self-dual
A category is self-dual if it is equivalent to its opposite (or dual) category.
Relevant implications
- binary copowers andself-dual implies binary powers
- binary coproducts andself-dual implies binary products
- binary powers andself-dual implies binary copowers
- binary products andself-dual implies binary coproducts
- co-Malcev andself-dual implies Malcev
- cocomplete andself-dual implies complete
- codistributive andself-dual implies distributive
- coequalizers andself-dual implies equalizers
- coextensive andself-dual implies extensive
- cofiltered limits andself-dual implies filtered colimits
- cogenerating set andself-dual implies generating set
- cogenerator andself-dual implies generator
- complete andself-dual implies cocomplete
- connected colimits andself-dual implies connected limits
- connected limits andself-dual implies connected colimits
- copowers andself-dual implies powers
- coproducts andself-dual implies products
- coreflexive equalizers andself-dual implies reflexive coequalizers
- coregular andself-dual implies regular
- cosifted limits andself-dual implies sifted colimits
- counital andself-dual implies unital
- countable copowers andself-dual implies countable powers
- countable coproducts andself-dual implies countable products
- countable powers andself-dual implies countable copowers
- countable products andself-dual implies countable coproducts
- directed colimits andself-dual implies directed limits
- directed limits andself-dual implies directed colimits
- disjoint coproducts andself-dual implies disjoint products
- disjoint finite coproducts andself-dual implies disjoint finite products
- disjoint finite products andself-dual implies disjoint finite coproducts
- disjoint products andself-dual implies disjoint coproducts
- distributive andself-dual implies codistributive
- epi-regular andself-dual implies mono-regular
- equalizers andself-dual implies coequalizers
- extensive andself-dual implies coextensive
- filtered colimits andself-dual implies cofiltered limits
- finite copowers andself-dual implies finite powers
- finite coproducts andself-dual implies finite products
- finite powers andself-dual implies finite copowers
- finite products andself-dual implies finite coproducts
- finitely cocomplete andself-dual implies finitely complete
- finitely complete andself-dual implies finitely cocomplete
- generating set andself-dual implies cogenerating set
- generator andself-dual implies cogenerator
- Grothendieck abelian andself-dual implies trivial
- groupoid implies directed colimits andepi-regular andpushouts andright cancellative andself-dual andwell-copowered
- groupoid implies directed limits andleft cancellative andmono-regular andpullbacks andself-dual andwell-powered
- infinitary codistributive andself-dual implies infinitary distributive
- infinitary coextensive andself-dual implies infinitary extensive
- infinitary distributive andself-dual implies infinitary codistributive
- infinitary extensive andself-dual implies infinitary coextensive
- initial object andself-dual implies terminal object
- left cancellative andself-dual implies right cancellative
- locally presentable andself-dual implies thin
- Malcev andself-dual implies co-Malcev
- mono-regular andself-dual implies epi-regular
- powers andself-dual implies copowers
- products andself-dual implies coproducts
- pullbacks andself-dual implies pushouts
- pushouts andself-dual implies pullbacks
- reflexive coequalizers andself-dual implies coreflexive equalizers
- regular andself-dual implies coregular
- right cancellative andself-dual implies left cancellative
- self-dual andsequential colimits implies sequential limits
- self-dual andsequential limits implies sequential colimits
- self-dual andsifted colimits implies cosifted limits
- self-dual andstrict initial object implies strict terminal object
- self-dual andstrict terminal object implies strict initial object
- self-dual andterminal object implies initial object
- self-dual andunital implies counital
- self-dual andwell-copowered implies well-powered
- self-dual andwell-powered implies well-copowered
- self-dual andwide pullbacks implies wide pushouts
- self-dual andwide pushouts implies wide pullbacks
- trivial implies essentially discrete andessentially finite andfinitary algebraic andGrothendieck topos andself-dual andsplit abelian
Examples
There are 16 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
- poset [0,1]
- trivial category
- walking commutative square
- walking composable pair
- walking idempotent
- walking isomorphism
- walking morphism
- walking parallel pair
Counterexamples
There are 51 categories without this property.
- category of abelian groups
- category of abelian sheaves
- category of algebras
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of commutative algebras
- category of commutative monoids
- 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 Hausdorff spaces
- category of left modules over a division ring
- category of left modules over a ring
- category of locally ringed spaces
- category of M-sets
- category of measurable spaces
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of metric spaces with ∞ allowed
- category of monoids
- category of non-empty sets
- category of pairs of sets
- category of pointed sets
- category of pointed topological spaces
- category of posets
- category of prosets
- category of rings
- category of rngs
- category of schemes
- category of sets
- category of sheaves
- 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
- dual of the category of sets
- poset of extended natural numbers
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- walking fork
- walking span
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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