nucleation

LIT-221: Yoneda lemma

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Properties6
StatusActive — read in full 2026-09-26 (NOTE-195); Worth reading as a compact, correct statement and proof of the lemma with the embedding and the iso-determination corollary. It also has a "Necessity of naturality" section, which textbooks rarely include and which any structuralist reading of the lemma needs. Two caveats: its classical statement does not state naturality in c and X (the HoTT section does), and its finite counterexample is ill-formed as written.
Version2
Tagsmathematics, individuation
SupportsCLAIM-142 — Two communicative realizations can agree on every probe in a restricted family without being structurally identical, so observational equivalence is relative to the probes, and only independently justified probes evaluated on held-out cases are evidence that structure was preserved, CLAIM-264 — Two objects can agree on every probe in a restricted family without being isomorphic, so equivalence under a probe family is coarser than identity unless the family is rich enough
Cited as “paper” byNOTE-195 — Yoneda lemma
Cited as “source” byTHEORY-032 — The Yoneda lemma determines an object only up to isomorphism, from its whole hom-functor, among objects the category already has; as a formal version of "an object is determined by its relations" it presupposes the relata, THEORY-038 — Category theory gives radical ontic structural realism no formal support: generalized elements exist in every category, morphisms relate objects the category presupposes, and Bain's physical cases eliminate spacetime points, not relata

Graph View

nLab contributors (principal author Urs Schreiber) (2009), nLab (ncatlab.org), a collaborative wiki for category theory and higher structures — https://ncatlab.org/nlab/show/Yoneda+lemma

Key takeaways

  • The page states that for a locally small category C, every presheaf X and every object c, there is a canonical bijection Hom_{[C^op,Set]}(y(c), X) ≅ X(c), given by η ↦ η_c(id_c). It proves the bijection by chasing id_c around a naturality square. From it the page derives that the Yoneda embedding y: c ↦ Hom_C(−,c) is fully faithful and that y(c) ≅ y(d) ⇔ c ≅ d. It also shows by counterexample that the hypothesis doing the work is naturality: objects whose hom-sets are pointwise in bijection need not be isomorphic.

Standing in the record

Filed at the owner’s request on 2026-09-26, for an account of the Yoneda lemma, connected where the literature allows to ontic structural realism. It was filed Deferred, unread. NOTE-195 is the close reading of 2026-09-26, and it placed the work: Active — Worth reading as a compact, correct statement and proof of the lemma with the embedding and the iso-determination corollary. It also has a “Necessity of naturality” section, which textbooks rarely include and which any structuralist reading of the lemma needs. Two caveats: its classical statement does not state naturality in c and X (the HoTT section does), and its finite counterexample is ill-formed as written.

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