By John C. Baez, Aristide Baratin, Laurent Freidel, Derek K. Wise

A "$2$-group" is a class built with a multiplication pleasant legislation like these of a bunch. simply as teams have representations on vector areas, $2$-groups have representations on "$2$-vector spaces", that are different types analogous to vector areas. regrettably, Lie $2$-groups mostly have few representations at the finite-dimensional $2$-vector areas brought via Kapranov and Voevodsky. as a result, Crane, Sheppeard and Yetter brought sure infinite-dimensional $2$-vector areas known as "measurable different types" (since they're heavily on the topic of measurable fields of Hilbert spaces), and used those to check infinite-dimensional representations of convinced Lie $2$-groups. the following they proceed this paintings. they start with a close examine of measurable different types. Then they provide a geometric description of the measurable representations, intertwiners and $2$-intertwiners for any skeletal measurable $2$-group. They research tensor items and direct sums for representations, and diverse options of subrepresentation. They describe direct sums of intertwiners, and sub-intertwiners--features no longer obvious in traditional staff illustration thought and learn irreducible and indecomposable representations and intertwiners. additionally they examine "irretractable" representations--another function no longer noticeable in traditional crew illustration conception. eventually, they argue that measurable different types built with a few additional constitution need to be thought of "separable $2$-Hilbert spaces", and examine this concept to a tentative definition of $2$-Hilbert areas as illustration different types of commutative von Neumann algebras.

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Recall that the horizontal composite β ◦ α is defined so that G UT H U αH U T Hp pp pp pp p (β◦α)pH βT H  U TH βT H pp pp pp p5  GUT H U αH commutes. Let us pick an element ψ ∈ U T H, which can be written in the form ⊕ ψz = ⊕ d(ut)z ψz,x , with ψz,x = X dkz,x ψz,y,x Y by definition of the composite field U T . Note that, thanks to Eq. (37) which defines the family of measures kz,x , the section ψz can also be written as ⊕ ψz = ⊕ duz ψz,y , with ψz,y = Y dty ψz,y,x X Having introduced all these notations, we now evaluate the image of ψ under the morphism (β ◦α)H : ((β ◦ α)H )z (ψz ) = (U αH )z ◦ (βT H )z (ψz ) ⊕ = Y ⊕ ⊕ duz ✶Uz,y ⊗ (αH )y duz [β˜z,y ⊗ ✶(T H)y ](ψz,y ) Y duz [β˜z,y ⊗ (αH )y ](ψz,y ) = Y ⊕ = ⊕ duz Y dty [β˜z,y ⊗ α ˜ y,x ⊗ ✶Hx ](ψz,y,x ) X Applying the disintegration theorem, we can rewrite this last direct integral as an integral over X with respect to the measure (u t )z = duz (y) ty Y We obtain ⊕ ((β ◦ α)H )z (ψz ) = ⊕ d(u t )z X ⊕ = dkz,x [β˜z,y ⊗ α ˜ y,x ⊗ ✶Hx ](ψz,y,x ) Y d(u t )z [(β ◦ α)z,x ⊗ ✶Hx ](ψz,x ) X where ⊕ dkz,x [β˜z,y ⊗ α ˜ y,x ](ψz,y,x ).

M=1 We call a natural transformation of this sort a matrix natural transformation. However: Theorem 13 Any natural transformation between matrix functors is a matrix natural transformation. Proof: Given matrix functors T, T : VectM → VectN , a natural transformation α : T ⇒ T gives for each basis object em ∈ VectM a morphism in VectN with components (αem )n : Tn,m ⊗ C → Tn,m ⊗ C. 33 Using the natural isomorphism between a vector space and that vector space tensored with C, these can be reinterpreted as operators αn,m : Tn,m → Tn,m .

This completes the proof of the theorem. This allows an easy definition for the 2-morphisms in Meas: Definition 36 A measurable natural transformation is a bounded natural transformation between measurable functors. For our work it will be useful to have explicit formulas for composition of matrix natural transformations. So, let us compute the vertical composite of two matrix natural transformations α and α: T,t HX T ,t α   α 2 G HY b T ,t For any object H ∈ H X , we get morphisms αH and αH in H Y .

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