2 Monad

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

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

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

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).