Alphabeta MathHow statement and proof provenance work
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Equivalence, quasi-inverse, and adjoint equivalence of categories
Definition
An equivalence of categories from to consists of functors and , called quasi-inverses, together with natural isomorphisms
The categories are then called equivalent. The notions of functor and natural isomorphism are those of Covariant functor, identity functor, composite functor, and contravariant functor and Natural isomorphism.
An adjoint equivalence is such data satisfying the triangle identities
Here , , , and are the whiskerings of Whiskering and horizontal composition of natural transformations. No triangle identity is required of a bare equivalence.
Depends on
Used by
- The Kleisli and Eilenberg–Moore categories of an idempotent monad are equivalent Corollary
- A two-object indiscrete preorder is equivalent but not isomorphic to its one-object poset reflection Counterexample
- Adjunction by unit, counit, and the triangle identities Definition
- Monadic and strictly monadic functors Definition
- Monoidal equivalence and monoidal quasi-inverse data Definition
- Chosen bases exhibit Mat_F as equivalent to finite-dimensional vector spaces Example
- Pointed sets are equivalent to sets and partial functions but not isomorphic as categories Example
- The Kleisli adjunction for the maybe monad is monadic but not strictly monadic Example
- Every equivalence of categories is an isomorphism of categories False statement
- FALSE: ordinary Beck creation characterizes strict monadicity False statement
- FALSE: The Kleisli and Eilenberg–Moore categories are equivalent for every monad False statement
- An adjoint equivalence is an adjunction whose unit and counit are natural isomorphisms Proposition
- An adjunction restricts to an equivalence on the subcategories fixed by its unit and counit Proposition
- Equivalence of categories is reflexive, symmetric, and transitive Proposition
- Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense Proposition
- A functor is an equivalence exactly when it is fully faithful and split essentially surjective, without Choice Theorem
- An equivalence between abelian categories is exact Theorem
- Every equivalence of categories can be equipped as an adjoint equivalence Theorem
- The matrix category is fully faithful in modules and, with chosen bases, equivalent to finite free modules Theorem
Dependency tree · two levels
9 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)
- Ahrens, Kapulkin and Shulman, Univalent categories and the Rezk completion, section 6 (standard reference, not scraped)