Defining the sheaf of bigraded homotopy groups in motivic homotopy theory.

Question

I have been learning motivic homotopy theory from these notes and on page 154 (page 8 of the pdf), the author defines $\pi_{p,q}(E)$ where $E$ is an $(s,t)$-bispectrum. He defines it as the sheaf of bigraded stable homotopy groups associated to the presheaf $$ U\mapsto \text{co}\hspace{-0.2cm}\lim_{m\geq -q}\text{Hom}_{SH^{\mathbb{A}^1}_s(k)}(S^{p-q}_S\wedge S_T^{q+m}\wedge\sum_s^\infty U_+, E_m) $$ However, this is a presheaf valued in sets, so even if we sheafify, we will get a sheaf valued in sets. Is there some group structure on the Hom sets that I am missing?

Answer 1

In classical homotopy theory, the stable homotopy category $\text{SH}$ is an additive category and so is its motivic analogue $\text{SH}(k)$ (they are even triangulated). One defines the group structure on the Hom-sets using $[X,Y] \simeq [X,\Omega\Sigma Y]$ coming from the equivalence between suspension and loop functors on the stable category (I use the common notation $[X,Y]$ for the Hom-sets in the stable category). Here one then needs to use that maps into loop spaces have a natural group structure for every homotopy category, see here on the nlab for example.

A reference for the statement that the (classical) stable category is additive can be found here on the nlab. I would guess that you should be able to find the corresponding statement in motivic homotopy theory in the original papers by Voevodsky and Morel.