2-Monad Skill

"A 2-monad is a monad in the 2-category of 2-categories — the natural habitat of algebraic structure on categories." — Blackwell, Kelly & Power (1989)

Trit: 0 (ERGODIC - coordinator) Color: #26D826 (Green) Status: Production Ready

---

Overview

A 2-monad is a monad internal to a 2-category K. Concretely, it consists of:

• An endo-2-functor T : K → K
• A 2-natural transformation η : Id_K → T (unit)
• A 2-natural transformation μ : T² → T (multiplication)
• Subject to the usual monad laws, now holding as equations between 2-natural transformations

The key insight: by varying the strictness of algebras and morphisms, a single 2-monad T generates a 4×4 grid of categories T-Alg_s, Ps-T-Alg, Lax-T-Alg, Colax-T-Alg with strict/pseudo/lax/colax morphisms between them.

Mathematical Definition

2-Monad (T, η, μ) on 2-category K:

  T : K → K           (endo-2-functor)
  η : Id → T          (unit, 2-natural)
  μ : TT → T          (multiplication, 2-natural)

  μ ∘ Tμ = μ ∘ μT     (associativity)
  μ ∘ Tη = id = μ ∘ ηT (unitality)

The Strictness Grid

The central organizing principle of BKP theory:

                    MORPHISMS
              strict  pseudo  lax   colax
           ┌────────┬────────┬──────┬───────┐
  strict   │ T-Alg_s│ T-Alg  │ T-Algl│ T-Algc│
           ├────────┼────────┼──────┼───────┤
A pseudo   │ Ps-s   │ Ps-T   │ Ps-l │ Ps-c  │
L          ├────────┼────────┼──────┼───────┤
G lax      │ Lax-s  │ Lax-ps │Lax-l │ Lax-c │
S          ├────────┼────────┼──────┼───────┤
  colax    │ Col-s  │ Col-ps │Col-l │ Col-c │
           └────────┴────────┴──────┴───────┘

BKP Default: T-Alg = (strict algebras, pseudo morphisms)
             This is the "right" category for most purposes.

GF(3) Mapping of Strictness

Strictness	Trit	Role	Justification
Colax	-1 (MINUS)	Validator	Weakens structure downward
Pseudo	0 (ERGODIC)	Coordinator	Equivalence-invariant
Lax	+1 (PLUS)	Generator	Adds structure upward
Strict	—	Fixed point	Degenerate case (all trits equal)

Key Theorems (BKP 1989)

1. Biadjunction

The inclusion J : T-Alg_s → T-Alg has a left biadjoint.
Equivalently: T-Alg has all PIE-limits and J preserves them.

2. Flexible Algebras

A strict T-algebra A is flexible iff it is a retract (in T-Alg)
of a free T-algebra TX for some X.

3. Coherence

Every pseudo T-algebra is equivalent (in T-Alg) to a strict one.
"Pseudo = Strict up to equivalence"

Canonical Examples

2-Monad T	K	T-Alg_s	T-Alg (pseudo)
Free monoid	Cat	Strict monoidal cats	Monoidal cats
Free coproduct	Cat	Cats with strict coproducts	Cats with coproducts
Free symmetric monoidal	Cat	Permutative cats	Symmetric monoidal cats
Idempotent completion	Cat	Cauchy-complete cats	Cauchy-complete cats
Presheaf construction	Cat^op	—	Topoi (as 2-algebras)

Julia/Catlab Integration

using Catlab.CategoricalAlgebra

# 2-Monad as endo-2-functor on Cat
@present Sch2Monad(FreeSchema) begin
    TwoCategory::Ob
    Algebra::Ob
    Morphism::Ob
    TwoCell::Ob

    alg_carrier::Hom(Algebra, TwoCategory)     # Underlying object
    alg_action::Hom(Algebra, Algebra)          # T-action
    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)
end

Why This Skill Was Missing

The concept of 2-monads was distributed across multiple skills without a single owner:

1. synthetic-adjunctions handles adjunctions but not their monad-generating capacity in 2-categories
2. kan-extensions covers Lan/Ran but not the 2-monad T whose algebras are the extended structures
3. infinity-operads connects to operads-as-2-monads (Weber) but doesn't own the BKP theory
4. free-monad-gen generates free monads in 1-categories, not free 2-monads
5. elements-infinity-cats works at the ∞-level but doesn't specialize to strict 2-monads

Gap: No skill owned the strictness grid, the BKP biadjunction theorem, or the T-Alg construction. This skill fills that gap as the central coordinator for 2-categorical algebraic structure.

