CatDat

quotient object classifier

A category $\C$ has a quotient object classifier if its dual has a subobject classifier. This means that it has finite colimits and an epimorphism* $\top : \Psi \twoheadrightarrow 0$ such that for every epimorphism $e : A \twoheadrightarrow B$ there is a unique morphism $\psi_e : \Psi \to A$ such that $$\begin{CD} \Psi @>{\top}>> 0 \\ @V{\psi_e}VV @VV{!}V \\ A @>>{e}> B \end{CD}$$ is a pushout diagram. Equivalently, the functor $\Quot : \C \to \Set^+$ is representable. *Every morphism $\Psi \to 0$ is a split epimorphism anyway.

• Dual property: subobject classifier
• Related properties: finitely cocomplete, regular quotient object classifier

Relevant implications

• co-Malcev andquotient object classifier implies quotient-trivial
• epi-regular andregular quotient object classifier implies quotient object classifier
• locally essentially small andquotient object classifier implies well-copowered
• pointed andquotient object classifier implies conormal
• quotient object classifier andself-dual implies subobject classifier
• quotient object classifier implies epi-regular andfinitely cocomplete
• quotient object classifier implies regular quotient object classifier
• self-dual andsubobject classifier implies quotient object classifier

Examples

There are 3 categories with this property.

• dual of the category of sets
• trivial category
• walking isomorphism

Counterexamples

There are 78 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 compact Hausdorff spaces
• category of countable groups
• category of countable sets
• category of fields
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• 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 Jónsson-Tarski algebras
• 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 pointed topological spaces
• category of posets
• category of prosets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sets
• category of sets and relations
• category of sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of vector spaces
• category of Z-functors
• delooping of a non-trivial finite group
• delooping of an infinite countable 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 topological spaces
• 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
• simplex category
• walking commutative square
• walking composable pair
• walking coreflexive pair
• walking fork
• walking idempotent
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Unknown

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

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