Alphabeta MathHow 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.
FALSE: every category has all small limits
Statement refuted
Every category has all small limits.
Facts & Assumptions
Given: The category of nonempty sets and all functions.
A category is complete when every small diagram in it has a limit (Finite, small, and large limits and colimits; complete and cocomplete categories).
An equalizer of must receive every map on which and agree (Equalizers and coequalizers as limits and colimits of a parallel pair).
Refutation
Let be constant at and . For any nonempty set , the unique function has composites constant at different values, so it does not equalize .
Thus this parallel-pair diagram has no cone and in particular no equalizer [F2]. Its indexing category is finite and hence small.
By [F1], the category of nonempty sets is not complete. This category refutes the universal statement.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
5 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, Definition 3.2.1 (standard reference, not scraped)