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This article is cited in 3 scientific papers (total in 3 papers) Framed motivic G. A. Garkushaa, I. A. Paninbc, P. Østværdc a Department of Mathematics, Swansea University, United Kingdom b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences c Department of Mathematics, University of Oslo, Oslo, Norway d Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2023, Volume 87, Issue 1, DOI: https://doi.org/10.4213/im9246 
Bibliographic databases:
    In memory of Vladimir Voevodsky § 1. IntroductionThe category $\Gamma$ of correspondences or multivalued functions on finite sets is of fundamental importance in topology [1]. Following Boardman and Vogt [2], Segal’s work on $\Gamma$-spaces give convenient models for $E_{\infty}$-spaces — these being spaces with multiplications that are unital, associative, and commutative up to higher coherent homotopies — and for infinite loop spaces. Segal applied his ideas to prove the celebrated Barratt–Priddy–Quillen theorem, identifying the group completion of the disjoint union $\bigsqcup_n B\Sigma_n$ of classifying spaces of symmetric groups with the infinite loop space of $\mathbb S$ (the topological sphere). Soon afterwards, Bousfield and Friedlander carried out their homotopical identification of connective spectra and $\Gamma$-space, which was an early striking success in the development of stable homotopy theory [3]. Moreover, $\Gamma$-spaces have the advantage of being simple to define, as well as being intrinsically tied to $K$-theory and topological Hochschild homology [4]. In this paper, we introduce the concept of framed motivic $\Gamma$-spaces together with a few axioms. The main purpose of our set-up is to advance our practical understanding of infinite loop spaces along with new viewpoints on connective and very effective spectra in the algebro-geometric setting of ${\mathbb A}^1$-homotopy theory [5], [6]. Voevodsky [7] envisioned this new direction of development in his work on framed correspondences in motivic homotopy theory. Working over a field $k$, our approach combines Segal’s category $\Gamma$ with Voevodsky’s symmetric monoidal category $\mathrm{Sm}/k_+$ of framed correspondences of level zero [7], which is a slight enlargement of $\mathrm{Sm}/k$, the category of smooth separated schemes of finite type over $\operatorname{Spec}(k)$. Recall from [8] that a framed motivic space is a pointed simplicial Nisnevich sheaf on the category of framed correspondences $\mathrm{Fr}_+(k)$. As noted in § 2, $\mathrm{Sm}/k_+$, the opposite category $\Gamma^{\mathrm{op}}$ of pointed finite sets and the category of framed motivic spaces $\mathcal{M}^{\mathsf{fr}}$, are enriched in the closed symmetric monoidal category of pointed motivic spaces $\mathcal{M}$ [9]. With respect to the said enrichments, we shall consider “continuous functors in two variables” for the monoidal product of $\Gamma^{\mathrm{op}}$ and $\mathrm{Sm}/k_+$ taking values in framed motivic spaces
$$
\begin{equation}
\mathcal X \colon \Gamma^{\mathrm{op}} \boxtimes \mathrm{Sm}/k_+ \to \mathcal{M}^{\mathsf{fr}}
\end{equation}
\tag{1}
$$
and call them framed motivic $\Gamma$-spaces (see Defintion 2.1). We should note that there is a canonically induced faithful functor obtained from the composite
$$
\begin{equation}
\mathrm{Sm}/k \to \mathrm{Sm}/k_+ \to \mathrm{Fr}_+(k).
\end{equation}
\tag{3}
$$
The definition of framed correspondences, as introduced in [7], employs an algebro-geometric analogue of a framing on the stable normal bundle of a manifold. The corresponding prerequisite will be recalled in (1), (2) and (3) (see § 2). In our quest to carry over Segal’s programme for $\Gamma$-spaces to $\mathbb{A}^1$-homotopy theory we begin by formulating some homotopical axioms for framed motivic $\Gamma$-spaces. These axioms concern both of the variables $\Gamma^{\mathrm{op}}$ and $\mathrm{Sm}/k_+$ in (1). Informally speaking, the pointed finite sets accounts for the $S^1$-suspension whereas the framed correspondences accounts for the $\mathbb G_m$-suspension in stable motivic homotopy theory. We may and will view $\Gamma^{\mathrm{op}}$ as the full subcategory of pointed finite sets with objects $n_+=\{0,\dots,n\}$, pointed at $0$ for every integer $n\geqslant 0$. Every $\Gamma$-space gives rise to a simplicial functor and hence an associated $S^1$-spectrum (for details, see Ch. 2 in [4]). Similarly in the motivic setting (see (9)) we show that every $U\in \mathrm{Sm}/k_+$ and $\mathcal X$ as in (1), give rise to a presheaf of $S^1$-spectra $\mathcal X(\mathbb S,U)$. We refer to [10] for a comprehensive introduction to the homotopical algebra of such presheaves. In axioms below we employ the notions of local equivalences for simplicial presheaves (see Ch. 4 in [10]) and stable local equivalences for presheaves of $S^1$-spectra (see Ch. 10 in [10]). For $n\geqslant 0$ and every finitely generated field extension $K/k$, we write $\widehat{\Delta}^n_{K/k}$ for the semilocalization of the standard algebraic $n$-simplex
