self-dual
A category is self-dual if it is equivalent to its opposite (or dual) category.
Relevant implications
- binary coproducts and self-dual implies binary products
- binary products and self-dual implies binary coproducts
- cocomplete and self-dual implies complete
- coequalizers and self-dual implies equalizers
- cogenerator and self-dual implies generator
- complete and self-dual implies cocomplete
- connected colimits and self-dual implies connected limits
- connected limits and self-dual implies connected colimits
- coproducts and self-dual implies products
- countable coproducts and self-dual implies countable products
- countable products and self-dual implies countable coproducts
- epi-regular and self-dual implies mono-regular
- equalizers and self-dual implies coequalizers
- filtered colimits and self-dual implies filtered limits
- filtered limits and self-dual implies filtered colimits
- finite coproducts and self-dual implies finite products
- finite products and self-dual implies finite coproducts
- finitely cocomplete and self-dual implies finitely complete
- finitely complete and self-dual implies finitely cocomplete
- generator and self-dual implies cogenerator
- Grothendieck abelian and self-dual implies trivial
- groupoid implies epi-regular and filtered colimits and pushouts and right cancellative and self-dual and well-copowered
- groupoid implies filtered limits and left cancellative and mono-regular and pullbacks and self-dual and well-powered
- initial object and self-dual implies terminal object
- left cancellative and self-dual implies right cancellative
- locally presentable and self-dual implies thin
- mono-regular and self-dual implies epi-regular
- products and self-dual implies coproducts
- pullbacks and self-dual implies pushouts
- pushouts and self-dual implies pullbacks
- right cancellative and self-dual implies left cancellative
- self-dual and sequential colimits implies sequential limits
- self-dual and sequential limits implies sequential colimits
- self-dual and strict initial object implies strict terminal object
- self-dual and strict terminal object implies strict initial object
- self-dual and terminal object implies initial object
- self-dual and well-copowered implies well-powered
- self-dual and well-powered implies well-copowered
- self-dual and wide pullbacks implies wide pushouts
- self-dual and wide pushouts implies wide pullbacks
- trivial implies essentially discrete and essentially finite and finitary algebraic and Grothendieck topos and self-dual and split abelian
Examples
There are 13 categories with this property.
- 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
There are 38 categories without this property.
- 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
There are 0 categories for which the database has no information on whether they satisfy this property.
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