Section (category theory)¶
In category theory, a branch of mathematics, a section is a right inverse of some morphism.
Core Idea¶
Section (category theory) is treated here as the recurring crossdomainmodelsstructuresrepresentations identity summarized by this source-grounded definition: In category theory, a branch of mathematics, a section is a right inverse of some morphism. In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f: X\to Y and g: Y\to X are morphisms whose composition f \circ g: Y\to Y is the identity morphism on Y , then g is a section of f , and.
Scope of Application¶
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Terminology. Borsuk's student, Samuel Eilenberg, was with Saunders Mac Lane the founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected this.
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Terminology. In fact, their earlier publications, up to, e.g., Mac Lane (1963)'s Homology, used the term right inverse.
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Examples. In the category of sets, every monomorphism (injective function) with a non-empty domain is a section, and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the.
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Terminology. It was not until 1965 when Eilenberg and John Coleman Moore coined the dual term 'coretraction' that Borsuk's term was lifted to category theory in general.
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Terminology. The term coretraction gave way to the term section by the end of the 1960s.
Clarity¶
A clear use of Section (category theory) names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In category theory, a branch of mathematics, a section is a right inverse of some morphism.
Manages Complexity¶
Section (category theory) compresses multiple crossdomainmodelsstructuresrepresentations details into a stable diagnostic relation. The source shows both the central mechanism—in the category of vector spaces over a field K, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis.—and the practical consequence—it was not until 1965 when Eilenberg and John Coleman Moore coined.
Abstract Reasoning¶
- Type the carrier. Identify the crossdomainmodelsstructuresrepresentations entities to which the claim applies.
- State the relation. Use the source-grounded identity: In category theory, a branch of mathematics, a section is a right inverse of some morphism.
- Check operation and conditions. The concept in topology was defined by Karol Borsuk in 1931.
- Demand recognition evidence. Borsuk's student, Samuel Eilenberg, was with Saunders Mac Lane the founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected this.
Knowledge Transfer¶
Within the home domain. Knowledge about Section (category theory) transfers literally when a new case preserves the same carrier type, relation, and recognition test. Borsuk's student, Samuel Eilenberg, was with Saunders Mac Lane the founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected this term to have initially be used. In fact, their earlier publications, up to, e.g., Mac Lane (1963)'s Homology, used the term right inverse. Beyond the home domain. No canonical parent is asserted for Section (category theory).
Neighborhood in Abstraction Space¶
Section (category theory) sits in a moderately populated region (44th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Category Theory & Homotopical Algebra (18 abstractions)
Nearest neighbors
- Adequate subcategory — 0.88
- Topological Algebra — 0.88
- Filling radius — 0.86
- Social networks — 0.86
- Characterization (mathematics) — 0.86
Computed from structural-signature embeddings · 2026-10-08