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 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.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 3rd 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWatching a part of Lurie's Arizona State [lecture](https://vimeo.com/120708135), I realise I had never thought before that a representation of a group $G$, can be thought as a $G$-invariant of a linear category under the trivial action of $G$. But that was just what we are being told in the slick homotopy type theoretic account of [[representation theory]], right? Work in the context $\mathbf{B} G$. So then what he takes to be the categorification of the Fourier transform, the equivalence between the category of representations of $G$ and that of the representations of the dual (for some kinds of group), is what in the HoTT picture? Is it better to be in a linear HoTT setting?

   Watching a part of Lurie’s Arizona State lecture, I realise I had never thought before that a representation of a group $G$, can be thought as a $G$-invariant of a linear category under the trivial action of $G$.

   But that was just what we are being told in the slick homotopy type theoretic account of representation theory, right? Work in the context $\mathbf{B} G$.

   So then what he takes to be the categorification of the Fourier transform, the equivalence between the category of representations of $G$ and that of the representations of the dual (for some kinds of group), is what in the HoTT picture? Is it better to be in a linear HoTT setting?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 3rd 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItex> But that was just what we are being told in the slick homotopy type theoretic account of [[representation theory]], right? Work in the context $\mathbf{B} G$. Yes, but now at a further nested level: First there is the trivial $G$-action on the linear category $\mathcal{C}$ (take, without restriction of generality, its core maximal groupoid if we want to stay within homotopy theory proper) and that is exhibited by the split homotopy fiber sequence $$ \array{ \mathcal{C} &\longrightarrow& (\mathbf{B}G) \times\mathcal{C} \\ && \downarrow \\ && \mathbf{B}G } \,. $$ The homotopy invariants of the action are equivalently the sections of this fibration, so here these are equivalently the maps $$ \rho \colon \mathbf{B}G \longrightarrow \mathcal{C} \,. $$ This is generally the expression for $G$-homotopy invariants of $G$ acting trivially on any $\mathcal{C}$. One then observes that this is of course itself a $G$ action, now on whatever $V \in \mathcal{C}$ the point of $\mathbf{B}G$ goes to. > So then what he takes to be the categorification of the Fourier transform, the equivalence between the category of representations of G and that of the representations of the dual (for some kinds of group), is what in the HoTT picture? This I haven't thought about.

   But that was just what we are being told in the slick homotopy type theoretic account of representation theory, right? Work in the context $\mathbf{B} G$.

   Yes, but now at a further nested level:

   First there is the trivial $G$-action on the linear category $\mathcal{C}$ (take, without restriction of generality, its core maximal groupoid if we want to stay within homotopy theory proper) and that is exhibited by the split homotopy fiber sequence

   $$\array{ \mathcal{C} &\longrightarrow& (\mathbf{B}G) \times\mathcal{C} \\ && \downarrow \\ && \mathbf{B}G } \,.$$

   The homotopy invariants of the action are equivalently the sections of this fibration, so here these are equivalently the maps

   $$\rho \colon \mathbf{B}G \longrightarrow \mathcal{C} \,.$$

   This is generally the expression for $G$-homotopy invariants of $G$ acting trivially on any $\mathcal{C}$. One then observes that this is of course itself a $G$ action, now on whatever $V \in \mathcal{C}$ the point of $\mathbf{B}G$ goes to.

   So then what he takes to be the categorification of the Fourier transform, the equivalence between the category of representations of G and that of the representations of the dual (for some kinds of group), is what in the HoTT picture?

   This I haven’t thought about.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 4th 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIt sounds like something we ought to want to do. Right at the end, Lurie suggests that the self-duality of Laurent series as shown by the Weil symbol gives rise to some part of geometric Langlands.

   It sounds like something we ought to want to do. Right at the end, Lurie suggests that the self-duality of Laurent series as shown by the Weil symbol gives rise to some part of geometric Langlands.