self-dual

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

Relevant implications

• accessible andself-dual implies coaccessible
• Barr-coexact andself-dual implies Barr-exact
• Barr-exact andself-dual implies Barr-coexact
• 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
• cartesian closed andself-dual implies cocartesian coclosed
• cartesian filtered colimits andself-dual implies cocartesian cofiltered limits
• CIP andself-dual implies CSP
• co-Malcev andself-dual implies Malcev
• coaccessible andself-dual implies accessible
• cocartesian coclosed andself-dual implies cartesian closed
• cocartesian cofiltered limits andself-dual implies cartesian filtered colimits
• cocomplete andself-dual implies complete
• codistributive andself-dual implies distributive
• coequalizers of kernel pairs andself-dual implies equalizers of cokernel pairs
• coequalizers andself-dual implies equalizers
• coextensive andself-dual implies extensive
• cofiltered limits andself-dual implies filtered colimits
• cofiltered andself-dual implies filtered
• cofiltered-limit-stable epimorphisms andself-dual implies filtered-colimit-stable monomorphisms
• cogenerating collection andself-dual implies generating collection
• cogenerator andself-dual implies generator
• cokernel pairs andself-dual implies kernel pairs
• cokernels andself-dual implies kernels
• complete andself-dual implies cocomplete
• connected colimits andself-dual implies connected limits
• connected limits andself-dual implies connected colimits
• conormal andself-dual implies normal
• copowers andself-dual implies powers
• coproducts andself-dual implies products
• coquotients of cocongruences andself-dual implies quotients of congruences
• coreflexive equalizers andself-dual implies reflexive coequalizers
• coregular andself-dual implies regular
• cosifted limits andself-dual implies sifted colimits
• cosifted andself-dual implies sifted
• cototal andself-dual implies total
• 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
• countably codistributive andself-dual implies countably distributive
• countably coextensive andself-dual implies countably extensive
• countably distributive andself-dual implies countably codistributive
• countably extensive andself-dual implies countably coextensive
• CSP andself-dual implies CIP
• 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
• effective cocongruences andself-dual implies effective congruences
• effective congruences andself-dual implies effective cocongruences
• epi-regular andself-dual implies mono-regular
• equalizers of cokernel pairs andself-dual implies coequalizers of kernel pairs
• equalizers andself-dual implies coequalizers
• exact cofiltered limits andself-dual implies exact filtered colimits
• exact filtered colimits andself-dual implies exact cofiltered limits
• extensive andself-dual implies coextensive
• extremal cogenerating collection andself-dual implies extremal generating collection
• extremal cogenerator andself-dual implies extremal generator
• extremal generating collection andself-dual implies extremal cogenerating collection
• extremal generator andself-dual implies extremal cogenerator
• filtered colimits andself-dual implies cofiltered limits
• filtered andself-dual implies cofiltered
• filtered-colimit-stable monomorphisms andself-dual implies cofiltered-limit-stable epimorphisms
• 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 collection andself-dual implies cogenerating collection
• generator andself-dual implies cogenerator
• groupoid implies self-dual
• groupoid implies self-dual
• 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
• kernel pairs andself-dual implies cokernel pairs
• kernels andself-dual implies cokernels
• left cancellative andself-dual implies right cancellative
• locally cartesian closed andself-dual implies locally cocartesian coclosed
• locally cocartesian coclosed andself-dual implies locally cartesian closed
• locally copresentable andself-dual implies locally presentable
• locally presentable andself-dual implies locally copresentable
• Malcev andself-dual implies co-Malcev
• mono-regular andself-dual implies epi-regular
• multi-cocomplete andself-dual implies multi-complete
• multi-complete andself-dual implies multi-cocomplete
• multi-initial object andself-dual implies multi-terminal object
• multi-terminal object andself-dual implies multi-initial object
• normal andself-dual implies conormal
• powers andself-dual implies copowers
• products andself-dual implies coproducts
• pullbacks andself-dual implies pushouts
• pushouts andself-dual implies pullbacks
• quotient object classifier andself-dual implies subobject classifier
• quotient-trivial andself-dual implies subobject-trivial
• quotients of congruences andself-dual implies coquotients of cocongruences
• reflexive coequalizers andself-dual implies coreflexive equalizers
• regular quotient object classifier andself-dual implies regular subobject classifier
• regular subobject classifier andself-dual implies regular quotient object classifier
• regular andself-dual implies coregular
• regular-quotient-trivial andself-dual implies regular-subobject-trivial
• regular-subobject-trivial andself-dual implies regular-quotient-trivial
• 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 andsifted implies cosifted
• self-dual andstrict initial object implies strict terminal object
• self-dual andstrict terminal object implies strict initial object
• self-dual andsubobject classifier implies quotient object classifier
• self-dual andsubobject-trivial implies quotient-trivial
• self-dual andterminal object implies initial object
• self-dual andtotal implies cototal
• 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
• self-dual andℵ₁-cofiltered limits implies ℵ₁-filtered colimits
• self-dual andℵ₁-cofiltered implies ℵ₁-filtered
• self-dual andℵ₁-filtered colimits implies ℵ₁-cofiltered limits
• self-dual andℵ₁-filtered implies ℵ₁-cofiltered
• self-dual andℵ₂-small copowers implies ℵ₂-small powers
• self-dual andℵ₂-small coproducts implies ℵ₂-small products
• self-dual andℵ₂-small powers implies ℵ₂-small copowers
• self-dual andℵ₂-small products implies ℵ₂-small coproducts
• trivial implies self-dual

