# Alternative definitions of Weibel's homotopy K-theory

Consider a sort of $$\mathbb{A}^1$$-homotopy-stable algebraic $$K$$-theory for rings constructed as follows.

For $$K_0$$ we take a symmetrization subject to natural direct sum operation of $$\mathbb{A}^1$$-homotopy classes of finite-dimensional idempotents in $$\mathbb{M}_\infty(B)$$. Higher and lower groups are defined using looping/suspension. I hypothesize that this construction coincides with Weibel's homotopy algebraic K-theory $$KH$$, which is originally defined via spectra. This my be a trivial fact not even worth describing in the literature, or quite possibly I'm terribly wrong and missing some simple counterexample.

(Sorry if the question is too simple or too unclear - I haven't been doing real math for years.)

• What kind of $\mathbb{A}^1$-homotopy are we talking about? Do you mean the $\mathbb{A}^1$-localization of the sheaf or are you just taking naive $\mathbb{A}^1$-classes? The second approach doesn't seem very likely to work out. And what do you mean by "suspension"? – Denis Nardin Jan 6 '20 at 12:06
• It is the naive approach - my rings are in general not commutative, so there is no way to build sheaves. Suspension in this terms is a particular $\Sigma$ fitting into the short exact sequence $\Sigma A \to A[t, t^{-1}] \xrightarrow{ev_1} A$. – Kolya Ivankov Jan 6 '20 at 12:14
• Thing is, I get this kind of theory as a byproduct of another construction, and it seems to be related to Cortinas-Thom bivariant algebraic theory $kk$, for which, given an $R$-aglebra $B$, one has $kk(E,B)\cong KH(B)$. I'm trying to prove the coincidence of my group with $KH_0$ via this sideway, but wonder if I'm inventing a bicycle by downgrading a jet fighter :) – Kolya Ivankov Jan 6 '20 at 12:19