CatDat

zero morphisms

A category has zero morphisms if for every pair of objects $A,B$ there is a distinugished morphism $0_{A,B} : A \to B$, called the zero morphism, such that we have $f \circ 0_{A,B} = 0_{A,C}$ and $0_{B,C} \circ g = 0_{A,C}$ for all morphisms $f : B \to C$ and $g : A \to B$. The zero morphisms are unique if they exist, hence this is actually a property of the category.

• Dual property: zero morphisms (self-dual)
• Related properties: pointed, preadditive
• nLab Link

Relevant implications

• inhabited and  zero morphisms implies   connected
• pointed is equivalent to   initial object and  zero morphisms
• pointed is equivalent to   terminal object and  zero morphisms
• preadditive implies   locally essentially small and  zero morphisms

Examples

There are 15 categories with this property.

• category of abelian groups
• category of Banach spaces with linear contractions
• category of finite abelian groups
• category of finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of left R-modules
• category of monoids
• category of pointed sets
• category of rngs
• category of sets and relations
• category of vector spaces
• empty category
• trivial category
• walking isomorphism

Counterexamples

There are 36 categories without this property.

• category of combinatorial species
• 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 locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of non-empty sets
• category of posets
• category of rings
• category of schemes
• category of sets
• 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
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• walking morphism
• walking parallel pair of morphisms

Unknown

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

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