (2019). "Towards a formal theory of graded monads." FSCD 4:1-17
- Katsumata, S. (2014). "Parametric effect monads." POPL
- Street, R. (1972). "The formal theory of monads." JPAA 2:149-168
- Weber, M. (2015). "Operads as polynomial 2-monads." TAC 30:1659-1712
SDF Interleaving
Primary Chapter: 6. Layering (shared with 2-monad + graded-monad)
The BKP hub layers all 5 skills. Each layer adds structure:
Layer 0: Raw type (no monad)
Layer 1: 2-monad T identified [2-monad]
Layer 2: Strictness grid position [2-monad]
Layer 3: Bar resolution formed [codescent]
Layer 4: Cocycle verified [codescent]
Layer 5: Flexibility checked [flexible-algebra]
Layer 6: Adjunctions lifted [doctrinal-adjunction]
Layer 7: Grades decomposed [graded-monad]
GF(3) Balanced Triad
codescent (-1) + bkp-interleaving (0) + flexible-algebra (+1) = 0 ✓
Skill Trit: 0 (ERGODIC - meta-coordination)