Natural transformation and its components

Depends on

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

Used by

• 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
• A balanced derived bifunctor Definition
• Canonical morphisms between parenthesised tensor words Definition
• Codensity monad Definition
• Constant diagrams, cones, cocones, and their morphisms Definition
• Dinatural transformation between functors on CᵒᵖtimesC Definition
• Functor category [C,D] Definition
• Identity natural transformation and vertical composition Definition
• Lax, strong, and strict monoidal functors Definition
• Left and right Kan extensions 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
• Natural isomorphism Definition
• Presheaves, covariantly and contravariantly representable functors, and representations Definition
• Set-weighted limits and colimits Definition
• Stable natural cohomology operation Definition
• The category of right-module endofunctors Definition
• Whiskering and horizontal composition of natural transformations Definition
• A weighted limit computing a kernel pair Example
• For a monoid action, Yoneda says that an equivariant map from the regular action is determined by the identity element Example
• For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
• Singletons define a natural transformation from the identity functor on sets to the covariant power-set functor Example
• The end formula checked by hand against natural transformations on the walking arrow Example
• A natural transformation is determined by its component at one object False statement
• FALSE: every weighted limit is the ordinary limit of the diagram it weights False statement
• A partial morphism of delta functors extends through one dimension shift Lemma
• Evaluation at the identity gives Nat(C(a,-),F)≅ F(a) and proves that the natural-transformation collection is a set Lemma
• A natural transformation induces natural transformations of left derived functors Proposition
• A natural transformation induces natural transformations of right derived functors Proposition
• A weighted limit of a set-valued diagram is the set of natural transformations from the weight Proposition
• Composing a dinatural transformation with a natural transformation on either side gives a dinatural transformation Proposition
• A family into a parametrised end is natural, or dinatural, in the parameter exactly when its composite with the counit is Theorem
• A monad morphism induces restriction of algebras and a natural comparison of free algebras Theorem
• A natural transformation of functors induces a unique morphism of their ends and of their coends Theorem

…and 8 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)