Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 14th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexFor completeness I have added pointer to * [[Emily Riehl]], [[Dominic Verity]], Section 6 of: _Fibrations and Yoneda's lemma in an $\infty$-cosmos_, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 ([arXiv:1506.05500](https://arxiv.org/abs/1506.05500), [doi:10.1016/j.jpaa.2016.07.003](https://doi.org/10.1016/j.jpaa.2016.07.003)) though there should really be some accompanying discussion of how this form of the statement is related to the usual one in terms of presheaves. <a href="https://ncatlab.org/nlab/revision/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/Yoneda+lemma+for+%28infinity%2C1%29-categories/13">v13</a>, <a href="https://ncatlab.org/nlab/show/Yoneda+lemma+for+%28infinity%2C1%29-categories">current</a>

   For completeness I have added pointer to

   • Emily Riehl, Dominic Verity, Section 6 of: Fibrations and Yoneda’s lemma in an $\infty$-cosmos, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 (arXiv:1506.05500, doi:10.1016/j.jpaa.2016.07.003)

   though there should really be some accompanying discussion of how this form of the statement is related to the usual one in terms of presheaves.

   diff, v13, current

2. • CommentRowNumber2.
   • CommentAuthornLab edit announcer
   • CommentTimeApr 16th 2021
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexAdded the fact the yoneda embedding is a natural transformation. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/Yoneda+lemma+for+%28infinity%2C1%29-categories/16">v16</a>, <a href="https://ncatlab.org/nlab/show/Yoneda+lemma+for+%28infinity%2C1%29-categories">current</a>

   Added the fact the yoneda embedding is a natural transformation.

   Anonymous

   diff, v16, current

3. • CommentRowNumber3.
   • CommentAuthorHurkyl
   • CommentTimeMay 24th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexCorrected the explanation of naturality. <a href="https://ncatlab.org/nlab/revision/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/Yoneda+lemma+for+%28infinity%2C1%29-categories/17">v17</a>, <a href="https://ncatlab.org/nlab/show/Yoneda+lemma+for+%28infinity%2C1%29-categories">current</a>

   Corrected the explanation of naturality.

   diff, v17, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 9th 2021
   • (edited Oct 9th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis old entry needs some attention concerning how it does or does not speak regarding size issues. Where the fully faithfulness is stated first ([here](https://ncatlab.org/nlab/show/Yoneda+lemma+for+infinity1-categories#YonedaLemma)) the entry does speak as if large $\infty$-categories are being considered, and I guess it should indeed hold in this generality, but the reference offered, namely HTT Prop. 5.1.3.1, speaks of small $\infty$-categories (namely simplicial sets, regarded as stand-ins for small quasi-categories). Further down under "Naturalness" ([here](https://ncatlab.org/nlab/show/Yoneda+lemma+for+infinity1-categories#Naturality)) large $\infty$-categories are mentioned more explicitly, and that's now also what the reference considers (HTT, Prop. 5.3.6.10), but neither talks about fully-faithfulness at this point. <a href="https://ncatlab.org/nlab/revision/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/Yoneda+lemma+for+%28infinity%2C1%29-categories/18">v18</a>, <a href="https://ncatlab.org/nlab/show/Yoneda+lemma+for+%28infinity%2C1%29-categories">current</a>

   This old entry needs some attention concerning how it does or does not speak regarding size issues.

   Where the fully faithfulness is stated first (here) the entry does speak as if large $\infty$-categories are being considered, and I guess it should indeed hold in this generality, but the reference offered, namely HTT Prop. 5.1.3.1, speaks of small $\infty$-categories (namely simplicial sets, regarded as stand-ins for small quasi-categories).

