Adjunction by unit, counit, and the triangle identities

Depends on

• Equivalence, quasi-inverse, and adjoint equivalence of categories
• Whiskering and horizontal composition of natural transformations

Used by

• A right adjoint preserves ends and a left adjoint preserves coends Corollary
• With the objectwise SAFT universal arrows supplied, a continuous Set-valued functor from a chosen-well-powered SAFT category is representable Corollary
• A componentwise family of morphisms need not be a natural transformation and hence need not be a unit Counterexample
• A wrong counit can be natural while both triangle identities fail Counterexample
• Adjoint triple L⊣ M⊣ R Definition
• Adjuncts and transposition under an adjunction Definition
• Global Kan extensions as adjoints to restriction Definition
• Left-closed, right-closed, and biclosed monoidal categories Definition
• Reflective full subcategory and reflector Definition
• The unit inserts basis vectors and the counit evaluates formal linear combinations in the free-vector-space adjunction Example
• The unit inserts generators as one-letter words and the counit evaluates words in the free-group adjunction Example
• A unit and counit determine an adjunction without the triangle identities False statement
• FALSE: A continuous functor on a complete category necessarily has a left adjoint False statement
• The hom-set form of an adjunction needs no size hypothesis False statement
• The unit and counit transpose formulas are mutually inverse Lemma
• An adjoint equivalence is an adjunction whose unit and counit are natural isomorphisms Proposition
• An adjunction induces postcomposition and precomposition adjunctions on legitimate functor categories Proposition
• An adjunction restricts to an equivalence on the subcategories fixed by its unit and counit Proposition
• The unit-counit definition imposes no local-smallness hypothesis Remark
• A dual object in the endofunctor category is an adjoint functor Theorem
• Adjoints are unique up to a unique natural isomorphism compatible with the adjunction data Theorem
• Adjunctions as absolute Kan extensions, with the preserved converse Theorem
• Adjunctions compose with the composite unit and counit formulas Theorem
• An ambient object lies in the essential image of a reflective inclusion exactly when its reflection unit is invertible Theorem
• Duality yields adjunctions of tensoring functors Theorem
• Every adjunction induces a monad on the domain of its left adjoint Theorem
• Natural transformations have mates under a pair of adjunctions Theorem
• Right adjoints preserve every limit that exists Theorem
• The comparison functor to the Eilenberg–Moore category exists and is unique Theorem
• The counit of a reflection is an isomorphism Theorem
• The free–forgetful Eilenberg–Moore adjunction induces the given monad Theorem
• The Kleisli adjunction induces the given monad Theorem
• The Kleisli factorisation functor for an adjunction inducing a monad exists and is unique Theorem
• The unit-counit, hom-set, unit-universal, and counit-universal encodings of an adjunction are equivalent Theorem
• Unit components are initial in comma categories, and counit components are terminal Theorem

Dependency tree · two levels

5 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, 2nd ed., Definition 4.2.5 (standard reference, not scraped)
• Tom Leinster, Basic Category Theory, Theorem 2.2.5 (standard reference, not scraped)