$$
\begin{equation*}
\Delta^n_K = \operatorname{Spec}\bigl(K[x_0,\dots,x_n]/(x_0+\dots+x_n-1)\bigr)
\end{equation*}
\notag
$$
with closed points $v_{0},\dots,v_n\in \Delta^n_K$ as vertices (see [6], § 3) for the colimit preserving realization functor from simplicial sets to Nisnevich sheaves. We recall that $v_{i}$ is a closed subscheme of $\Delta^n_K$, as defined by $x_{j}=0$, for $j\neq i$, $0\leqslant i\leqslant n$. Following § 2 in [11], we write $\widehat{\Delta}^{\bullet}_{K/k}$ for the corresponding cosimplicial semilocal scheme. We are now ready to introduce the main objects of study of our paper. Axiom. A framed motivic $\Gamma$-space $\mathcal X$ is called special if the following conditions (1)–(5) are met. 1. We have $\mathcal X(0_+,U)=\ast=\mathcal X(n_+,\varnothing)$ for all $n\geqslant 0$ and $U\in \mathrm{Sm}/k_+$, while for all $n\geqslant 1$ and non-empty $U\in \mathrm{Sm}/k_+$, the naturally induced morphism
$$
\begin{equation*}
\mathcal X(n_+,U) \to \mathcal X(1_+,U) \times \overset{n}{\cdots} \times \mathcal X(1_+,U)
\end{equation*}
\notag
$$
is a local equivalence of pointed motivic spaces. 2. For all $n\geqslant 0$ and $U\in \mathrm{Sm}/k_+$ the framed presheaf of stable homotopy groups is ${\mathbb A}^1$-invariant, radditive and $\sigma$-stable (see Remark 1.1).3. (Cancellation.) Let $\mathbb{G}$ denote the cone of the $1$-section $\mathrm{Spec}(k)\to\mathbb G_m $ in the category $\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$. For all $n\geqslant 0$ and $U\in \mathrm{Sm}/k_+$, the canonical morphism
$$
\begin{equation*}
\mathcal X(\mathbb S,\mathbb{G}^{\wedge n}\times U) \to \underline{\mathrm{Hom}} (\mathbb{G},\mathcal X(\mathbb S,\mathbb G^{\wedge n+1}\times U))
\end{equation*}
\notag
$$
is a stable local equivalence 4. (${\mathbb A}^1$-invariance.) For all $U\in \mathrm{Sm}/k_+$, the induced morphism
$$
\begin{equation*}
\mathcal X(\mathbb S,U\times\mathbb A^1)\to\mathcal X(\mathbb S,U)
\end{equation*}
\notag
$$
is a naturally induced stable local equivalence. 5. (Nisnevich excision.) For every elementary Nisnevich square in the category $\mathrm{Sm}/k$ there is a homotopy cartesian square in the stable local model structure.Moreover, a special framed motivic $\Gamma$-space $\mathcal X$ is called very effective if (6) holds and, and very special, if (7) holds. 6. (Suslin contractibility.) For all $U\in \mathrm{Sm}/k_+$ and any finitely generated field extension $K/k$, the geometric realization of the simplicial $S^1$-spectrum
$$
\begin{equation*}
\mathcal X(\mathbb S,\mathbb G\times U)(\widehat{\Delta}^{\bullet}_{K/k})
\end{equation*}
\notag
$$
is stably equivalent to the trivial spectrum. 7. (Grouplikeness.) For all $U\in \mathrm{Sm}/k_+$, the Nisnevich sheaf $\pi^{\mathsf{nis}}_{0}\mathcal X(1_+,U)$, which is associated with the presheaf of connected components on $\mathrm{Sm}/k$, takes values in abelian groups.Remark 1.1. The reader will recognize axioms 1 and 7 as sheaf versions of special and very special Segal $\Gamma$-spaces respectively [3], [1]. Axiom 2 makes use of the assumption that $\mathcal X$ is a framed motivic $\Gamma$-space. A framed presheaf $\mathcal F$ is $\sigma$-stable if $\mathcal F(\sigma_{V})=\operatorname{id}_{\mathcal F(V)}$ for all $V\in\mathrm{Sm}/k$. Here, the level $1$ explicit framed correspondence $(\{0\}\times V,{\mathbb A}^1\times V,\operatorname{pr}_{\mathbb A^1}, \operatorname{pr}_{V})\in\mathrm{Fr}_{1}(V,V)$ defines a map $\sigma_{V}\colon V\to V$ in $\mathrm{Fr}_+(k)$ (see § 2 in [8]). A presheaf $\mathcal F$ on $\mathrm{Sm}/k$ is radditive if $\mathcal F(\varnothing)=*$ and $\mathcal F(X_1\sqcup X_2)=\mathcal F(X_1)\times\mathcal F(X_2)$ for all $X_1,X_2\in\mathrm{Sm}/k$. In axiom 3, $\mathbb{G}$ is a simplicial object in $\mathrm{Sm}/k_+$ with smash product $\mathbb G^{\wedge n}$, formed in $\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$ (see Notation 8.1 in [8]). Axioms 2–5 are concerned with presheaves of $S^1$-spectra, as in [10], Part IV. Axiom 6 traces back to Suslin’s work on rationally contractible presheaves in [11] (see also [12], [13]). Example 1.1. An example of a quintessential special framed motivic $\Gamma$-space is given by The evaluation functor in (15) associates with every framed motivic $\Gamma$-space $\mathcal X$ an object in the category of framed motivic spectra in the sense of Definition 2.1 in [13]:
$$
\begin{equation*}
\mathcal X_{S^1,\mathbb G} \in \mathbf{Sp}^{\mathsf{fr}}_{S^1,\mathbb G}(k).
