Equivalence, quasi-inverse, and adjoint equivalence of categories

Depends on

• Natural isomorphism
• Covariant functor, identity functor, composite functor, and contravariant functor
• Whiskering and horizontal composition of natural transformations

Used by

• Finite Eilenberg–Watts is a biequivalence Corollary
• The center is Morita invariant, via natural endomorphisms of the identity Corollary
• 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
• Bicategories, pseudofunctors, and biequivalences Definition
• Monadic and strictly monadic functors Definition
• Monoidal equivalence and monoidal quasi-inverse data Definition
• The Deligne product of finite linear categories 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 Deligne product of finite vector spaces is finite vector spaces 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
• Derived comparisons give unique normalized homotopy maps Lemma
• Equivalences preserve small projective generators Lemma
• Finite module duality is exact with commuting bimodule actions Lemma
• The opposite Deligne product is the category of finite bimodules Lemma
• 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
• Categorical Eilenberg–Watts equivalences for finite linear categories Theorem
• Closed oriented two-dimensional TQFTs are commutative Frobenius algebras Theorem
• Eilenberg-Watts schematic biequivalence between the Morita bicategory and module categories Theorem
• Every equivalence of categories can be equipped as an adjoint equivalence Theorem
• Finite abelian categories admit finite-dimensional module models Theorem
• Finite Deligne products exist via tensor-product algebras Theorem
• Finite Eilenberg–Watts for right exact linear functors Theorem
• Finite left exact functors are Hom functors with dual bimodule kernels Theorem
• Finite locally free sheaves and geometric vector bundles Theorem
• Graded Eilenberg-Watts theorem with coherent shifts Theorem
• Module reconstruction from a small projective generator with supplied copowers and cokernels Theorem
• Morita equivalence is invertibility of a bimodule Theorem
• The matrix category is fully faithful in modules and, with chosen bases, equivalent to finite free modules Theorem
• Typed colored knowledgeable Frobenius data classify open-closed sewing 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)