ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-13

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.

Pullbacks in Set are fibre products and pushouts are quotients of tagged disjoint unions

Example

For $X \overset{f}{\to} Z \overset{g}{\leftarrow} Y$ , the pullback in $S e t$ is ${(x, y) \in X \times Y : f (x) = g (y)}$ . For $X \overset{a}{\leftarrow} W \overset{b}{\to} Y$ , the pushout is the tagged union $X ⨿ Y$ modulo the least equivalence relation identifying $a (w)$ with $b (w)$ for every $w$ .

Facts & Assumptions

Given: The displayed cospan and span of sets.

[F1]

Pullbacks and pushouts have their compatible-pair universal properties (Pullbacks and pushouts as limits and colimits of cospans and spans).

[F2]

Morphisms in $S e t$ are functions (Sets and functions form the large locally small category $S e t$ ).

[F3]

Equivalence relations are reflexive, symmetric, and transitive, and maps constant on classes factor uniquely through the quotient (Equivalence relation, equivalence class, and the quotient set $A / \sim$ , Let $\sim$ be an equivalence relation on $A$ with quotient map $π : A \to A / \sim$ , and let $f : A \to B$ . There is a function $g : A / \sim \to B$ with $g \circ π = f$ if and only if $a \sim a^{'}$ implies $f (a) = f (a^{'})$ ; and such a $g$ is then unique).

Verification

technique · construction

1.1

The coordinate projections of the displayed subset satisfy the cospan equation. If $r : T \to X$ and $s : T \to Y$ satisfy $f r = g s$ , then $t \mapsto (r (t), s (t))$ is the unique function into the subset with those two projections. This is [F1].

F1F2

1.2

In the tagged union, generate an equivalence relation from $(0, a (w)) \sim (1, b (w))$ . The quotient injections agree on $W$ .

2.1

Compatible maps $r : X \to T$ and $s : Y \to T$ define a function on the tagged union by cases. It is equal on every generating pair and hence constant on classes, so [F3] gives a unique quotient factor. Conversely any quotient map restricts to such a compatible pair. This is the pushout property [F1].

F1F2F3step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

E. Riehl, Category Theory in Context, Examples 3.1.24 and 3.1.25 (standard reference, not scraped)