Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$

Depends on

• Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Used by

• For a presheaf P, Nat(C(-,a),P)≅ P(a) naturally in a and P Corollary
• On a preorder the comonads are exactly the monotone contractive maps with Gp below G(Gp); on a poset they are exactly the interior operators Corollary
• The hom-functor turns a coend into an end and carries an end to an end Corollary
• Comonad on a category Definition
• Covariant functor, identity functor, composite functor, and contravariant functor Definition
• Dinatural transformation between functors on CᵒᵖtimesC Definition
• Final and initial functors via nonempty connected comma categories Definition
• Pointwise Kan extensions as those preserved by representables Definition
• Presheaves, covariantly and contravariantly representable functors, and representations Definition
• Set-weighted limits and colimits Definition
• The category of elements of a covariant functor or a presheaf Definition
• The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category Definition
• The reverse and the opposite of a monoidal category Definition
• The tensor product of a presheaf and a covariant set-valued functor Definition
• The twisted arrow category and its projection to CᵒᵖtimesC Definition
• Wedges and cowedges, and the categories they form Definition
• The opposite-group functor is naturally isomorphic to the identity functor by inversion Example
• FALSE: under this page's convention a coend is the colimit of the same twisted-arrow diagram whose limit is the end False statement
• A wedge on a product index category is exactly a family dinatural in each variable separately Lemma
• A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category Proposition
• Composing a dinatural transformation with a natural transformation on either side gives a dinatural transformation Proposition
• Contravariant derived functors are derived on the opposite category Proposition
• The end of a functor made mute in its contravariant variable is the ordinary limit of that functor Proposition
• Orientation and notation conventions in force on this page Remark
• A coend is a colimit weighted by the hom-bifunctor, and an end a limit weighted by it Theorem
• A family into a parametrised end is natural, or dinatural, in the parameter exactly when its composite with the counit is Theorem
• A representable functor carries a weighted limit to the weighted limit of the composed diagram Theorem
• A weighted limit is an end of powers and a weighted colimit a coend of copowers Theorem
• A weighted limit is an ordinary limit over the category of elements of the weight, and a weighted colimit an ordinary colimit over it Theorem
• An end is a limit over the twisted arrow category, and a coend is a colimit over its opposite Theorem
• Dinatural transformations do not compose in general Theorem
• Every theorem about categories has a formal dual obtained by reversing morphisms and composition Theorem
• Fubini: an end over a product index category and the two iterated ends exist together and agree Theorem
• The co-Yoneda isomorphisms: a set-valued functor is a coend against a representable Theorem
• The contravariant power-set functor is monadic Theorem
• The end of the function-set functor on a representable is evaluation Theorem
• The opposite of a preadditive category is preadditive Theorem
• The opposite of an abelian category is abelian Theorem

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)