   Further down under “Naturalness” (here) large $\infty$-categories are mentioned more explicitly, and that’s now also what the reference considers (HTT, Prop. 5.3.6.10), but neither talks about fully-faithfulness at this point.

   diff, v18, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 9th 2021
   • (edited Oct 9th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Emily Riehl]], [[Dominic Verity]], Def. 6.2.3 and Thm. 7.2.22 in: *The comprehension construction*, Higher Sttructures **2** 1 (2018) ([arXiv:1706.10023](https://arxiv.org/abs/1706.10023), [hs:39](http://137.111.162.45/index.php/higher_structures/article/view/39)) but that also speaks about Yoneda for small $\infty$-categories... <a href="https://ncatlab.org/nlab/revision/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/Yoneda+lemma+for+%28infinity%2C1%29-categories/19">v19</a>, <a href="https://ncatlab.org/nlab/show/Yoneda+lemma+for+%28infinity%2C1%29-categories">current</a>

   added pointer to:

   • Emily Riehl, Dominic Verity, Def. 6.2.3 and Thm. 7.2.22 in: The comprehension construction, Higher Sttructures 2 1 (2018) (arXiv:1706.10023, hs:39)

   but that also speaks about Yoneda for small $\infty$-categories…

   diff, v19, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2022
   • (edited May 28th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Kerodon]], *The Yoneda Embedding* $[$[kerodon:03JA](https://kerodon.net/tag/03JA)$]$ This does seem to be formulated in the generality of large (but locally small, of course) $\infty$-categories. <a href="https://ncatlab.org/nlab/revision/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/Yoneda+lemma+for+%28infinity%2C1%29-categories/20">v20</a>, <a href="https://ncatlab.org/nlab/show/Yoneda+lemma+for+%28infinity%2C1%29-categories">current</a>

   added pointer to:

   • Kerodon, The Yoneda Embedding $[$kerodon:03JA$]$

   This does seem to be formulated in the generality of large (but locally small, of course) $\infty$-categories.

   diff, v20, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 8th 2022
   • (edited Jul 8th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have made more explicit in the text pointer to * [[Kerodon]] [Prop. 8.2.1.3](https://kerodon.net/tag/03M5) for the $\infty$-Yoneda lemma over possibly large $\infty$-categories * [[Kerodon]] [Thm. 8.2.5.4](https://kerodon.net/tag/03NJ) for the $\infty$-Yoneda embedding over possibly large $\infty$-categories. <a href="https://ncatlab.org/nlab/revision/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/Yoneda+lemma+for+%28infinity%2C1%29-categories/21">v21</a>, <a href="https://ncatlab.org/nlab/show/Yoneda+lemma+for+%28infinity%2C1%29-categories">current</a>

   I have made more explicit in the text pointer to

   • Kerodon Prop. 8.2.1.3

   for the $\infty$-Yoneda lemma over possibly large $\infty$-categories

   • Kerodon Thm. 8.2.5.4

   for the $\infty$-Yoneda embedding over possibly large $\infty$-categories.

   diff, v21, current

8. • CommentRowNumber8.
   • CommentAuthorperezl.alonso
   • CommentTimeSep 18th 2023
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexPointer to recent article: * Nikolai Kudasov, [[Emily Riehl]], [[Jonathan Weinberger]]. *Formalizing the $\infty$-categorical Yoneda lemma*. (2023). ([arXiv:2309.08340](https://arxiv.org/abs/2309.08340)). <a href="https://ncatlab.org/nlab/revision/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/Yoneda+lemma+for+%28infinity%2C1%29-categories/24">v24</a>, <a href="https://ncatlab.org/nlab/show/Yoneda+lemma+for+%28infinity%2C1%29-categories">current</a>

   Pointer to recent article:

   • Nikolai Kudasov, Emily Riehl, Jonathan Weinberger. Formalizing the $\infty$-categorical Yoneda lemma. (2023). (arXiv:2309.08340).

   diff, v24, current

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 15th 2024
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded * Shay Ben-Moshe, _Uniqueness and $(\infty,2)$-Naturality of Yoneda_ &lbrack;[arXiv:2405.08799](https://arxiv.org/abs/2405.08799)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories/26">diff</a>, <a href="https://ncatlab.org/nlab/revision/Yoneda+lemma+for+%28infinity%2C1%29-categories/26">v26</a>, <a href="https://ncatlab.org/nlab/show/Yoneda+lemma+for+%28infinity%2C1%29-categories">current</a>

   Added

   • Shay Ben-Moshe, Uniqueness and $(\infty,2)$-Naturality of Yoneda [arXiv:2405.08799]

   diff, v26, current