\end{equation*}
\notag
$$
Recall that the triangulated category of framed bispectra $\mathbf{SH}^{\mathsf{fr}}_{\mathsf{nis}}(k)$, whose objects are those of $\mathbf{Sp}^{\mathsf{fr}}_{S^1,\mathbb G}(k)$, is equivalent to the stable motivic homotopy category $\mathbf{SH}(k)$ via the identity with quasi-inverse equal to the big framed motive functor (see Theorem 2.2 in [13]). The big framed motive functor is closely related to Example 1.1 (for details, we refer to [8], § 12). For the purposes of this paper, it is not necessary to discuss model structures on framed motivic $\Gamma$-spaces. Our next definition is inspired by Segal’s homotopy category of $\Gamma$-spaces [1]. Definition 1.1. The homotopy category of framed motivic $\Gamma$-spaces is defined as the category whose objects are special framed motivic $\Gamma$-spaces and for which the morphisms are given by In § 3, we will discuss how $\mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)$ is related to the unstable pointed motivic homotopy category $\mathbf{H}(k)$ and to connective motivic spectra $\mathbf{SH}(k)_{\geqslant 0}$ via the commutative (up to equivalence of functors) diagram of adjunctions: Here, $\mathcal X\in \mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)$ is mapped to its underlying motivic space $\mathcal X (1_+,\mathsf{pt})\in \mathbf{H}(k)$, and to its framed motivic spectrum $\mathcal X_{S^1,\mathbb G}\in\mathbf{SH}(k)_{\geqslant 0}$ under the equivalence between $\mathbf{SH}^{\mathsf{fr}}_{\mathsf{nis}}(k)$ and $\mathbf{SH}(k)$ in [13]. We refer to Remark 3.1 for the definition of $\Gamma\mathbb M_{\mathsf{fr}}$, a version of the big framed motive functor introduced in § 12 of [8].Theorem 1.1. For every infinite perfect field $k$, there is an equivalence of categories Its quasi-inverse functor $\mathbf{SH}(k)_{\geqslant 0} \xrightarrow{\simeq} \mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)$ maps $\mathcal E \in \mathbf{SH}(k)_{\geqslant 0}$ to an explicitly constructed framed motivic $\Gamma$-space $\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}} \in \mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)$. Let $\mathbf{SH}^{\mathsf{veff}}(k)$ be the full subcategory of $\mathbf{SH}(k)$ generated under homotopy colimits and extensions by motivic ${\mathbb P}^1$-suspension spectra of smooth schemes. This category is of interest since it gives rise to the very effective slice filtration introduced in [14]. We note $\mathbf{SH}^{\mathsf{veff}}(k)$ is contained in the triangulated category $\mathbf{SH}(k)_{\geqslant 0}$, which is generated under homotopy colimits and extensions by motivic ${\mathbb P}^1$-suspension spectra $\Sigma^{p,q}U_+$ where $p\geqslant q$ and $U\in\mathrm{Sm}/k$, and closed under homotopic colimits and extensions. We shall study $\mathbf{SH}^{\mathsf{veff}}(k)$ from the point of view of framed motivic $\Gamma$-spaces. Definition 1.2. The homotopy category of very effective framed motivic $\Gamma$-spaces is the full subcategory of $\mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)$ comprised of very effective special framed motivic $\Gamma$-spaces. We will now show that Axiom 6 on Suslin contractibility of special framed motivic $\Gamma$-spaces captures precisely the difference between $\mathbf{SH}^{\mathsf{veff}}(k)$ and $\mathbf{SH}(k)_{\geqslant 0}$. Finally, we employ Axiom 7 in our recognition principle for infinite motivic loop spaces. Theorem 1.3. For every infinite perfect field $k$ and every $\mathcal E\in \mathbf{SH}(k)$, there exists a very special framed motivic $\Gamma$-space $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}$ and a local equivalence of pointed motivic spaces: Moreover, if $\mathcal X$ is a very special framed motivic $\Gamma$-space, then $\mathcal X(1_+,\mathsf{pt})$ is an infinite motivic loop space. Guide to the paper. For the convenience of the reader we begin § 2 by reviewing background on enriched categories with the aim at introducing framed motivic $\Gamma$-spaces. As prime examples, we discuss the motivic sphere spectrum $\mathbf{1}$, algebraic cobordism $\mathbf{MGL}$, motivic cohomology $\mathbf{MZ}$, and Milnor–Witt motivic cohomology $\widetilde{\mathbf{M}}\mathbf{Z}$. Our main results (Theorems 1.1–1.3) are formulated in § 3. Finally, in § 4 we record some novel homotopical properties of framed motivic $\Gamma$-spaces. Throughout the paper, we