Rigid category¶
A monoidal category in which every object has a left and right dual, with evaluation and coevaluation morphisms satisfying triangular identities.
Core Idea¶
A rigid category extends finite-dimensional-style duality to every object in a monoidal setting. Cups and caps create and eliminate dual pairs, allowing morphisms to transpose across tensor factors coherently. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
The load-bearing residual is not the broad topic of category theory. It is A monoidal category in which every object has a left and right dual, with evaluation and coevaluation morphisms satisfying triangular identities.
Scope of Application¶
Rigid category belongs to category theory and is useful where the analyst can specify monoidal category, tensor unit, objects, left and right dual objects, evaluation and coevaluation maps and snake identities, then evaluate every object has the declared left and right dual data satisfying both triangular identities. The scope is broad within that domain but bounded by the need for every object has the declared left and right dual data satisfying both triangular identities. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making every object has the declared left and right dual data satisfying both triangular identities the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Rigid category can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Rigid category. Rigid category compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: monoidal category, tensor unit, objects, left and right dual objects, evaluation and coevaluation maps and snake identities. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express every object has the declared left and right dual data satisfying both triangular identities independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of category theory because they reuse monoidal category, tensor unit, objects, left and right dual objects, evaluation and coevaluation maps and snake identities, Cups and caps create and eliminate dual pairs, allowing morphisms to transpose across tensor factors coherently., and type the carrier, state every parameter and convention in the definition, test that every object has the declared left and right dual data satisfying both triangular identities, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Rigid category Domain-specific
Parents (1) — more general patterns this builds on
-
Rigid category is a kind of Duality Prime
The proposed strict upward parent is
prime:duality.
Hierarchy path (1) — routes to 1 parentless root
- Rigid category → Duality
Neighborhood in Abstraction Space¶
Rigid category sits in a crowded region of the domain-specific corpus (4th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Category-Theoretic Structures (79 abstractions)
Nearest neighbors
- Compact closed category — 0.96
- Monoid (category theory) — 0.94
- Unitary modular tensor category — 0.94
- Opposite category — 0.93
- Closed monoidal category — 0.93
Computed from structural-signature embeddings · 2026-09-08