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.

Under the definable-class diagram convention, the empty set is the product of the large family of all sets

Example

Under the library's definable-class diagram convention, for the discrete large diagram in $S e t$ containing every set as a factor, the empty set is a product apex.

Facts & Assumptions

Given: The class-indexed discrete diagram of all sets.

[F1]

For a family indexed by a set, a product cone consists of one map to every factor and is terminal among such cones (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations). The diagram here is indexed by a proper class, which that definition does not cover, so "product" is used below in the extended sense: an apex with one map to every factor, terminal among all such cones over the definable-class diagram. That extension is stipulated here rather than cited, and it is the whole point of the example — the pathology below is a consequence of leaving set-sized indexing, not a statement about any product the definition supplies.

[F2]

A large diagram is one whose indexing category is not small; completeness does not assert that such diagrams have no limits (Finite, small, and large limits and colimits; complete and cocomplete categories).

[F3]

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

Verification

technique · universal property

1.1

There is one empty function $\emptyset \to X$ for every set $X$ , so these functions form a cone with apex $\emptyset$ .

1.2

Any cone with apex $Y$ includes a function from $Y$ to the empty-set factor. Such a function exists only when $Y = \emptyset$ . Hence every cone has empty apex.

2.1

Between any two empty apices there is exactly one function, and all leg equations hold automatically because the functions are empty. Therefore the cone of step 1.1 is terminal among cones and is a product by [F1].

F1F3step 1.1step 1.2

3.1

The indexing family is a proper class, so [F2] classifies the diagram as large. Its having this limit is compatible with completeness being a claim only about all small diagrams.

F2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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, Example 3.7.4 (standard reference, not scraped)