employ the following notation: $k$ – infinite perfect field of exponential characteristic $e$; $\mathsf{pt}$ – the scheme $\mathrm{Spec}(k)$; $\mathrm{Sm}/k$ – smooth separated schemes of finite type; $\mathrm{Sm}/k_+$ – framed correspondences of level zero; $\mathrm{Shv}_{\bullet}(\mathrm{Sm}/k)$ – closed symmetric monoidal category of pointed Nisnevich sheaves; $\mathcal{M}=\Delta^{\mathrm{op}}\mathrm{Shv}_{\bullet}(\mathrm{Sm}/k)$ – pointed motivic spaces, that is, pointed simplicial Nisnevich sheaves; $\mathrm{Fr}_+(k)$ – the category of framed Voevodsky correspondences; $\mathrm{Pre}^{\mathsf{fr}}(k)$ – framed presheaves, a.k.a. presheaves of sets on $\mathrm{Fr}_+(k)$; $i\colon \mathrm{Sm}/k\to\mathrm{Fr}_+(k)$ – the composite functor $\mathrm{Sm}/k\to\mathrm{Sm}/k_+\to\mathrm{Fr}_+(k)$; $S^{s,t}$, $\Omega^{s,t}$, $\Sigma^{s,t}$ – motivic $(s,t)$-sphere, loop space, and suspension; $\mathbf{S}_{\bullet}$ – pointed simplicial sets. Our standard convention for motivic spheres is that $S^{2,1}\simeq\mathbb{P}^1\simeq T$ and $S^{1,1}\simeq\mathbb{A}^1\setminus \{0\}$, as in [5]. Our approach in this paper is a homage to Segal’s work on categories and cohomology theories [1]. Along the same line we use minimal machinery to achieve concrete models for infinite motivic loop spaces and motivic spectra with prescribed properties. Based on Voevodsky’s notes [7], the machinery of framed motives is developed in [8]. As an application, explicit computations of infinite motivic loop spaces are given as follows: namely, $\Omega_{\mathbb P^1}^{\infty}\Sigma_{\mathbb P^1}^{\infty}A$, $A\in\mathcal M$, is locally equivalent to the space $C_{\ast}\mathrm{Fr}(A^c)^{\mathrm{gp}}$ (“gp” stands for group completion), where $A^c$ is a projective cofibrant replacement of $A$ (see § 10 in [8]). Based on [8], [15]–[17], a motivic recognition principle for infinite motivic loop spaces using the language of infinity categories is given in [18]. § 2. Framed motivic $\Gamma$-spacesWe refer to [19] and [20] for the projective motivic model structure on the closed symmetric monoidal category of pointed motivic spaces $\mathcal{M}$. This model structure is combinatorial, proper, simplicial, symmetric monoidal, and weakly finitely generated. Let $\Delta[\,{\bullet}\,]$ be the standard cosimplicial simpicial set $n\mapsto\Delta[n]$. If there is no likelihood of confusion, we sometimes regard it as a cosimplicial smooth scheme, where each $\Delta[n]$ is regarded as the disjoint union $\bigsqcup_{\Delta[n]}\mathsf{pt}$. The simplicial function $\mathrm{Hom}$-object between pointed motivic spaces $A$ and $B$ is given by
$$
\begin{equation*}
\mathbf{S}_{\bullet}(A,B)=\mathrm{Hom}_{\mathcal{M}}(A\wedge \Delta[\,{\bullet}\,]_+,B)=\mathrm{Hom}_{\mathcal{M}} \bigl(A,B(\Delta[\,{\bullet}\,]\times-)\bigr).
\end{equation*}
\notag
$$
For every $U\in\mathrm{Sm}/k$, the Yoneda lemma identifies $\mathbf{S}_{\bullet}(U_+,A)$ with the pointed simplicial set of sections $A(U)$. Recall that $A\in\mathcal{M}$ is finitely presentable if the functor $\mathrm{Hom}_{\mathcal{M}}(A,-)$ preserves directed colimits. For example, the representable pointed motivic space $U_+$ is finitely presentable for every $k$-smooth scheme $U\in \mathrm{Sm}/k$. A collection $\mathcal C$ of finitely presentable pointed motivic spaces can be enriched in $\mathcal{M}$ by means of the $\mathcal{M}$-enriched $\mathrm{Hom}$-functor:
$$
\begin{equation}
\begin{aligned} \, [A,B](X) &:= \underline{\mathrm{Hom}}_{\,\mathcal{M}}(A,B)(X)\,{=}\, \mathbf{S}_{\bullet}(A\wedge X_+,B) \,{=}\,\mathrm{Hom}_{\mathcal{M}} \bigl(A\wedge\Delta[\,{\bullet}\,]_+,B(X\times-)\bigr) \nonumber \\ &\,=\mathrm{Hom}_{\mathcal{M}} \bigl(A,B(X\times\Delta[\,{\bullet}\,]\times-)\bigr), \qquad A,B\in\mathcal C,\quad X\in \mathrm{Sm}/k. \end{aligned}
\end{equation}
\tag{8}
$$
The enriched composition in $\mathcal C$ is inherited from the enriched composition in $\mathcal{M}$. We write $[\mathcal C,\mathcal{M}]$ for the category of $\mathcal{M}$-enriched covariant functors from $\mathcal C$ to $\mathcal{M}$ (refer to [21], § 4, for its projective model structure), in which the weak equivalences and fibrations are defined pointwise. Voevodsky [7] defined the morphisms in $\mathrm{Sm}/k_+$, by setting
$$
\begin{equation*}
\mathrm{Sm}/k_+(X,Y) := \mathrm{Hom}_{\mathrm{Shv}_{\bullet} (\mathrm{Sm}/k)}(X_+,Y_+), \qquad X,Y\in \mathrm{Sm}/k.
