CatDat

trivial

A category is trivial if it is equivalent to the trivial category (with just one object and just one morphism). 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 and  pullbacks and  subobject classifier implies   trivial
• binary coproducts and  groupoid and  inhabited implies   trivial
• binary products and  groupoid and  inhabited implies   trivial
• cartesian closed and  pointed implies   trivial
• connected and  essentially discrete implies   trivial
• disjoint finite coproducts and  thin implies   trivial
• Grothendieck abelian and  self-dual implies   trivial
• groupoid and  initial object implies   trivial
• groupoid and  terminal object implies   trivial
• pointed and  strict initial object implies   trivial
• pointed and  strict terminal object implies   trivial
• trivial implies   essentially discrete and  essentially finite and  finitary algebraic and  Grothendieck topos and  self-dual and  split abelian

Examples

There are 2 categories with this property.

• trivial category
• walking isomorphism

Counterexamples

There are 49 categories without this property.

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• 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 left R-modules
• 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 monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets and relations
• 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
• empty category
• 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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