CatDat

counital

A category is counital if its dual is unital, i.e., it has a zero object, finite colimits, and for all objects $X,Y$ the two morphisms $(\mathrm{id}_X;0) : X \sqcup Y \twoheadrightarrow X$ and $(0;\mathrm{id}_Y) : X \sqcup Y \twoheadrightarrow Y$ are jointly strongly monomorphic. When products exist, the canonical morphism $X \sqcup Y \to X \times Y$ therefore must be a strong monomorphism.

• Dual property: unital
• Related properties: co-Malcev, finitely cocomplete, pointed

Relevant implications

• biproducts andfinitely cocomplete andpointed implies counital
• co-Malcev andpointed implies counital
• counital andself-dual implies unital
• counital implies finitely cocomplete andpointed
• self-dual andunital implies counital

Examples

There are 11 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of commutative monoids
• category of finite abelian groups
• category of finitely generated abelian groups
• category of left modules over a division ring
• category of left modules over a ring
• category of pointed sets
• category of vector spaces
• trivial category
• walking isomorphism

Counterexamples

There are 53 categories without this property.

• category of algebras
• category of combinatorial species
• category of commutative algebras
• category of commutative rings
• category of fields
• category of finite orders
• category of finite 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 Hausdorff spaces
• 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 posets
• category of prosets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets and relations
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of Z-functors
• 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
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• walking commutative square
• walking composable pair
• walking fork
• walking morphism
• 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 Banach spaces with linear contractions