CatDat

self-dual

A category is self-dual if it is equivalent to its opposite (or dual) category.

• Dual property: self-dual (self-dual)
• nLab Link

Relevant implications

• trivial implies   finitary algebraic and  Grothendieck topos and  split abelian and  self-dual and  essentially discrete and  essentially finite

  trivial
• locally presentable and  self-dual implies   thin

  This follows from Adamek-Rosicky, Thm. 1.64.
• Grothendieck abelian and  self-dual implies   trivial

  This follows since the dual of a non-trivial Grothendieck abelian category cannot be Grothendieck abelian. See Peter Freyd, Abelian categories, p. 116.
• groupoid implies   self-dual and  mono-regular and  pullbacks and  filtered limits and  left cancellative and  well-powered

  easy
• groupoid implies   self-dual and  epi-regular and  pushouts and  filtered colimits and  right cancellative and  well-copowered

  [dualized] easy
• self-dual and  complete implies   cocomplete

  trivial by self-duality
• self-dual and  cocomplete implies   complete

  trivial by self-duality
• self-dual and  finitely complete implies   finitely cocomplete

  trivial by self-duality
• self-dual and  finitely cocomplete implies   finitely complete

  trivial by self-duality
• self-dual and  well-powered implies   well-copowered

  trivial by self-duality
• self-dual and  well-copowered implies   well-powered

  trivial by self-duality
• self-dual and  initial object implies   terminal object

  trivial by self-duality
• self-dual and  terminal object implies   initial object

  trivial by self-duality
• self-dual and  products implies   coproducts

  trivial by self-duality
• self-dual and  coproducts implies   products

  trivial by self-duality
• self-dual and  finite products implies   finite coproducts

  trivial by self-duality
• self-dual and  finite coproducts implies   finite products

  trivial by self-duality
• self-dual and  binary products implies   binary coproducts

  trivial by self-duality
• self-dual and  binary coproducts implies   binary products

  trivial by self-duality
• self-dual and  countable products implies   countable coproducts

  trivial by self-duality
• self-dual and  countable coproducts implies   countable products

  trivial by self-duality
• self-dual and  equalizers implies   coequalizers

  trivial by self-duality
• self-dual and  coequalizers implies   equalizers

  trivial by self-duality
• self-dual and  filtered limits implies   filtered colimits

  trivial by self-duality
• self-dual and  filtered colimits implies   filtered limits

  trivial by self-duality
• self-dual and  sequential limits implies   sequential colimits

  trivial by self-duality
• self-dual and  sequential colimits implies   sequential limits

  trivial by self-duality
• self-dual and  connected limits implies   connected colimits

  trivial by self-duality
• self-dual and  connected colimits implies   connected limits

  trivial by self-duality
• self-dual and  strict initial object implies   strict terminal object

  trivial by self-duality
• self-dual and  strict terminal object implies   strict initial object

  trivial by self-duality
• self-dual and  pullbacks implies   pushouts

  trivial by self-duality
• self-dual and  pushouts implies   pullbacks

  trivial by self-duality
• self-dual and  generator implies   cogenerator

  trivial by self-duality
• self-dual and  cogenerator implies   generator

  trivial by self-duality
• self-dual and  left cancellative implies   right cancellative

  trivial by self-duality
• self-dual and  right cancellative implies   left cancellative

  trivial by self-duality
• self-dual and  wide pullbacks implies   wide pushouts

  trivial by self-duality
• self-dual and  wide pushouts implies   wide pullbacks

  trivial by self-duality
• self-dual and  mono-regular implies   epi-regular

  trivial by self-duality
• self-dual and  epi-regular implies   mono-regular

  trivial by self-duality

Examples

• category of finite abelian groups
• category of finite sets and bijections
• category of sets and relations
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• discrete category on two objects
• empty category
• partial order [0,1]
• trivial category
• walking isomorphism
• walking morphism
• walking parallel pair of morphisms

Counterexamples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• category of fields
• category of finite orders
• category of finite sets
• category of finite sets and injections
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of left R-modules
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of vector spaces
• category of Z-functors
• delooping of the additive monoid of ordinal numbers
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity

Unknown

For these categories the database has no info if they satisfy this property or not.

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