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.
   • CommentAuthornLab edit announcer
   • CommentTimeNov 8th 2020
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexonly removing the latex markers Valeria de Paiva <a href="https://ncatlab.org/nlab/revision/diff/G-crossed+braided+fusion+category/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-crossed+braided+fusion+category/7">v7</a>, <a href="https://ncatlab.org/nlab/show/G-crossed+braided+fusion+category">current</a>

   only removing the latex markers

   Valeria de Paiva

   diff, v7, current

2. • CommentRowNumber2.
   • CommentAuthornLab edit announcer
   • CommentTimeMar 12th 2024
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexAdded references and links Benjamin Haake <a href="https://ncatlab.org/nlab/revision/diff/G-crossed+braided+fusion+category/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-crossed+braided+fusion+category/8">v8</a>, <a href="https://ncatlab.org/nlab/show/G-crossed+braided+fusion+category">current</a>

   Added references and links

   Benjamin Haake

   diff, v8, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 12th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have completed the identifiable bibitems <a href="https://ncatlab.org/nlab/revision/diff/G-crossed+braided+fusion+category/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-crossed+braided+fusion+category/9">v9</a>, <a href="https://ncatlab.org/nlab/show/G-crossed+braided+fusion+category">current</a>

   I have completed the identifiable bibitems

   diff, v9, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 12th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave touched formatting, hyperlinking and formatting <a href="https://ncatlab.org/nlab/revision/diff/G-crossed+braided+fusion+category/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-crossed+braided+fusion+category/9">v9</a>, <a href="https://ncatlab.org/nlab/show/G-crossed+braided+fusion+category">current</a>

   have touched formatting, hyperlinking and formatting

   diff, v9, current

5. • CommentRowNumber5.
   • CommentAuthorperezl.alonso
   • CommentTimeMar 6th 2025
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexIf I don't replace H by a fusion category but just by a $\mathbb{C}$-algebra, does that algebra have a name? It's essentially ``$G$-commutative'' but I'm wondering if there's an established name.

   If I don’t replace H by a fusion category but just by a $\mathbb{C}$-algebra, does that algebra have a name? It’s essentially “$G$-commutative” but I’m wondering if there’s an established name.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 6th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere is a notion of "[crossed G-algebras](https://ncatlab.org/nlab/show/crossed+G-algebra#CrossedGAlgebras)".

   There is a notion of “crossed G-algebras”.

7. • CommentRowNumber7.
   • CommentAuthorperezl.alonso
   • CommentTimeMar 6th 2025
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexAh, of course, thanks. On a different note. Usually, when we talk about crossed modules, we think of group exact sequences $1\to K\to A\to G \to \Gamma\to 1$ where $(A\to G)$ is the crossed module. In particular, if $A\to G$ is surjective, then $\Gamma=1$, and this crossed module is equivalent to $(K\to 1)$. Now, thinking of $A$ as a fusion category, $K$ would be $A_1$ the fusion subcategory of degree $1\in \Gamma$. Moreover, if the $\Gamma$-grading is faithful, then this map $A\to G$ is surjective, which somewhat hints that this $G$-crossed braided fusion category $A$ is equivalent to the braided fusion category $A_1$. In what sense is this equivalence true?

   Ah, of course, thanks.

   On a different note. Usually, when we talk about crossed modules, we think of group exact sequences $1\to K\to A\to G \to \Gamma\to 1$ where $(A\to G)$ is the crossed module. In particular, if $A\to G$ is surjective, then $\Gamma=1$, and this crossed module is equivalent to $(K\to 1)$. Now, thinking of $A$ as a fusion category, $K$ would be $A_1$ the fusion subcategory of degree $1\in \Gamma$. Moreover, if the $\Gamma$-grading is faithful, then this map $A\to G$ is surjective, which somewhat hints that this $G$-crossed braided fusion category $A$ is equivalent to the braided fusion category $A_1$. In what sense is this equivalence true?

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMar 7th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexI don't know, haven't really thought about it. But it sounds like there should be an evident comparison functor, in which case you could explicitly check whether it's essentially surjective and fully faithful.

   I don’t know, haven’t really thought about it. But it sounds like there should be an evident comparison functor, in which case you could explicitly check whether it’s essentially surjective and fully faithful.

9. • CommentRowNumber9.
   • CommentAuthorperezl.alonso
   • CommentTimeMar 7th 2025
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexAdded classification of $G$-crossed braided fusion categories with faithful grading. <a href="https://ncatlab.org/nlab/revision/diff/G-crossed+braided+fusion+category/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/G-crossed+braided+fusion+category/12">v12</a>, <a href="https://ncatlab.org/nlab/show/G-crossed+braided+fusion+category">current</a>

   Added classification of $G$-crossed braided fusion categories with faithful grading.

   diff, v12, current

10. • CommentRowNumber10.
    • CommentAuthorperezl.alonso
    • CommentTimeMar 7th 2025
    • (edited Mar 7th 2025)
    • PermaLink
    Author: perezl.alonso
    Format: MarkdownItex(rolled back an edit, I overlooked and repeated a pointer)

    (rolled back an edit, I overlooked and repeated a pointer)