Examples

There are 27 categories with this property.

• category of finite abelian groups
• category of finite sets and bijections
• category of finite-dimensional vector spaces
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• category of finitely generated free abelian groups
• category of finitely generated free modules over Z x Z
• category of finitely generated projective modules over the ring of dual numbers
• category of sets and relations
• delooping of a group
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• delooping of the additive monoid of natural numbers
• discrete category of sets
• discrete category on two objects
• empty category
• real interval [0,1]
• trivial category
• walking commutative square
• walking composable pair
• walking idempotent
• walking isomorphism
• walking morphism
• walking parallel pair
• walking splitting

Counterexamples

There are 91 categories without this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of cochain complexes of abelian groups
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of coproducts of Euclidean spaces
• category of countable groups
• category of countable sets
• category of countable-dimensional vector spaces
• category of directed graphs
• category of fields
• category of fields of characteristic zero
• category of filtered vector spaces
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and injections
• category of finite sets and surjections
• category of finite sets of cardinality a power of 3
• category of finite sets of even cardinality
• category of finite sets of odd cardinality
• category of finitely generated abelian groups
• category of free abelian groups
• category of graded abelian groups
• category of graded modules over a graded ring
• category of groups
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of large families of abelian groups
• category of large families of sets
• category of large families of sets which are mostly empty
• category of large families of sets which are mostly singletons
• category of large families of vector spaces which are mostly zero
• category of large vector spaces over a large field with a small basis
• category of left modules over a division ring
• category of left modules over a non-semisimple ring
• category of left modules over a ring
• category of locally ringed spaces
• category of long transfinite sequences of abelian groups
• 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 partially ordered sets
• category of partially ordered sets without isolated points
• category of pointed sets
• category of pointed topological spaces
• category of preordered sets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sequences of abelian groups
• category of set functions and commutative squares
• category of sets
• category of sets with a binary relation
• category of sets with a distinguished subset
• category of sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of vector spaces
• category of Z-functors
• cocompletion of a discrete–pair join
• delooping of the additive monoid of ordinal numbers
• dual of the category of sets
• dual of the category of topological spaces
• forked commutative square
• partially ordered collection of ordinal numbers
• partially ordered set of extended natural numbers
• partially ordered set of natural numbers
• preordered set of integers w.r.t. divisibility
• simplex category
• walking coreflexive pair
• walking fork
• walking span

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

—