Weak n-category¶
Organize cells through dimension n with composition and units that satisfy associativity and unitality up to coherently related higher cells rather than by strict equality.
Core Idea¶
A weak n-category is an n-dimensional categorical structure in which composition and identity laws hold through specified coherent higher equivalences or isomorphisms rather than all holding as literal equalities. Failure of strict equality at one dimension is witnessed by a cell in the next dimension, and coherence laws constrain the possible composites of those witnesses so that alternative parenthesizations and unit insertions agree at the appropriate higher level.
Its autonomous residual is dimension-indexed categorical composition governed by higher coherence data, including explicit model dependence, rather than ordinary category theory or the unrestricted idea of approximate equality.
Scope of Application¶
Weak n-category applies when the analyst can specify a dimensionally graded collection of objects and k-cells for k from one through n, with source, target, identity, and composition data in a declared model and establish that the chosen model supplies typed cells and compositions through dimension n together with coherence data that weakens categorical laws without leaving composites arbitrary. The entry identifies the shared weak-higher-categorical architecture and maps model dependence; it does not declare all proposed definitions equivalent or choose one as universally canonical.
Clarity¶
A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because weak means laws hold coherently up to higher structure, not that axioms are optional, approximate, probabilistic, or logically deficient. The disciplined statement is that the object counts as Weak n-category exactly when the chosen model supplies typed cells and compositions through dimension n together with coherence data that weakens categorical laws without leaving composites arbitrary
Manages Complexity¶
The abstraction compresses bicategorical, globular-operadic, opetopic, simplicial, Segal-type, and enriched models; finite n and omega-dimensional limits into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.
Compression can hide assumptions. A responsible use therefore declares dimension n, cell shape, strict versus weak compositions, coherence data, invertibility level, algebraic versus simplicial presentation, truncation, and equivalence notion and returns to the full diagnostic whenever a convention or boundary case changes.
Abstract Reasoning¶
- Type the carrier. Establish a dimensionally graded collection of objects and k-cells for k from one through n, with source, target, identity, and composition data in a declared model and reject examples from a different problem. 2. Lock the rule. Express that the chosen model supplies typed cells and compositions through dimension n together with coherence data that weakens categorical laws without leaving composites arbitrary independently of one notation or implementation.
Knowledge Transfer¶
Transfer within higher category theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from A bicategory is the weak two-dimensional case: composition of 1-cells is associative and unital through specified natural isomorphisms satisfying pentagon and triangle coherence. to Models of higher homotopy types use weak higher-groupoidal structures in which higher cells are invertible in the relevant sense. demonstrates that continuity.
Relationships to Other Abstractions¶
Current abstraction Weak n-category Domain-specific
Parents (1) — more general patterns this builds on
-
Weak n-category is a kind of Category Prime
The proposed strict upward parent is
prime:category.
Hierarchy paths (3) — routes to 3 parentless roots
- Weak n-category → Category → Associativity → Invariance
- Weak n-category → Category → Closure
- Weak n-category → Category → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Weak n-category 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 — Relations, Definability & Constraint Structure (11 abstractions)
Nearest neighbors
- Tetracategory — 0.93
- Interchange law — 0.91
- Mathematical structure — 0.90
- Traced monoidal category — 0.90
- Linear group — 0.89
Computed from structural-signature embeddings · 2026-09-08