\end{equation*}
\notag
$$
In case $X$ is connected we have $\mathrm{Sm}/k_+(X,Y)=\mathrm{Hom}_{\mathrm{Sm}/k}(X,Y)_+$ by Example 2.1 of [7]. Lemma 2.1. With the above notation, the constant simplicial sets are identified, where $U,V,X\in \mathrm{Sm}/k$. Proof. By definition, we have It is evident that $\mathrm{Hom}_{\mathcal{M}}((U\times X)_+,V_+)= \mathrm{Sm}/k_+(U\times X,V)$. This proves the lemma. Remark 2.1. In fact, $\mathrm{Sm}/k_+(-,V)$, $V\in \mathrm{Sm}/k$, is the Nisnevich sheaf associated with the presheaf $U\mapsto\mathrm{Hom}_{\mathrm{Sm}/k}(U,V)\sqcup \mathsf{pt}$. Our first example is Segal’s category $\Gamma^{\mathrm{op}}$ of pointed finite sets and pointed maps. Example 2.1. As in § 5 of [8], we view $\Gamma^{\mathrm{op}}$ as a full subcategory of $\mathcal{M}$ by sending $K\in\Gamma^{\mathrm{op}}$ to $(\bigsqcup_{K\setminus\ast} \mathsf{pt})_+$, where the coproduct is indexed by the non-based elements in $K$. This turns $\Gamma^{\mathrm{op}}$ into a symmetric monoidal $\mathcal{M}$-category. Hence, $[\Gamma^{\mathrm{op}},\mathcal{M}]$ is a closed symmetric monoidal category by [22]. We claim that $[\Gamma^{\mathrm{op}},\mathcal{M}]$ can be identified with the category $\Gamma\mathcal{M}$ of covariant functors from $\Gamma^{\mathrm{op}}$ to $\mathcal{M}$, sending $0_+$ to the basepoint $*$ of $\mathcal M$. In this case, $\mathcal C=\bigl\{\bigsqcup_{K\setminus *} \mathsf{pt}\bigm| K\in \Gamma^{\mathrm{op}}\bigr\}$. An $\mathcal{M}$-enriched functor $\mathcal X\in[\Gamma^{\mathrm{op}},\mathcal{M}]$ sends $K\in\Gamma^{\mathrm{op}}$ to $\mathcal X\bigl(\bigsqcup_{K\setminus *} \mathsf{pt}\bigr)\in\mathcal M$, and for $K,L\in\Gamma^{\mathrm{op}}$, there is a morphism
$$
\begin{equation*}
\alpha_{K,L} \colon \mathcal X\biggl(\bigsqcup_{K\setminus *} \mathsf{pt}\biggr) \bigwedge_{\mathcal{M}}\biggl[\bigsqcup_{K\setminus *} \mathsf{pt},\bigsqcup_{L\setminus *}\mathsf{pt}\biggr] \to \mathcal X\biggl(\bigsqcup_{L\setminus *}\mathsf{pt}\biggr).
\end{equation*}
\notag
$$
Here, the motivic space $\bigl[\bigsqcup_{K\setminus*}\mathsf{pt}, \bigsqcup_{L\setminus*}\mathsf{pt}\bigr]$ is given by
$$
\begin{equation*}
U \mapsto \Gamma^{\mathrm{op}}\biggl(\bigsqcup_{K\times n(U)_+\setminus (*,+)}\mathsf{pt},\bigsqcup_{L\setminus*}\mathsf{pt}\biggr),
\end{equation*}
\notag
$$
where $n(U)$ is the number of connected components of $U \in \mathrm{Sm}/k$, and $n(U)_+ = \{0,1,\dots,n{(U)}\}$. Since $\mathcal X\in[\Gamma^{\mathrm{op}},\mathcal{M}]$ takes values in simplicial sheaves it follows that
$$
\begin{equation*}
\mathcal X(K)(U) = \mathcal X(K)(U_1)\times\overset{n(U)}{\cdots} \times\mathcal X(K)(U_{n{(U)}}),
\end{equation*}
\notag
$$
and consequently, we have
$$
\begin{equation*}
\alpha_{K,L}(U) = \alpha_{K,L}(U_1)\times\overset{n(U)}{\cdots}\times\alpha_{K,L}(U_{n{(U)}}).
