The Grothendieck ring of higher (underived) stacks does not exhibit the same phenomenon as varieties vs schemes, where the class of a stack is directly related to its truncation via ( [\mathcal{X}] = [\pi_0(\mathcal{X})] ). Here's the detailed reasoning:

Background Recap

• Grothendieck Ring of Varieties/Schemes: For schemes, the relation ( [X] = [X_{\text{red}}] ) holds because the re...

Short answer

• Allowing higher (underived) Artin stacks does not change the Grothendieck ring: the truncation functor τ≤1 induces an isomorphism on Grothendieck rings. In particular, for any higher Artin stack X of finite type with affine stabilizers (over a field of characteristic 0), one has [X] = [τ≤1 X] in the Grothendieck ring.
• The statement [X] = [π0(X)] is false in general for underived s...

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 th...

Below is a sketch of why nothing “new’’ appears once you pass from 1-stacks (Artin stacks) to genuinely higher underived stacks. In fact one shows by a Postnikov induction that for each (n\ge1) the natural map [ K_0\bigl(\text{Artin (1)-stacks}\bigr);\longto;K_0\bigl(\text{Artin (n)-stacks}\bigr) ] induced by “forgetting all homotopy above degree 1’’ is an isomorphism. Consequently [ ...

This is an excellent question that touches on the foundational principles of motivic invariants for increasingly complex geometric objects. The answer involves distinguishing between "linear" structures (like those in derived geometry) and "non-linear" structures (like the higher homotopy groups of underived stacks).

Summary of the Known Landscape

Let's first recap the situation for 0-stacks...

Let k be a field of characteristic 0.
For every integer n≥0 (and also for n = ∞) let Artin≤n(k) denote the n-Artin stacks of finite type over k whose inertia (stabiliser) groups are affine.
Following Toën one defines the Grothendieck group

 K₀(Artin≤n(k))

as the free abelian group generated by isomorphism classes [X] of such stacks, modulo

 (a) scissor relations [X] = [U] + [Z] for every cl...

This is an interesting question about extending the Grothendieck ring construction to higher stacks. Let me address the main points:

The Current State

For higher (underived) stacks, there isn't a well-established theory of the Grothendieck ring in the literature, unlike the cases you mentioned (varieties, schemes, algebraic spaces, and Artin stacks). The main obstacles are:

1. **Lack of reas...

The Grothendieck Ring of Higher Stacks

This question explores an interesting intersection of algebraic geometry and higher category theory. Let me address the key points about Grothendieck rings and higher stacks.

Background on Grothendieck Rings

The Grothendieck ring of varieties has the "scissor relation" that allows us to write when is a closed subvari...