CatDat

subobject classifier

A category $\C$ has a subobject classifier if it has finite limits and a monomorphism* $\top : 1 \hookrightarrow \Omega$ such that for every monomorphism $m : A \hookrightarrow B$ there is a unique morphism $\chi_m : B \to \Omega$ such that $$\begin{CD} A @>{m}>> B \\ @V{!}VV @VV{\chi_m}V \\ 1 @>>{\top}> \Omega \end{CD}$$ is a pullback diagram. Equivalently, the functor $\Sub : \C^{\op} \to \Set^+$ is representable. *Every morphism $1 \to \Omega$ is a split monomorphism anyway.

• Dual property: quotient object classifier
• Related properties: elementary topos, finitely complete, regular subobject classifier
• nLab Link

Relevant implications

• elementary topos is equivalent to cartesian closed andfinitely complete andsubobject classifier
• locally essentially small andsubobject classifier implies well-powered
• Malcev andsubobject classifier implies subobject-trivial
• mono-regular andregular subobject classifier implies subobject classifier
• pointed andsubobject classifier implies normal
• quotient object classifier andself-dual implies subobject classifier
• self-dual andsubobject classifier implies quotient object classifier
• subobject classifier implies finitely complete andmono-regular
• subobject classifier implies regular subobject classifier

Examples

There are 12 categories with this property.

• category of combinatorial species
• category of countable sets
• category of finite sets
• category of Jónsson-Tarski algebras
• category of M-sets
• category of pairs of sets
• category of pointed sets
• category of sets
• category of sheaves
• category of simplicial sets
• trivial category
• walking isomorphism

Counterexamples

There are 67 categories without this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of countable groups
• category of fields
• category of finite abelian groups
• category of finite groups
• category of finite ordered 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 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 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 and relations
• category of sets with finite-to-one maps
• 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
• 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 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
• 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 is 1 category for which the database has no information on whether it satisfies this property. Please help us fill in the gaps by contributing to this project.

• category of Z-functors