Supermanifold¶
In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables.
Core Idea¶
Supermanifold is treated here as the recurring formal models and representations identity summarized by this source-grounded definition: In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables. In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables. In the standard mathematical formulation, a smooth supermanifold is a locally ringed space whose structure sheaf is locally isomorphic to the tensor product of the ring of ordinary smooth functions C^\infty(\mathbb.
Scope of Application¶
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Definition. A more concrete coordinate-based formalism, used especially in parts of the physics literature, describes supermanifolds by charts with values in the even and odd parts of a Grassmann algebra.
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Definition. Variants of this definition are used in the smooth, complex-analytic, and algebraic categories.
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Definition. In the terminology used by DeWitt and Rogers, the even elements are called c-numbers and the odd elements a-numbers.
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Definition. As in the case of an ordinary manifold, a supermanifold is then described by an atlas of charts with suitably smooth transition functions.
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Definition. With suitable restrictions on transition functions, the two approaches are equivalent; more generally, the comparison is best understood using the functor of points.
Clarity¶
A clear use of Supermanifold names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables.
Manages Complexity¶
Supermanifold compresses multiple formal models and representations details into a stable diagnostic relation. The source shows both the central mechanism—the relationship between this formalism and the locally ringed-space definition requires some care, and is often clarified using the functor of points.—and the practical consequence—in the terminology used by DeWitt and Rogers, the even elements are called c-numbers and the odd elements a-numbers.
Abstract Reasoning¶
- Type the carrier. Identify the formal models and representations entities to which the claim applies.
- State the relation. Use the source-grounded identity: In mathematics and mathematical physics, supermanifolds are generalizations of manifolds in which the algebra of functions includes both commuting and anticommuting variables.
- Check operation and conditions. The underlying ordinary manifold |M| , sometimes called the reduced manifold or body, is obtained by quotienting \mathcal OM by the sheaf of ideals generated by its odd elements.
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Supermanifold transfers literally when a new case preserves the same carrier type, relation, and recognition test. A more concrete coordinate-based formalism, used especially in parts of the physics literature, describes supermanifolds by charts with values in the even and odd parts of a Grassmann algebra. Variants of this definition are used in the smooth, complex-analytic, and algebraic categories. Beyond the home domain. No canonical parent is asserted for Supermanifold.
Neighborhood in Abstraction Space¶
Supermanifold sits in a moderately populated region (47th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Algebraic Structures, Groups & Operators (42 abstractions)
Nearest neighbors
- Character variety — 0.87
- Steenrod problem — 0.86
- Stable manifold theorem — 0.86
- A∞-operad — 0.86
- Hochschild homology — 0.86
Computed from structural-signature embeddings · 2026-10-08