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The other day on the preprint server
are:
asking readers to help invent the upper right corner of this “push out”:
$\array{ \mathclap{\text{Quantum information}} \\ \mathllap{ {}^{\text{Add tensor}} _{\text{structure}} } \Big\uparrow \\ \text{Linear algebra} &\underset{\text{Add parameters}}{\longrightarrow}& \text{Bundle theory} }$
$\,$
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:
$\array{ \big( Vect, \otimes \big) &\longrightarrow& \big( Vect_{Set} , \sqcup , \boxtimes \big) \\ \big\uparrow && \big\uparrow \\ Vect &\longrightarrow& \big( Vect_{Set}, \sqcup \big) }$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.
$\mathbf{C} \;\; \colon \;\; \array{ (1,0) &\longrightarrow& (1,1) \\ \big\uparrow && \big\uparrow \\ (0,0) &\longrightarrow& (1,0) } \;\;\;\; \mapsto \;\;\;\; \array{ MonCat &\longleftarrow& DistMonCat \\ \big\downarrow && \big\downarrow \\ Cat &\longleftarrow& CoCartCat }$Here in $\int \mathbf{C}$ 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.
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.
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’?
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 $KU$ and $8$ for $KO$.
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.
Flat vector bundles over spaces are the pushout of $Vect \to (VectBund, \sqcup)$ and $Vect \to (Vect, \otimes)$?
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 $\mathcal{V}_{(-)} \colon S \to Vect$ from sets to groupoids, and use that these are all disjoint unions of $\mathbf{B} G$-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…
I have typed up a note on this and related ideas on $\infty$-local systems in a pdf kept here.
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