Pseudomonad (category theory)¶
A higher-categorical monad whose unit and multiplication laws hold up to coherent invertible cells.
Core Idea¶
A pseudomonad weakens ordinary monad equations in a controlled higher-dimensional way. An endo action T still has unit η and multiplication μ. Instead of requiring the unit and associativity composites to be literally equal, it supplies specified invertible higher cells between them and demands further coherence. 'Up to isomorphism' is not license to ignore the laws: the witnesses and their compatibility are part of the object.
The diagram pseudomonad sends a category to its diagrams, puts an object into a one-object diagram, and flattens diagrams of diagrams. The small-presheaf pseudomonad uses Yoneda and weighted-colimit multiplication. Both come from an authored construction with size restrictions; they show why an exact ambient 2-category and coherence convention matter. A strict 2-monad fits as a special case, but not every pseudomonad has strictly equal laws in its given presentation.
Scope of Application¶
The ambient higher category and size convention must be stated; 'up to isomorphism' alone omits coherence.
- Higher category theory. Study monad-like algebra under weak associativity and units.
- Categorical completion. Model free diagram and small-presheaf completion operations.
- Pseudoalgebra theory. Relate chosen colimit structures to actions of pseudomonads.
- Formal semantics. Check coherence and size before transferring monadic constructions.
Clarity¶
A pseudomonad has an endo action, unit, and multiplication, with unit and associativity laws witnessed by coherent invertible higher cells. The diagram pseudomonad uses one-object diagram insertion and flattening; the small-presheaf pseudomonad uses Yoneda and weighted colimits. A bare 'up to isomorphism' claim is too weak without coherence and size conditions.
Manages Complexity¶
Coherence packages many possible rebracketings and unit insertions into compatible transformations. The abstraction lets weakly associative constructions be handled systematically, while size and ambient-category choices prevent a concise formula from becoming ill-typed.
Abstract Reasoning¶
Choose the ambient higher category, define T, η, and μ, supply invertible law comparisons, verify coherence, and check the construction is well-typed under its size convention.
Knowledge Transfer¶
The pseudomonad architecture applies to several categorical completion processes, but an ordinary weak analogy about repeated operations is not a literal pseudomonad. Higher-cell data, well-typed transformations, and coherence must survive transfer.
Relationships to Other Abstractions¶
Current abstraction Pseudomonad (category theory) Domain-specific
Parents (1) — more general patterns this builds on
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Pseudomonad (category theory) is a kind of Mathematical structure Domain-specific
Pseudomonad (category theory) is a strict kind of Mathematical structure: its frozen identity entails the parent's defining structure while adding domain-specific restrictions.
Hierarchy path (1) — routes to 1 parentless root
- Pseudomonad (category theory) → Mathematical structure → Set and Membership
Neighborhood in Abstraction Space¶
Pseudomonad (category theory) sits in a crowded region of the domain-specific corpus (39th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Category Theory & Higher Structures (18 abstractions)
Nearest neighbors
- 3-Category — 0.89
- Simplicial Localization — 0.88
- Complement (group theory) — 0.88
- Join of Categories — 0.87
- Monoidal Natural Transformation — 0.87
Computed from structural-signature embeddings · 2026-10-08