CatDat

subobject classifier

A category $\mathcal{C}$ has a subobject classifier if it has finite limits and a monomorphism $\top : 1 \to \Omega$ from the terminal object such that for every monomorphism $m : A \to B$ there is a unique morphism $\chi_m : B \to \Omega$ such that $B \leftarrow A \rightarrow 1$ is the pullback of $B \rightarrow \Omega \leftarrow 1$. Equivalently, the functor $\mathrm{Sub} : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}^+$ is representable.

• Related properties: elementary topos
• nLab Link

Relevant implications

• elementary topos is equivalent to   cartesian closed and  finitely complete and  subobject classifier

  by definition
• subobject classifier and  locally essentially small implies   well-powered

  Mac Lane & Moerdijk, Prop. I.3.1
• subobject classifier implies   finitely complete and  mono-regular

  The first part holds by convention, and the second part: any monomorphism $U \to X$ is the equalizer of $$\chi_U,\chi_X : X \to \Omega$.
• additive and  pullbacks and  subobject classifier implies   trivial

  see MSE/4086192

Examples

• category of combinatorial species
• category of finite sets
• category of M-sets
• category of pointed sets
• category of sets
• category of simplicial sets
• trivial category
• walking isomorphism

Counterexamples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of commutative rings
• category of fields
• category of finite abelian groups
• category of finite orders
• 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 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 posets
• category of rings
• category of rngs
• category of sets and relations
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of vector spaces
• 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.

• category of locally ringed spaces
• category of schemes
• category of Z-functors