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.
The Yoneda lemma requires its category to be small
Statement
False claim: the Yoneda lemma can be stated and proved only when its category is small.
Facts & Assumptions
Given: The false claim above and the category .
For every locally small category , object , and functor , evaluation at the identity is a bijection , with an explicit inverse; smallness is not a hypothesis (Evaluation at the identity gives and proves that the natural-transformation collection is a set).
The same bijection is natural in both and for every locally small , without forming a functor category on a large source (The Yoneda bijection is natural in both and ).
The category is large and locally small (Sets and functions form the large locally small category ).
A category is small when its objects and morphisms form sets, locally small when each hom-collection is a set, and large when it is not small (Small, locally small, and large categories).
Refutation
By [L3] and [F1], is a locally small category that is not small.
Apply [L1] and [L2] to , any set , and any functor . The Yoneda bijection exists and has both naturalities despite the category being large.
Thus local smallness, which makes each hom-collection a set, suffices for the pointwise Yoneda lemma; the large locally small category refutes the asserted need for smallness.
Depends on
- Evaluation at the identity gives $\operatorname{Nat}(\mathcal C(a,-),F)\cong F(a)$ and proves that the natural-transformation collection is a set
- The Yoneda bijection $\operatorname{Nat}(\mathcal C(a,-),F)\cong F(a)$ is natural in both $a$ and $F$
- Sets and functions form the large locally small category $\mathbf{Set}$
- Small, locally small, and large categories
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
15 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, Theorem 2.2.4 and Remark 2.2.7 (standard reference, not scraped)