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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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology 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.
   • CommentAuthorAlec Rhea
   • CommentTimeMar 6th 2019
   • (edited Mar 6th 2019)
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   Author: Alec Rhea
   Format: MarkdownItexIt seems like there's some confusion in the definition section of the [[2-limit]] page -- it asserts the existence of a pseudonatural equivalence $K(X,L)\simeq Cat^\mathcal{C}(J,K(X,F-)),$ but these are hom-categories and not 2-categories themselves so pseudonatural transformations aren't defined on them. The two obvious routes I see to rectify this are to require the equivalence on the full $2$-categories instead of the hom-categories, or to promote the hom-categories to discrete $2$-categories, but I'm not sure which (or if either) is implicitly intended. Any clarification is greatly appreciated.

   It seems like there’s some confusion in the definition section of the 2-limit page – it asserts the existence of a pseudonatural equivalence $K(X,L)\simeq Cat^\mathcal{C}(J,K(X,F-)),$ but these are hom-categories and not 2-categories themselves so pseudonatural transformations aren’t defined on them. The two obvious routes I see to rectify this are to require the equivalence on the full $2$-categories instead of the hom-categories, or to promote the hom-categories to discrete $2$-categories, but I’m not sure which (or if either) is implicitly intended. Any clarification is greatly appreciated.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 7th 2019
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   Author: Mike Shulman
   Format: MarkdownItex$Cat$ is a 2-category, so $Cat$-valued (hom-)functors have pseudonatural transformations between them. Perhaps the confusing part is that $X$ is the argument of the functors, i.e. the equivalence is "pseudonatural in $X$"?

   $Cat$ is a 2-category, so $Cat$-valued (hom-)functors have pseudonatural transformations between them. Perhaps the confusing part is that $X$ is the argument of the functors, i.e. the equivalence is “pseudonatural in $X$”?

3. • CommentRowNumber3.
   • CommentAuthorAlec Rhea
   • CommentTimeMar 7th 2019
   • PermaLink
   Author: Alec Rhea
   Format: MarkdownItexAh -- the part confusing me was how a pseudonatural equivalence could have $K(X,L)$ (the hom category) as a domain if $K$ is just a $2$-category with hom $1$-categories, but if I understand you correctly this is actually the representable 2-presheaf $K(-,L):K\to Cat$ at $X$ on the left and we're asserting a pseudonatural equivalence to another pseudofunctor?

   Ah – the part confusing me was how a pseudonatural equivalence could have $K(X,L)$ (the hom category) as a domain if $K$ is just a $2$-category with hom $1$-categories, but if I understand you correctly this is actually the representable 2-presheaf $K(-,L):K\to Cat$ at $X$ on the left and we’re asserting a pseudonatural equivalence to another pseudofunctor?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 7th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexRight.

   Right.

5. • CommentRowNumber5.
   • CommentAuthorAlec Rhea
   • CommentTimeMar 7th 2019
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   Author: Alec Rhea
   Format: MarkdownItexMuch appreciated Mike :^).

   Much appreciated Mike :^).

6. • CommentRowNumber6.
   • CommentAuthorDELETED_USER_2018
   • CommentTimeJun 17th 2020
   • (edited Apr 11th 2023)
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   Author: DELETED_USER_2018
   Format: MarkdownItex[deleted]

   [deleted]

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJun 17th 2020
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   Author: Mike Shulman
   Format: MarkdownItexYes. There are really only two basic notions of limit that include all the others as special cases: weighted 2-limits and weighted bilimits. Weighted 2-limits are 2-functorial and weighted bilimits are pseudofunctorial.

   Yes. There are really only two basic notions of limit that include all the others as special cases: weighted 2-limits and weighted bilimits. Weighted 2-limits are 2-functorial and weighted bilimits are pseudofunctorial.

8. • CommentRowNumber8.
   • CommentAuthorDELETED_USER_2018
   • CommentTimeJun 17th 2020
   • (edited Apr 11th 2023)
   • PermaLink
   Author: DELETED_USER_2018
   Format: MarkdownItex[deleted]

   [deleted]

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJun 18th 2020
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   Author: Mike Shulman
   Format: MarkdownItexHave you looked at the references on the [[2-limit]] page? Particularly Steve Lack's *2-categories companion* is a good overview.

   Have you looked at the references on the 2-limit page? Particularly Steve Lack’s 2-categories companion is a good overview.

10. • CommentRowNumber10.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeJun 18th 2020
    • (edited Apr 11th 2023)
    • PermaLink
    Author: DELETED_USER_2018
    Format: MarkdownItex[deleted]

    [deleted]