CatDat

trivial

A category is trivial if it is equivalent to the trivial category. Equivalently, there is an initial object $0$ such that for every object $A$ the unique morphism $0 \to A$ is an isomorphism. Notice that we do not demand that the category is isomorphic to the trivial category. As a consequence, every inhabited indiscrete category is trivial in our sense.

• Dual property: trivial (self-dual)
• nLab Link

Relevant implications

• additive andregular subobject classifier implies trivial
• binary coproducts andgroupoid andinhabited implies trivial
• binary products andgroupoid andinhabited implies trivial
• cartesian closed andpointed implies trivial
• connected andessentially discrete implies trivial
• disjoint finite coproducts andthin implies trivial
• disjoint finite products andthin implies trivial
• Grothendieck abelian andself-dual implies trivial
• groupoid andinhabited andzero morphisms implies trivial
• groupoid andinitial object implies trivial
• groupoid andterminal object implies trivial
• pointed andstrict initial object implies trivial
• pointed andstrict terminal object implies trivial
• subobject classifier andthin implies trivial
• trivial implies essentially discrete andessentially finite andfinitary algebraic andGrothendieck topos andself-dual andsplit abelian

Examples

There are 2 categories with this property.

• trivial category
• walking isomorphism

Counterexamples

There are 63 categories without this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of fields
• category of finite abelian groups
• 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 finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of Hausdorff spaces
• category of left modules over a division ring
• category of left modules over a ring
• 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 pointed 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 vector 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 are 0 categories for which the database has no information on whether they satisfy this property.

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