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 $\mathcal{C}$ is co-Malcev if and only if the category of representable functors $\mathcal{C} \to \mathbf{Set}^+$ is Malcev.

• Dual property: Malcev
• Related properties: finitely cocomplete

Relevant implications

• additive andfinitely cocomplete implies co-Malcev
• co-Malcev andpointed implies counital
• co-Malcev andself-dual implies Malcev
• co-Malcev implies finitely cocomplete
• elementary topos implies co-Malcev
• finitely cocomplete andthin implies co-Malcev
• Malcev andself-dual implies co-Malcev

Examples

There are 28 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of combinatorial species
• category of finite abelian groups
• category of finite sets
• category of finitely generated abelian groups
• category of Hausdorff spaces
• category of left modules over a division ring
• category of left modules over a ring
• category of M-sets
• category of metric spaces with continuous maps
• category of pairs of sets
• category of pointed sets
• category of sets
• category of sheaves
• category of simplicial sets
• category of vector spaces
• category of Z-functors
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 36 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 fields
• category of finite orders
• 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 non-expansive maps
• category of metric spaces with ∞ allowed
• category of monoids
• category of non-empty sets
• category of posets
• category of prosets
• category of rings
• category of rngs
• category of schemes
• category of sets and relations
• category of small categories
• category of smooth manifolds
• category of topological spaces
• delooping of a non-trivial finite group
• delooping of an infinite 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
• empty category
• walking fork
• walking parallel pair of morphisms
• walking span

Unknown

There is 1 category for which the database has no information on whether it satisfies this property. Please help us fill in the gaps by contributing to this project.

• category of measurable spaces