\end{equation*}
\notag
$$
To a morphism $f\colon K\to L$ in $\Gamma^{\mathrm{op}}$ we associate the morphism $\mathcal X\bigl(\bigsqcup_{K\setminus *}\mathsf{pt}\bigr)\to \mathcal X\bigl(\bigsqcup_{L\setminus *}\mathsf{pt}\bigr)$ with $U$-sections Clearly, this yields the identification of $[\Gamma^{\mathrm{op}},\mathcal{M}]$ with pointed functors from $\Gamma^{\mathrm{op}}$ to $\mathcal{M}$.We are passing to the definition of the category of framed motivic spaces $\mathcal{M}^{\mathsf{fr}}$ and to its natural enrichment over $\mathcal{M}$. Let $\mathrm{Fr}_+(k)$ be the category of framed correspondences, as in § 2 of [8]. Let $\mathrm{Pre}^{\mathsf{fr}}(k)$ be the category of framed presheaves, that is, the category of presheaves of sets on $\mathrm{Fr}_+(k)$. Let $i\colon \mathrm{Sm}/k\to\mathrm{Sm}/k_+\to\mathrm{Fr}_+(k)$ be the composite functor. Recall from § 2 of [8] that a framed Nisnevich sheaf on $\mathrm{Sm}/k$ is a framed presheaf such that its restriction to $\mathrm{Sm}/k$ via the functor $i$ is a Nisnevich sheaf. Let $\mathrm{Shv}_\bullet^{\mathsf{fr}}(k)$ denote the category of pointed framed Nisnevich sheaves. The morphisms in this category are just morphisms of pointed framed presheaves. The category of framed motivic spaces $\mathcal M^{\mathsf{fr}}$ is the category of simplicial objects in $\mathrm{Shv}_\bullet^{\mathsf{fr}}(k)$. There is a canonically induced faithful functor $\iota\colon\mathcal{M}^{\mathsf{fr}}\to\mathcal{M}$, obtained from the composite $i\colon\mathrm{Sm}/k\to\mathrm{Sm}/k_+\to\mathrm{Fr}_+(k)$. Following § 6 of [7], there is a natural pairing $\mathrm{Sm}/k_+ \times \mathrm{Fr}_+(k)\xrightarrow{\otimes} \mathrm{Fr}_+(k)$ taking $(X,Y)$ to $X\times Y$ and $(f,\alpha)$ to $f\times \alpha$. In what follows, this pairing will be used systematically without special mention. We also use it in the natural enrichment of $\mathcal{M}^{\mathsf{fr}}$ over $\mathcal M$. First, with each framed Nisnevich sheaf $\mathcal F$ and each $X\in \mathrm{Sm}/k_+$ we associate the framed Nisnevich sheaf $\mathcal F(X\times -)$. In detail, given $\alpha\in \mathrm{Fr}_n(U',U)$, we put $\alpha^*\colon \mathcal F(X\times U)\to \mathcal F(X\times U')$ to be $(\operatorname{id}_X\times \alpha)^*$. If $\mathcal F$ is a pointed framed Nisnevich sheaf, then the framed Nisnevich sheaf $\mathcal F(X\times -)$ is also pointed. Second, every morphism $f\colon X'\to X$ in $\mathrm{Sm}/k_+$ induces a morphism of framed sheaves $f^*\colon \mathcal F(X\times -)\to \mathcal F(X'\times -)$. Namely, if $U\in \mathrm{Fr}_+(k)$, one sets $f^*\colon \mathcal F(X\times U)\to \mathcal F(X'\times U)$ to be $(f\times \operatorname{id}_U)^*$. If $\mathcal F$ is a pointed framed Nisnevich sheaf, then the morphism of framed sheaves $f^*\colon \mathcal F(X\times -)\to \mathcal F(X'\times -)$ is a morphism of pointed framed Nisnevich sheaves. Finally, similarly to (8), $\mathcal{M}^{\mathsf{fr}}$ is naturally enriched over $\mathcal M$. Namely,
$$
\begin{equation*}
{\mathcal{M}}(A,B)(X):=\mathrm{Hom}_{\mathcal{M}^{\mathsf{fr}}} \bigl(A,B(X\times\Delta[\,{\bullet}\,]\times-)\bigr), \qquad A,B\in\mathcal{M}^{\mathsf{fr}}, \quad X\in \mathrm{Sm}/k.
