CatDat

co-Malcev

A category is co-Malcev when its dual is Malcev, i.e., it has finite colimits and if $X \sqcup X \twoheadrightarrow R$ is a coreflexive corelation, then it is cosymmetric and cotransitive.
This terminology is not standard, but we have added it to properly formulate the interesting theorem that the dual of an elementary topos is Malcev, i.e., that every elementary topos is co-Malcev.
To settle this property, we often use that $\C$ is co-Malcev if and only if the category of representable functors $\C \to \Set^+$ is Malcev.

• Dual property: Malcev
• Related properties: finitely cocomplete

Relevant implications

• additive andfinitely cocomplete implies co-Malcev
• co-Malcev andpointed implies counital
• co-Malcev andregular quotient object classifier implies regular-quotient-trivial
• co-Malcev andself-dual implies Malcev
• co-Malcev implies finitely cocomplete
• epi-regular andextensive andregular implies co-Malcev
• finitely cocomplete andthin implies co-Malcev
• Malcev andself-dual implies co-Malcev

Examples

There are 46 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of cochain complexes of abelian groups
• category of combinatorial species
• category of compact Hausdorff spaces
• category of countable sets
• category of filtered vector spaces
• category of finite abelian groups
• category of finite sets
• 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 abelian groups
• category of graded abelian groups
• category of graded modules over a graded ring
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• 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 M-sets
• category of measurable spaces
• category of pairs of sets
• category of pointed sets
• category of sequences of abelian groups
• category of set functions and commutative squares
• category of sets
• category of sets with a distinguished subset
• category of sheaves
• category of simplicial sets
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of vector spaces
• category of Z-functors
• 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
• real interval [0,1]
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 51 categories without this property.

• category of algebras
• category of Banach spaces with linear contractions
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of countable groups
• category of fields
• category of finite groups
• category of finite ordered sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of free abelian groups
• category of groups
• category of locally ringed 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 partially ordered 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 sets and relations
• category of sets with finite-to-one maps
• category of small categories
• category of smooth manifolds
• category of topological spaces
• 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
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• dual of the category of sets
• dual of the category of topological spaces
• empty category
• forked commutative square
• simplex category
• walking coreflexive pair
• walking fork
• walking idempotent
• walking parallel pair
• walking span
• walking splitting

Unknown

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

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