TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

The two-element set is a subobject classifier for Set

Statement

In $S e t$ , the inclusion

$true : 1 = {*} \to 2 = {0, 1}, * \mapsto 1,$

is a subobject classifier.

Facts & Assumptions

Given: A monomorphism $m : A \to X$ in $S e t$ .

[L1]

In $S e t$ , morphisms are functions (Sets and functions form the large locally small category $S e t$ ).

[L2]

Monomorphisms in $S e t$ are exactly injections (In $S e t$ , monomorphisms are exactly injections and epimorphisms are exactly surjections).

[L3]

A subobject classifier is a monomorphism $1 \to Ω$ whose pullbacks classify all subobjects uniquely (Subobject classifier).

Proof

technique · direct

1.1

By [L1] and [L2], $m$ is an injection. Its image $m [A] \subseteq X$ is a subset, and the bijection $A \to m [A]$ identifies $m$ with the inclusion $m [A] ↪ X$ . Define $χ_{m} : X \to 2$ by $χ_{m} (x) = 1$ for $x \in m [A]$ and $χ_{m} (x) = 0$ otherwise.

givenL1L2construct

2.1

The pullback of $true : 1 \to 2$ along $χ_{m}$ has underlying set $χ_{m}^{- 1} (1) = m [A]$ , so its inclusion into $X$ represents the same subobject as $m$ via the bijection from step 1.1.

step 1.1L1algebra

3.1

If $f : X \to 2$ has a pullback representing the same subobject as $m$ , then $f^{- 1} (1) = m [A]$ , so $f$ agrees pointwise with $χ_{m}$ . Hence the classifying map is unique. By [L3], $true : 1 \to 2$ is a subobject classifier.

step 2.1L3algebra∎

Depends on

Used by

The subobject classifier in a presheaf category on the walking arrow Example

Dependency tree · two levels

17 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)