False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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FALSE: A subobject is a monomorphism rather than an equivalence class of representatives

Statement

False claim. A subobject of an object $C$ is an individual monomorphism into $C$ , rather than an equivalence class of monomorphisms under mutual factorisation.

Facts & Assumptions

Given: The set $X = {0, 1}$ and singleton sets $A = {0}$ and $B = {*}$ .

[L1]

A subobject is an equivalence class of monomorphisms into a fixed object under mutual factorisation (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms).

[L2]

Mutually factoring monomorphisms have unique inverse factor maps and represent the same subobject (Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it).

Refutation

technique · counterexample

1.1

Let $m : A \to X$ send $0$ to $0$ , and let $n : B \to X$ send $*$ to $0$ . Both maps are injective, and an injection $f$ in $S e t$ is monic: if $f \circ g = f \circ h$ then $f (g (x)) = f (h (x))$ for every $x$ , so $g (x) = h (x)$ by injectivity and $g = h$ . They are nonetheless different morphisms, because their domains differ.

given

2.1

The unique bijections $u : A \to B$ and $v : B \to A$ satisfy $m = n \circ u$ and $n = m \circ v$ . Thus $m$ and $n$ mutually factor and [L2] makes their factor maps inverse isomorphisms.

step 1.1L2

3.1

Consequently the two different monomorphisms determine one equivalence class $[m] = [n]$ , which is the subobject prescribed by [L1]. The individual monomorphisms are representatives, not the subobject itself.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, definition 4.7.5 (standard reference, not scraped)