Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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 infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic manifolds mathematics 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 5th 2023
    • (edited Apr 5th 2023)

    The other day on the preprint server


    asking readers to help invent the upper right corner of this “push out”:

    Quantum information structure Add tensor Linear algebra Add parameters Bundle theory \array{ \mathclap{\text{Quantum information}} \\ \mathllap{ {}^{\text{Add tensor}} _{\text{structure}} } \Big\uparrow \\ \text{Linear algebra} &\underset{\text{Add parameters}}{\longrightarrow}& \text{Bundle theory} }


    In reply, we want to say that this pushout is “parameterized quantum information theory” exhibited by the category VectBund of vector bundles equipped with its external tensor product of vector bundles, the 1-categorical semantics for dependent linear type theory which exhibits it as a classically controlled quantum language. (K-theory would come in only as a second step of group completion…)

    To make this precise, at least in the case that the parameters form discrete sets, we want to say that there is a pushout diagram of the following form, in a suitable ambient (2,1)-category:

    (Vect,) (Vect Set,,) Vect (Vect Set,) \array{ \big( Vect, \otimes \big) &\longrightarrow& \big( Vect_{Set} , \sqcup , \boxtimes \big) \\ \big\uparrow && \big\uparrow \\ Vect &\longrightarrow& \big( Vect_{Set}, \sqcup \big) }

    In order to make sense of this diagram and to see that it is indeed a pushout, we need an ambient (2,1)-category inside which we have 1. categories, 2. monoidal categories, 3. co-cartesian cateories and 4. distributive monoidal categories, so that functors may go from categories with some number of monoidal structures to categories with at least these monoidal structures, maybe more.

    Take that (2,1)-category to be the (2,1)-Grothendieck construction on the evident square of forgetful functors.

    C:(1,0) (1,1) (0,0) (1,0)MonCat DistMonCat Cat CoCartCat \mathbf{C} \;\; \colon \;\; \array{ (1,0) &\longrightarrow& (1,1) \\ \big\uparrow && \big\uparrow \\ (0,0) &\longrightarrow& (1,0) } \;\;\;\; \mapsto \;\;\;\; \array{ MonCat &\longleftarrow& DistMonCat \\ \big\downarrow && \big\downarrow \\ Cat &\longleftarrow& CoCartCat }

    Here in C\int \mathbf{C} the above square is a pushout, as desired. I think.

    Have made a first note about this at VectBund in a new section Amalgamation of monoidal and parameter structures. For the moment this is a little experimental, will try to polish this up.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 6th 2023

    I see the authors remark

    We can think of two ways, there may be others, in which quantum problems come with continuous parameters. First, in periodic systems, the dof in the unit cell constitute the fiber of a bundle over the momentum torus, or Brillion zone. This point of view was important in understanding the “10-fold way” a topological classification of free fermion states [Kit09,Has13].

    So this arises at what you call “a second step of group completion” in their quantum K-theory, I imagine. John Baez has a page of resources on the ten-fold way.

    From Tenfold topology of crystals: Unified classification of crystalline topological insulators and superconductors

    One of the insightful ways to derive and decipher the structure of the periodic table of TISCs is to naturally encapsulate the tenfold way within “complex K-theory” and “real K-theory,” the former of which is sometimes referred to as “the simplest generalized cohomology theory.”

    Worth a page then. Any preference for ’10-fold’, ’tenfold’, or ’ten-fold’?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2023

    We have a page already for K-theory classification of topological phases of matter, of which the “10-fold way” is a small subsector (namely that of global crystal symmetry, which is quite unrealistic), e.g. pp. 13 here, p. 48 here: The 10 here is simply the 2 + 8 distinct K-theory groups of the point, 2 for KUKU and 88 for KOKO.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2023

    On the topic of their question: What I don’t see yet is how to make a pushout of the above kind for topological vector bundles over topological spaces. What I do see, as a mild generalization of the previous argument, is how to get flat vector bundles over spaces.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 6th 2023
    • (edited Apr 6th 2023)

    Flat vector bundles over spaces are the pushout of Vect(VectBund,)Vect \to (VectBund, \sqcup) and Vect(Vect,)Vect \to (Vect, \otimes)?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2023

    No, in the bottom right corner we need to use flat vector bundles already. (This kind of pushout can at most add a monoidal structure, not change the underlying category.)

    Namely, for flat vector bundles we simply enhance, in the previous argument, the domains of the families of vector bundles 𝒱 ():SVect\mathcal{V}_{(-)} \colon S \to Vect from sets to groupoids, and use that these are all disjoint unions of BG\mathbf{B} G-s.

    To get non-flat vector bundles into the picture, one will need to consider a stacky enhancement of the diagram. This may be a little fiddly, since – I guess – one will need to realize the items on the left as “stacks of trivial vector bundles” — which requires that the ambient comparison 2-functor forgets not just monoidal structure but also Grothendieck topologies. Haven’t worked this out yet…

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2023

    I have typed up a note on this and related ideas on \infty-local systems in a pdf kept here.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 8th 2023

    Looking good.

    Some typos:

    homtopy quasi-cocartesian categories; linear typs; P. Cagne and P.-A. Melliés (wrong accent)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2023

    Thanks! Fixed now.