Let k be a field of characteristic 0 and let

Stkᵑ(k) (n = 0,1,2,…,∞)

denote the category of n-Artin algebraic stacks of finite type over k whose diagonal is affine (0-stacks are varieties, 1-stacks are the usual Artin stacks, etc.).
For every n≥0 we put a “scissor’’–Grothendieck group on these stacks as follows (this is exactly the set-up used by B. Toën).

Definition. K₀(Stkᵑ) is the free abelian group generated by isomorphism classes [X] of n-stacks X∈Stkᵑ(k) subject to the three relations

(1) Cut–paste: if Z⊂X is closed with open complement U,
[ X ] = [ Z ] + [ U ].

(2) g-equivalence: if f:X→Y is representable and induces a bijection X( k̄ ) → Y( k̄ ) on geometric points, then [X]=[Y].

(3) Vector bundles: if E→X is a vector bundle of rank r,
[ E ] = 𝔏ʳ ·[ X ] with 𝔏=[𝔸¹].

These are the relations already needed for 1-stacks; no new relations are introduced when n>1. The product is given by fibre product over Spec k, so we obtain a commutative ring K₀(Stkᵑ).

Notice that for n=0 the third relation is automatic (it is a consequence of 1), so K₀(Stk⁰) is the usual Grothendieck ring of varieties K₀(Varₖ).

1. Toën’s theorem

The key result is the following theorem of Bertrand Toën (Grothendieck rings of n-Artin stacks, 2005, Th. 3.10).

Theorem (Toën).
For every n≥1 the obvious truncation functor
τ_{\le1}: Stkᵑ → Stk¹
induces an isomorphism of rings

 τ_{\le1*}: K₀(Stkᵑ)  ≅→  K₀(Stk¹).

In particular K₀(Stk¹) ≅ K₀(Stk²) ≅ … ≅ K₀(Stk^∞); the Grothendieck ring stabilises as soon as one allows 1-stacks.

Sketch of proof idea.
Write an n-stack F as the successive homotopy-fibre product along its Postnikov tower

 F → τ_{\le n-1}F → … → τ_{\le1}F.

Each stage is obtained from the previous one by gluing a higher K(A,i)–bundle for some affine group scheme A (A is unipotent when i>1). These K(A,i)–bundles are Zariski-locally trivial, and the vector-bundle relation (3) together with the g-equivalence relation allows one to kill the contribution of K(A,i) in the Grothendieck ring. Inductively one gets [F] = [τ_{\le1}F].

2. Identification with a localisation of K₀(Varₖ)

Long before one reaches “higher’’ stacks, already 1-stacks force one to invert certain classes. If we set

S = { 𝔏, 𝔏ᵐ−1  (m≥1), [GL_r]  (r≥1) } ⊂ K₀(Varₖ),

then Toën proves that the natural map

K₀(Varₖ)[S^{-1}] → K₀(Stk¹)

is an isomorphism. Combining this with the theorem above we get

K₀(Stkᵑ) ≅ K₀(Varₖ)[ 𝔏, 𝔏ᵐ−1, [GL_r] ]^{-1}      for every n≥1.

Thus allowing 2-, 3-, …, ∞-stacks does not enlarge the ring any further.

3. What information is really lost?

For an n-stack F, its class only depends on τ_{\le1}F, not merely on π₀(F). Example:

BGₘ (1-stack)  π₀(BGₘ)=pt,
but [BGₘ] = (𝔏−1)^{-1} ≠ 1.

So higher homotopy groups (π₂,π₃,…) do not contribute, but the group-action data already present in the 1-truncation does.

4. Answer to the original question

• Yes, one can define a Grothendieck ring for underived higher Artin stacks exactly as for varieties or ordinary stacks.

• The ring stabilises at level 1: for any n≥1 K₀(Stkᵑ) ≅ K₀(Stk¹).

• Equivalently, every class [F] of an n-stack satisfies [F] = [τ_{\le1}F] in the Grothendieck ring. It is false that [F]=[π₀(F)] in general (BGₘ is the standard counter-example).

• Consequently

 K₀(Stkᵑ) ≅ K₀(Varₖ)[ 𝔏, 𝔏ᵐ−1, [GL_r] ]^{-1}     (n≥1).

So, passing from ordinary (1-) stacks to higher (underived) stacks introduces no new elements in the Grothendieck ring beyond those that were already forced by 1-stacks.