\end{equation*}
\notag
$$
The enriched composition in $\mathcal{M}^{\mathsf{fr}}$ is inherited from the enriched composition in $\mathcal M$. Our second example is Voevodsky’s category of framed correspondences of level zero. Example 2.2. We enrich $\mathrm{Sm}/k_+$ in $\mathcal{M}$ by setting The monoidal product $\Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+$ is the $\mathcal{M}$-category with objects $\mathrm{Ob}\Gamma^{\mathrm{op}}\times \mathrm{Ob}\mathrm{Sm}/k_+$ and $\mathcal{M}$-morphisms Note that $\Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+$ is a symmetric monoidal $\mathcal{M}$-category.Definition 2.1. 1) A motivic $\Gamma$-space is an $\mathcal{M}$-enriched functor in two variables $\mathcal X\colon \Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+\to \mathcal{M}$. 2) A framed motivic $\Gamma$-space is an $\mathcal{M}$-enriched functor $\mathcal X\colon \Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+\to \mathcal{M}^{\mathsf{fr}}$. Remark 2.2. Let $\Gamma^{\mathrm{op}}\times\mathrm{Sm}/k_+$ denote the underlying category of the $\mathcal M$-category $\Gamma^{\mathrm{op}}\boxtimes\mathrm{Sm}/k_+$. Every motivic $\Gamma$-space $\mathcal X\colon \Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+\to\mathcal{M}$ gives rise to a functor $\mathcal X\colon \Gamma^{\mathrm{op}}\times \mathrm{Sm}/k_+\to\mathcal{M}$, denoted by the same letter. Unravelling the previous definition, a framed motivic $\Gamma$-space is equivalent to giving the following data: – an $\mathcal M$-functor $\mathcal X\colon \Gamma^{\mathrm{op}} \boxtimes \mathrm{Sm}/k_+\to\mathcal{M}$; – a functor $\mathcal X'\colon \Gamma^{\mathrm{op}}\times \mathrm{Sm}/k_+\to\mathcal{M}^{\mathsf{fr}}$; – the induced functor $\mathcal X\colon \Gamma^{\mathrm{op}} \times\mathrm{Sm}/k_+\to\mathcal M$ equals the composite functor $\Gamma^{\mathrm{op}}\times\mathrm{Sm}/k_+\xrightarrow{\mathcal X'} \mathcal M^{\mathsf{fr}}\xrightarrow{\iota}\mathcal M$ such that the canonical morphism factors through $\mathrm{Hom}_{\mathcal M^{\mathsf{fr}}} (\mathcal X'(K,U),\mathcal X'(K,V)(Y\times-))$ for all $K\in\Gamma^{\mathrm{op}}$, $U,V,Y\in\mathrm{Sm}/k_+$.Evaluation FunctorsEvery motivic $\Gamma$-space, namely, $\mathcal X\in [\Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+,\mathcal{M}]$ and every $U\in \mathrm{Sm}/k_+$ give rise to an enriched functor $\mathcal X(U)\in [\Gamma^{\mathrm{op}},\mathcal{M}]$. In Example 2.1 we identified $\mathcal X(U)$ with the datum of a pointed functor from $\Gamma^{\mathrm{op}}$ to $\mathcal{M}$. Following Example 2.1.2.1 in [4], by the sphere spectrum we mean the inclusion $\mathbb S\colon \Gamma^{\mathrm{op}}\hookrightarrow \mathbf{S}_{\bullet}$. By taking the left Kan extension along the sphere spectrum $\mathbb S\colon \Gamma^{\mathrm{op}}\hookrightarrow \mathbf{S}_{\bullet}$, we obtain the evaluation functor with values in motivic $S^1$-spectra
$$
\begin{equation}
\begin{gathered} \, \mathsf{ev}_{S^1} \colon [\Gamma^{\mathrm{op}},\mathcal{M}] \to \mathbf{Sp}_{S^1}(k), \\ \mathcal X(U)\mapsto \mathcal X(\mathbb S,U) = (\mathcal X(S^{0})(U),\mathcal X(S^1)(U),\mathcal X(S^2)(U),\dots). \end{gathered}
\end{equation}
\tag{9}
$$
We refer to $\mathcal X(\mathbb S,\mathsf{pt})$ as the underlying motivic $S^1$-spectrum of $\mathcal X$. On the other hand, for $K\in\Gamma^{\mathrm{op}}$ we have an enriched functor $\mathcal X(K)\in [\mathrm{Sm}/k_+,\mathcal{M}]$ (see Example 2.2). Moreover, for $U,V\in \mathrm{Sm}/k_+$, there are natural morphisms in $\mathcal{M}$
$$
\begin{equation*}
V_+ \to [U,U\times V]\to\underline{\mathrm{Hom}}_{\,\mathcal{M}} \bigl(\mathcal X(K)(U),\mathcal X(K)(U\times V)\bigr).
