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

• essentially discrete and  connected implies   trivial

  trivial
• trivial implies   finitary algebraic and  Grothendieck topos and  split abelian and  self-dual and  essentially discrete and  essentially finite

  trivial
• pointed and  cartesian closed implies   trivial

  We have $X \cong X \times 1 \cong X \times 0 \cong 0$ for every object $X$.
• disjoint finite coproducts and  thin implies   trivial

  For every object $A$ the two inclusions $A \rightrightarrows A + A$ must be equal, so their equalizer is $A$, but also $0$ since the coproduct is disjoint. Hence $A = 0$.
• strict initial object and  pointed implies   trivial

  If $0$ is the zero object, then for every object $A$ the unique morphism $A \to 0$ is an isomorphism by assumption.
• Grothendieck abelian and  self-dual implies   trivial

  This follows since the dual of a non-trivial Grothendieck abelian category cannot be Grothendieck abelian. See Peter Freyd, Abelian categories, p. 116.
• additive and  pullbacks and  subobject classifier implies   trivial

  see MSE/4086192
• groupoid and  binary products and  inhabited implies   trivial

  Let $\mathcal{C}$ be an inhabited groupoid with binary products. Then it is connected, so we may assume $\mathcal{C}=BG$ for a group $G$ with unique object $*$. But then $* \times * = *$, so there are $p,q \in G$ such that $G \to G \times G$, $x \mapsto (px,qx)$ is bijective. From here it is an easy exercise to deduce $G=1$.
• groupoid and  initial object implies   trivial

  easy
• strict terminal object and  pointed implies   trivial

  [dualized] If $0$ is the zero object, then for every object $A$ the unique morphism $A \to 0$ is an isomorphism by assumption.
• groupoid and  binary coproducts and  inhabited implies   trivial

  [dualized] Let $\mathcal{C}$ be an inhabited groupoid with binary products. Then it is connected, so we may assume $\mathcal{C}=BG$ for a group $G$ with unique object $*$. But then $* \times * = *$, so there are $p,q \in G$ such that $G \to G \times G$, $x \mapsto (px,qx)$ is bijective. From here it is an easy exercise to deduce $G=1$.
• groupoid and  terminal object implies   trivial

  [dualized] easy

Examples

• trivial category
• walking isomorphism

Counterexamples

• 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

For these categories the database has no info if they satisfy this property or not.

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