TITLE: What does it mean for a category to admit direct integrals?
QUESTION [30 upvotes]: Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals.
Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)_{x\in X}$, one can form a new representation
$$
H=\int^\oplus H_x = \left\{\text{$L^2$ functions $f:X \to \coprod_{x\in X} H_x$}\right\},
$$
called the direct integral of the $H_x$'s.
Conversely, given $H\in \mathrm{Rep}(G)$ and an abelian sub-von Neumann algabra $A\subset \mathrm{End}_G(H)$ of the von Neumann algebra of endomorphisms of $H$, one can find a measure space $X$ s.t. $A=L^\infty X$, and a measurable family of unitary representations $(H_x)_{x\in X}$ such that $H=\int^\oplus H_x$.
The category of unitary representation of $G$ is said to admit direct integral decompositions.

What does it mean for a $W^*$-category to admit direct integrals?
What does it mean for a $W^*$-category to admit direct integral decompositions?

Presumably, "admitting direct integrals" is not a property of a category, but is extra structure instead. What is this extra structure? What axioms does this extra structure satisfy?
Is "admitting direct integral decompositions" a property of a category that admits direct integrals?

Here are some examples of categories that admit direct integral decompositions.
Example1: The category of representations of a separable C* algebra.
Example2: Let $A$ be a separable C* algebra, let $(p_n)_{n\in\mathbb N}$ be an increasing sequence of projections in $A$, and let $p\in A$ be yet another projection which is bigger than all the $p_n$.
Then the category of all representations $(H,\pi)$ of $A$ such that $\pi(p_n)\to \pi(p)$ in the strong topology is another example of a category that admits direct integral decompositions.
Example3 (very slight generalisation of Example2): Let $A$ be a separable C* algebra. Let $S$ be a countable set. For every $\alpha\in S$, let $(p^\alpha_n)_{n\in\mathbb N}$ be an increasing sequence of projections, and let $p^\alpha\in A$ be another projection which is bigger than all the $p^\alpha_n$.
Then we can consider the category of all representations $(H,\pi)$ of $A$ such that $\forall \alpha\in S,\pi(p^\alpha_n)\to \pi^\alpha(p)$ in the strong topology.

REPLY [6 votes]: The following is an argument for showing that "having direct integral" is definitely not a property nor a "property-like-structure" of $W^*$-categegories, but a real, non-trivial additional structure. I.e. that in examples where they exists they are not intrinsic to the $W^*$ structure. That does not answer the question, but I felt like it was worth mentioning as it shows that lots of approach to this question are bound to fail, and hopefully give some hints for what kind of additional structure a solution should have.
Let $C=C([0,1])$, and I'm considering the category of representations of $C$, it admit direct integral, but the $W^*$-category have a lot of automorphisms that do not preserves these representations.
Indeed the category of representation of $C$ is equivalent to the category of normal representation of its enveloping Von Neuman algebra $C''$. As any commutative von Neuman algebra $C''$ can be split into an atomic part and a diffuse part.
One has one atom for each element of $[0,1]$, and they corresponds to the irreducible representations $\chi_x$ of $C$.
So for any bijection of $[0,1]$ there is an automorphism of $C''$ that justs permute the atom and leave the diffuse part unchanged. On the $W^*$-category it permutes the irreducible representation (and their orthogonal sums) while leaving the diffuse part of each representation unchanged.
As any representation is a direct integral of irreducible, these direct integral are clearly not going to be preserved by the automorphisms: 
for example if I simply chose the automorphism $1-x$ of $[0,1]$ then the representation $L^2([0,1/2]) = \int_0^{1/2} \chi_x$ should become (if integral were preserved) under the automorphism $L^2[1/2,1] = \int_{1/2}^{1} \chi_x$ and these representation are not even isomorphic.
Even worse (but I have not checked this last part in details), in the case (as above) where the automorphism of $[0,1]$ is chosen measurable, it seems to me that the isomorphisms of the $W^*$-category preserve the natural notions of measurable fields of representations and morphisms. Indeed unless I'm mistaken the splitting of representations in "atomic+diffuse" can be applied to the measurable fields and if the permutation of the atomic part is measurable it can be applied to fields as well. So this type of structure will be insuficient. (this seem to for example completely prevent what is suggested at the end of Tobias Fritz answer).