Let $K^H$ be the homotopy $K$-theory spectrum. It is defined as a colimit of the form $|\mathbb{K}(X\times \Delta^{\bullet})|$. Here $\Delta^{\bullet}$ is the co-simplicial scheme defined by the algebraic simplices. $\mathbb{K}$ denote the algebraic $K$-theory spectrum. Now you can define something similar but instead of the spectrum you replace it with the space (like loop space of $Q$-construction) we denote it by $K$. What is the relation of non-negative homotopy groups of $|\mathbb{K}(X\times \Delta^{\bullet})|$ and $|K(X\times \Delta^{\bullet})|$? When do they coincide? I think if $X$ is $K_0$-regular i.e. $K_0(X\times \mathbb{A}^n)\cong K_0(X\times \mathbb{A}^{(n+1)})$ which automatically implies $K_i$-regular for $i<0$, the isomorphism of non-negative homotopy groups happen. If it is not right or there are other conditions I'd like to know about them. Thanks.

Let $K^H$ be the homotopy $K$-theory spectrum. It is defined as a colimit of the form $|\mathbb{K}(X\times \Delta^{\bullet})|$. Here $\Delta^{\bullet}$ is the co-simplicial scheme defined by the algebraic simplices. $\mathbb{K}$ denote the algebraic $K$-theory spectrum. Now you can define something similar but instead of the spectrum you replace it with the space (like loop space of $Q$-construction) we denote it by $K$. What is the relation of non-negative homotopy groups of $|\mathbb{K}(X\times \Delta^{\bullet})|$ and $|K(X\times \Delta^{\bullet})|$? When do they coincide? I think if $X$ is $K_0$-regular i.e. $K_0(X\times \mathbb{A}^n)\cong K_0(X\times \mathbb{A}^{(n+1)})$ which automatically implies $K_i$-regular for $i<0$, the isomorphism of non-negative homotopy groups happen. If it is not right or there are other conditions I'd like to know about them. Thanks. BTW if my guess is correct I'd also like a reference for it!