False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01

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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.
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FALSE: a subobject classifier is any object representing monomorphisms

Statement

A subobject classifier is any object representing monomorphisms.

Facts & Assumptions

Given: The set $X = {0, 1}$ and the two singleton inclusions $m : {a} \to X$ , $n : {b} \to X$ with $m (a) = 0 = n (b)$ .

[L1]

A subobject classifier classifies subobjects, meaning mutual-factorization classes of monomorphisms, and the classifying map is unique for that class (Subobject classifier, Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms).

[L2]

Two monomorphisms representing the same subobject class need only be isomorphic, not literally equal as arrows (Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it).

Refutation

technique · direct

1.1

The monomorphisms $m$ and $n$ have different domains and are different arrows, but they factor through each other via the unique bijection ${a} ≅ {b}$ . So [L2] says that they represent the same subobject of $X$ .

givenL2

2.1

Any classifier must assign the same characteristic map to the subobject class represented by $m$ and $n$ , because [L1] classifies subobjects rather than literal presenting monomorphisms. Therefore no object representing monomorphisms as bare arrows can be the right notion.

step 1.1L1

3.1

So the statement is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Tom Leinster, Basic Category Theory, Exercise 6.3.26 (standard reference, not scraped)