DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Subobject classifier

Definition

Let $C$ be a category with a terminal object $1$ (Initial object, terminal object, and zero object) and pullbacks (Pullbacks and pushouts as limits and colimits of cospans and spans).

A subobject classifier is a monomorphism

$true : 1 \to Ω$

(Monomorphism and epimorphism by left and right cancellation) such that for every monomorphism $m : A \to X$ there exists a unique morphism $χ_{m} : X \to Ω$ for which $m$ is, up to isomorphism of pullback objects, the pullback of $true$ along $χ_{m}$ .

The map $χ_{m}$ is the classifying morphism of the subobject represented by $m$ . The uniqueness clause is part of the definition: the classifier classifies subobjects in the sense of Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms, not arbitrary representing monomorphisms.

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

S. Mac Lane, Categories for the Working Mathematician, 2nd ed., IV.9 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Exercise 6.3.26 (standard reference, not scraped)