Symplectic category¶
In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into M \times N^{-} , where the superscript minus means minus the given symplectic form (for example, the graph of a symplectomorphism; hence, minus).
Core Idea¶
Symplectic category is treated here as the recurring symplectic geometry identity summarized by this source-grounded definition: In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into M \times N^{-} , where the superscript minus means minus the given symplectic form (for example, the graph of a symplectomorphism; hence, minus). In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into M \times N^{-} , where the.
Scope of Application¶
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Documented setting. The notion was introduced by Alan Weinstein, according to whom "Quantization problems suggest that the category of symplectic manifolds and symplectomorphisms be augmented by the inclusion of canonical relations as morphisms.".
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Documented setting. Strictly speaking, the symplectic category is not a well-defined category (since the composition may not be well-defined) without some transversality conditions.
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Documented setting. In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into M \times N^{-} , where the.
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Documented setting. The notion was introduced by Alan Weinstein, according to whom "Quantization problems suggest that the category of symplectic manifolds and symplectomorphisms be augmented by the inclusion of canonical relations as morphisms.".
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Documented setting. Strictly speaking, the symplectic category is not a well-defined category (since the composition may not be well-defined) without some transversality conditions.
Clarity¶
A clear use of Symplectic category names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into M \times N^{-} , where the superscript minus means minus the given symplectic form (for example, the.
Manages Complexity¶
Symplectic category compresses multiple symplectic geometry details into a stable diagnostic relation. The source shows both the central mechanism—strictly speaking, the symplectic category is not a well-defined category (since the composition may not be well-defined) without some transversality conditions.—and the practical consequence—in mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into.
Abstract Reasoning¶
- Type the carrier. Identify the symplectic geometry entities to which the claim applies.
- State the relation. Use the source-grounded identity: In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into M \times N^{-} , where the superscript minus means minus the given symplectic form (for example, the graph of a symplectomorphism; hence, minus).
- Check operation and conditions.
Knowledge Transfer¶
Within the home domain. Knowledge about Symplectic category transfers literally when a new case preserves the same carrier type, relation, and recognition test. The notion was introduced by Alan Weinstein, according to whom "Quantization problems suggest that the category of symplectic manifolds and symplectomorphisms be augmented by the inclusion of canonical relations as morphisms." The composition of canonical relations is given by a fiber product. Strictly speaking, the.
Relationships to Other Abstractions¶
Current abstraction Symplectic category Domain-specific
Parents (1) — more general patterns this builds on
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Symplectic category is a kind of, conditional Mathematical Category Domain-specific
Supported under the stated category-like construction while recognizing that canonical relations can create composition and transversality subtleties.
Condition / exception Supported under the stated category-like construction while recognizing that canonical relations can create composition and transversality subtleties.
Hierarchy path (1) — routes to 1 parentless root
- Symplectic category → Mathematical Category → Mathematical structure → Set and Membership
Neighborhood in Abstraction Space¶
Symplectic category sits in a moderately populated region (59th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Algebraic Structures, Groups & Operators (42 abstractions)
Nearest neighbors
- Subfunctor — 0.87
- Hochschild homology — 0.85
- Small category — 0.85
- Canonical Sheaf — 0.85
- Minkowski space (number field) — 0.84
Computed from structural-signature embeddings · 2026-10-08