ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

A slice of Set computed as a locally cartesian closed category

Example

Let $f : X \to Y$ be the function with $X = {a, b, c}$ , $Y = {0, 1}$ , and

$f (a) = 0, f (b) = 0, f (c) = 1.$

Let $p : E \to X$ have fibers $E_{a} = {u, v}$ , $E_{b} = {w}$ , and $E_{c} = {z}$ .

Facts & Assumptions

Given: The displayed finite sets and maps over $X$ and $Y$ .

[L1]

$S e t$ is locally cartesian closed, so each pullback functor has a right adjoint $Π_{f}$ (Set is locally cartesian closed).

[L2]

Slice objects and pullback functors are defined as in Slice categories, composition, and pullback along a morphism.

Verification

technique · direct

1.1

The dependent product $Π_{f} E \to Y$ has fiber over $0$ equal to $E_{a} \times E_{b} = {(u, w), (v, w)}$ and fiber over $1$ equal to $E_{c} = {z}$ . So $Π_{f} E$ has three elements in total, two over $0$ and one over $1$ .

givenalgebra

2.1

If $q : B \to Y$ is the singleton fiber over $1$ , then $f^{*} q \to X$ is the singleton fiber over $c$ , and a map $f^{*} q \to p$ over $X$ is exactly a choice of an element of $E_{c}$ , namely $z$ . This matches maps $q \to Π_{f} E$ over $Y$ , which must choose the unique element over $1$ .

step 1.1algebra

3.1

This concrete fiber computation realizes the pullback/right-adjoint pattern guaranteed by Set is locally cartesian closed, using the slice formalism of Slice categories, composition, and pullback along a morphism.

step 2.1L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Emily Riehl, Category Theory in Context, 2nd ed., Section 4.6 (standard reference, not scraped)