DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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Natural transformation and its components

Definition

Let $F, G : C \to D$ be functors (Covariant functor, identity functor, composite functor, and contravariant functor). A natural transformation $α : F \Rightarrow G$ is a family of morphisms $α_{A} : F A \to G A$ , one for each object $A$ of $C$ , such that every $f : A \to B$ satisfies the naturality equation

$G f \circ α_{A} = α_{B} \circ F f .$

The morphism $α_{A}$ is the component of $α$ at $A$ .

Depends on

Covariant functor, identity functor, composite functor, and contravariant functor

Used by

Composition of Deligne kernels is balanced tensor product Corollary
Finite Eilenberg–Watts is a biequivalence Corollary
Graded bimodule maps classify shift-compatible transformations Corollary
The center is Morita invariant, via natural endomorphisms of the identity Corollary
A componentwise family between functors need not be a natural transformation Counterexample
A componentwise family of morphisms need not be a natural transformation and hence need not be a unit Counterexample
Unrestricted graded natural transformations are not determined by the regular module Counterexample
A balanced derived bifunctor Definition
Bicategories, pseudofunctors, and biequivalences Definition
Canonical morphisms between parenthesised tensor words Definition
Categories fibred in groupoids over a site Definition
Codensity monad Definition
Coherently shift-compatible functors and natural transformations Definition
Constant diagrams, cones, cocones, and their morphisms Definition
Dinatural transformation between functors on CᵒᵖtimesC Definition
Fppf sheaves of sets and sheafification Definition
Functor category [C,D] Definition
Geometric fibre functor and étale fundamental group Definition
Identity natural transformation and vertical composition Definition
Lax, strong, and strict monoidal functors Definition
Left and right Kan extensions Definition
Left and right Nakayama functors by finite kernel calculus Definition
Local systems and pullback Definition
Monad on a category Definition
Monoidal natural transformation Definition
Morphism of cohomological delta functors Definition
Morphism of homological delta functors Definition
Morphisms between monads on one category Definition
Morphisms of presheaves Definition
Morphisms representable by algebraic spaces Definition
Natural isomorphism Definition
Presheaves, covariantly and contravariantly representable functors, and representations Definition
Representable morphisms of presheaves and fibrewise properties Definition
Set-weighted limits and colimits Definition
Simplicial objects, simplicial commutative rings and homotopy groups Definition
Stable natural cohomology operation Definition
The category of right-module endofunctors Definition
The Deligne product of finite linear categories Definition
Weak linear 2-category Definition
Weak monoidal 2-category Definition

…and 49 more results.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)