---

Bidirectional Neighbor Index

Edge-Scoped Propagator Table

Edge	Direction	Scope	Fires When
2-monad → doctrinal-adjunction	outbound	scope:compose	Adjunction between T-algebras formed
doctrinal-adjunction → 2-monad	inbound	scope:change	Lax/colax structure on adjoint discovered
2-monad → flexible-algebra	outbound	scope:change	New T-algebra constructed
flexible-algebra → 2-monad	inbound	scope:verify	Flexibility of algebra verified
2-monad → codescent	outbound	scope:compose	Pseudoalgebra needs strictification
codescent → 2-monad	inbound	scope:verify	Codescent object validates coherence
2-monad → graded-monad	outbound	scope:change	Index category for grading identified
graded-monad → 2-monad	inbound	scope:compose	Graded monad embedded as 2-monad
2-monad → synthetic-adjunctions	outbound	scope:compose	Monad extracted from adjunction
synthetic-adjunctions → 2-monad	inbound	scope:change	Adjunction L ⊣ R generates monad RL
2-monad → kan-extensions	outbound	scope:compose	Lan/Ran gives 2-monad on presheaves
kan-extensions → 2-monad	inbound	scope:change	Migration via T-algebra structure
2-monad → free-monad-gen	outbound	scope:change	Free T-algebra needed
free-monad-gen → 2-monad	inbound	scope:compose	Free monad lifted to 2-monad
2-monad → infinity-operads	outbound	scope:compose	Operad realized as polynomial 2-monad
infinity-operads → 2-monad	inbound	scope:change	Dendroidal nerve of T-Alg
2-monad → acsets-algebraic-databases	outbound	scope:change	C-Set categories are 2-monadic
acsets-algebraic-databases → 2-monad	inbound	scope:verify	Schema migration preserves T-structure
2-monad → segal-types	outbound	scope:verify	Nerve of T-Alg is Segal
segal-types → 2-monad	inbound	scope:verify	Composition in T-Alg validated
2-monad → open-games	outbound	scope:compose	Game composition via 2-monadic structure
open-games → 2-monad	inbound	scope:change	Para(Optic) as 2-monad algebra
2-monad → bidirectional-lens-logic	outbound	scope:change	Lax/colax = covariant/contravariant
bidirectional-lens-logic → 2-monad	inbound	scope:verify	4-kind lattice validates strictness grid

Mutual Awareness Summary

         codescent (-1)
              ↑ verify
              │
flexible (+1) ←── 2-MONAD (0) ──→ doctrinal-adj (0)
              │         │
              ↓ compose ↓ change
      graded (0)   synthetic-adj (+1)

+ 8 additional edges to existing skills

Total edges: 24 (12 bidirectional pairs) Propagator balance: 8 scope:change + 8 scope:compose + 8 scope:verify = balanced

GF(3) Triads

codescent (-1) ⊗ 2-monad (0) ⊗ flexible-algebra (+1) = 0 ✓  [BKP Core]
sheaf-cohomology (-1) ⊗ 2-monad (0) ⊗ synthetic-adjunctions (+1) = 0 ✓  [Adjunction-Monad]
segal-types (-1) ⊗ 2-monad (0) ⊗ free-monad-gen (+1) = 0 ✓  [Free-Forgetful]
linear-logic (-1) ⊗ 2-monad (0) ⊗ operad-compose (+1) = 0 ✓  [Resource-Algebraic]
covariant-fibrations (-1) ⊗ 2-monad (0) ⊗ discopy-operads (+1) = 0 ✓  [Fibered-Operadic]

Commands

just 2-monad-grid              # Display full strictness grid
just 2-monad-algebras T        # List T-algebra categories
just 2-monad-from-adjunction L R  # Extract monad from adjunction
just 2-monad-coherence check   # Verify pseudo = strict up to equiv
just 2-monad-bkp-biadjoint T   # Compute BKP left biadjoint

References

• Blackwell, Kelly & Power (1989). "Two-dimensional monad theory." JPAA 59:1-41
• Kelly (1974). "Doctrinal adjunction." LNM 420:257-280
• Lack (2002). "Codescent objects and coherence." JPAA 175:223-241
• Lack (2010). "A 2-categories companion." IMA Vol. Math. Appl. 152:105-191
• Weber (2015). "Operads as polynomial 2-monads." TAC 30:1659-1712
• Street (1972). "The formal theory of monads." JPAA 2:149-168

SDF Interleaving

Primary Chapter: 6. Layering

Concepts: layered data, metadata, provenance

GF(3) Balanced Triad

2-monad (0) + SDF.Ch6 (0) + [balancer] (0) = 0

Skill Trit: 0 (ERGODIC - coordination)

Connection Pattern

Layering adds structure level by level. 2-monads layer algebraic structure on categories, with each strictness level a new layer.

CT lattice atlas

Part of: para-mensch-commons (CT lattice family).