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: the free-cocompletion theorem holds for an arbitrary large locally small source category with no change in meaning
Statement refuted
That the statement of The presheaf category on a small category is the free cocompletion remains true with no change in meaning when the source category is merely locally small and may be large.
Under this library's formation rules the smallness hypothesis is not cosmetic: without it, the presheaf category is not a category object on disk, and the Yoneda functor is not a functor into one.
Facts & Assumptions
Given: The large locally small category .
The free-cocompletion theorem is stated for a small source category (The presheaf category on a small category is the free cocompletion).
A functor category is formed only when the source is small; for an arbitrary large locally small , the notation is only metatheoretic shorthand (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding, If is small and is locally small then is locally small; if both are small it is small).
A category may be locally small without being small (Small, locally small, and large categories).
Refutation
The category is locally small and large, so it satisfies the weakened hypothesis of the false claim by [F2].
But [F1] says that for such a large source the notation is not formed as a category in this library, and [L1] is a theorem about that presheaf category and the Yoneda functor landing in it. So the unchanged large-source sentence is not even a legal instance of the theorem on disk.
Therefore the false claim fails under the house schema: the smallness hypothesis in [L1] is mathematically active here, not removable decoration.
Depends on
- The presheaf category on a small category is the free cocompletion
- If $\mathcal C$ is small and $\mathcal D$ is locally small then $[\mathcal C,\mathcal D]$ is locally small; if both are small it is small
- Small, locally small, and large categories
- The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding
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, 2nd ed., Theorem 6.5.11 and surrounding discussion (standard reference, not scraped)