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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.
   • CommentAuthorJohn Baez
   • CommentTimeJul 2nd 2010
   • PermaLink
   Author: John Baez
   Format: MarkdownItexI started two new entries: [[Gabriel-Ulmer duality]] and [[Lex]].

   I started two new entries: Gabriel-Ulmer duality and Lex.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 26th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI do not understand: the bottom of the page [[Gabriel-Ulmer duality]] lists many tens of entries which supposedly link to it, and I tested some of them, and they don't.

   I do not understand: the bottom of the page Gabriel-Ulmer duality lists many tens of entries which supposedly link to it, and I tested some of them, and they don’t.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 26th 2012
   • (edited May 26th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt is linked to from _[[category theory - contents]]_, which is included in all those entries listed there!

   It is linked to from category theory - contents, which is included in all those entries listed there!

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 26th 2012
   • (edited May 26th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexSo contents pages are transitive in the linking ? Strange (notice, on the more logical side, that the links via redirect names do not list), I would say suboptimal as clearly singles out entries which do not have direct connection. I have added the reference of Kelly from Cahiers which in section 9 has Gabriel-Ulmer duality for $V$-enriched categories, where $V$ is closed symmetric monoidal category whose underlying category $V_0$ is locally small, complete and cocomplete.

   So contents pages are transitive in the linking ? Strange (notice, on the more logical side, that the links via redirect names do not list), I would say suboptimal as clearly singles out entries which do not have direct connection.

   I have added the reference of Kelly from Cahiers which in section 9 has Gabriel-Ulmer duality for $V$-enriched categories, where $V$ is closed symmetric monoidal category whose underlying category $V_0$ is locally small, complete and cocomplete.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 26th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexYeah, the list of linked-to entries at the bottom of each page is not too useful.

   Yeah, the list of linked-to entries at the bottom of each page is not too useful.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeMay 26th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexRegarding that there is an enriched version (I do not like however the fact that it is proved only for $V$ _symmetric_ monoidal, as I would like at least to have it for bimodules over noncommutative ring instead of modules over commutative ring), is there a further generalization in the setup of formal category theory, say in the language of [[equipment]]s ?

   Regarding that there is an enriched version (I do not like however the fact that it is proved only for $V$ symmetric monoidal, as I would like at least to have it for bimodules over noncommutative ring instead of modules over commutative ring), is there a further generalization in the setup of formal category theory, say in the language of equipments ?

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeMay 26th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI don't know of any equipmenty way of defining "locally finitely presentable".

   I don’t know of any equipmenty way of defining “locally finitely presentable”.

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeMay 26th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexthat's a pity...

   that’s a pity…

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMay 27th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYeah. I'm not saying there isn't a way of doing it; I haven't tried very hard. Maybe someone has even already done it.

   Yeah. I’m not saying there isn’t a way of doing it; I haven’t tried very hard. Maybe someone has even already done it.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 22nd 2018
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI added a reference to an [MO question](https://mathoverflow.net/q/293031/447) on the $\infty$-version.

    I added a reference to an MO question on the $\infty$-version.

11. • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 24th 2018
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexWhat's likely to happen in this domain? In the 1-categorical case we have a general result for sound doctrines, $\mathbb{D}$, which produces a [duality](http://www.tac.mta.ca/tac/volumes/10/20/10-20.pdf) between $\mathbb{D}-\mathbf{Th}$ and $\mathbb{D}-\mathbf{LP}$. This includes Gabriel-Ulmer and Morita duality for presheaf categories. So moving to the $(\infty, 1)$-case, there will analogues for these, as in #10 for Gabriel-Ulmer. Presumably there will be plenty to say about $\infty$-Morita theory, and whatever we have at [[derived Morita equivalence]] will be a subsection of this. Then we'll speak of Morita-equivalent theories (perhaps in intensional type theory) with equivalent categories of models. Then we can have $(\infty, 1)$-toposes as 'bridges', equivalences in $(\infty, 2)$-categories, $(\infty, 2)$-Yoneda, etc. And then this should (at some stage) make contact with whatever Lurie's doing with conceptual completeness in A.9 of [[Spectral Algebraic Geometry]].

    What’s likely to happen in this domain? In the 1-categorical case we have a general result for sound doctrines, $\mathbb{D}$, which produces a duality between $\mathbb{D}-\mathbf{Th}$ and $\mathbb{D}-\mathbf{LP}$. This includes Gabriel-Ulmer and Morita duality for presheaf categories.

    So moving to the $(\infty, 1)$-case, there will analogues for these, as in #10 for Gabriel-Ulmer.

    Presumably there will be plenty to say about $\infty$-Morita theory, and whatever we have at derived Morita equivalence will be a subsection of this. Then we’ll speak of Morita-equivalent theories (perhaps in intensional type theory) with equivalent categories of models. Then we can have $(\infty, 1)$-toposes as ’bridges’, equivalences in $(\infty, 2)$-categories, $(\infty, 2)$-Yoneda, etc.

    And then this should (at some stage) make contact with whatever Lurie’s doing with conceptual completeness in A.9 of Spectral Algebraic Geometry.