\end{equation*}
\notag
$$
By adjunction, we obtain the morphisms
$$
\begin{equation}
\begin{gathered} \, \mathcal X(K)(U)\wedge V_+\to\mathcal X(K)(U\times V), \\ \mathcal X(K)(U)\to\underline{\mathrm{Hom}}_{\,\mathcal M} \bigl(V_+,\mathcal X(K)(U\times V)\bigr). \end{gathered}
\end{equation}
\tag{10}
$$
The simplices of $\mathbb{G}\in\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$ consist of finite disjoint unions $\mathbb{G}^{\bigsqcup_{<\infty}}_m$ of copies of the multiplicative group scheme $\mathbb{G}_m$ and $\mathsf{pt}$. Namely, the simplices are $\mathbb{G}_m$, $\mathbb{G}_m\sqcup\mathsf{pt}$, $\mathbb{G}_m\sqcup\mathsf{pt}\sqcup\mathsf{pt}$, $\dots$ (we also refer the reader to [8], Notation 8.1). As a special case of (10), we have
$$
\begin{equation}
\begin{gathered} \, \mathcal X(K)(U)\wedge (\mathbb{G}^{\bigsqcup_{<\infty}}_m)_+ \to \mathcal X(K)\bigl(U\times \mathbb{G}^{\bigsqcup_{<\infty}}_m\bigr), \\ \mathcal X(K)(U)\to\underline{\mathrm{Hom}}_{\,\mathcal M} \bigl(\bigl(\mathbb{G}^{\bigsqcup_{<\infty}}_m\bigr)_+, \mathcal X(K)\bigl(U\times \mathbb{G}^{\bigsqcup_{<\infty}}_m\bigr)\bigr). \end{gathered}
\end{equation}
\tag{11}
$$
For the smash powers of $\mathbb{G}$, we define the morphisms
$$
\begin{equation}
\begin{gathered} \, \mathcal X(K)(\mathbb{G}^{\wedge n})\wedge\mathbb{G}_+\to \mathcal X(K)(\mathbb G^{\wedge n+1}), \\ \mathcal X(K)(\mathbb{G}^{\wedge n})\to\underline{\mathrm{Hom}}_{\,\mathcal M} \bigl(\mathbb{G}_+,\mathcal X(K)(\mathbb G^{\wedge n+1})\bigr) \end{gathered}
\end{equation}
\tag{12}
$$
to be the geometric realization of
$$
\begin{equation*}
\begin{gathered} \, l \mapsto\bigl\{\mathcal X(K)((\mathbb G^{\wedge n})_l)\wedge (\mathbb{G}_+)_l \to \mathcal X(K)((\mathbb G^{\wedge (n+1)})_l)\bigr\}, \\ l \mapsto\bigl\{\mathcal X(K)((\mathbb G^{\wedge n})_l) \to \underline{\mathrm{Hom}}_{\,\mathcal M} \bigl((\mathbb{G}_+)_l, \mathcal X(K)((\mathbb G^{\wedge (n+1)})_l)\bigr)\bigr\}, \end{gathered}
\end{equation*}
\notag
$$
obtained from (11). Using (12), we obtain the evaluation functor with values in motivic $\mathbb{G}$-spectra
$$
\begin{equation}
\begin{gathered} \, \mathsf{ev}_\mathbb{G} \colon [\mathrm{Sm}/k_+,\mathcal{M}] \to \mathbf{Sp}_{\mathbb{G}}(k), \\ \mathcal X(K)\mapsto \bigl(\mathcal X(K)(\mathsf{pt}), \mathcal X(K)(\mathbb{G}),\mathcal X(K)(\mathbb G^{\wedge 2}),\dots\bigr). \end{gathered}
\end{equation}
\tag{13}
$$
We refer to [9], Ch. 3, § 2.3, for a discussion of the category $\mathbf{Sp}_{S^1,\mathbb{G}}(k)$ of motivic $(S^1,\mathbb{G})$-bispectra. Its associated homotopy category is equivalent to $\mathbf{SH}(k)$. Combining (9) and (13), we obtain the evaluation functor
$$
\begin{equation}
\begin{gathered} \, \mathsf{ev}_{S^1,\mathbb{G}} \colon [\Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+,\mathcal{M}] \to \mathbf{Sp}_{S^1,\mathbb{G}}(k), \\ \mathcal X \mapsto \mathcal X_{S^1,\mathbb G} = \mathsf{ev}_{S^1,\mathbb{G}}(\mathcal X). \end{gathered}
\end{equation}
\tag{14}
$$
More precisely, for $i,j\geqslant 0$ we have
$$
\begin{equation*}
\mathsf{ev}_{S^1,\mathbb{G}}(\mathcal X)_{i,j}= \mathcal X(S^{i},\mathbb G^{\wedge j})\in\mathcal{M}.
\end{equation*}
\notag
$$
The evident structure maps turn $\mathcal X_{S^1,\mathbb G}$ into a motivic $(S^1,\mathbb{G})$-bispectrum. In turn, let $[\Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+,\mathcal{M}^{\mathsf{fr}}]$ denote the category of $\mathcal M$-enriched functors from $\Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+$ to $\mathcal{M}^{\mathsf{fr}}$. Following Definition 2.1, uts objects are the framed motivic $\Gamma$-spaces. If $\mathcal X$ is a framed motivic $\Gamma$-space, then the structure morphisms
$$
\begin{equation*}
\begin{gathered} \, \mathcal X(S^{i},\mathbb G^{\wedge j})\to \underline{\mathrm{Hom}} \bigl(S^1,\mathcal X(S^{i+1},\mathbb G^{\wedge j})\bigr), \\ \mathcal X(S^{i},\mathbb G^{\wedge j})\to\underline{\mathrm{Hom}} \bigl(\mathbb G_+,\mathcal X(S^{i+1},\mathbb G^{\wedge j+1})\bigr) \end{gathered}
\end{equation*}
\notag
$$
are morphisms in $\mathcal M^{\mathsf{fr}}$. Therefore, $\mathcal X_{S^1,\mathbb G}\in \mathbf{Sp}^{\mathsf{fr}}_{S^1,\mathbb{G}}(k)$ is a framed motivic $(S^1,\mathbb G)$-bispectrum in the sense of [13], Definition 2.1. Similarly to (14), we obtain the evaluation functor
$$
\begin{equation}
\begin{gathered} \, \mathsf{ev}_{S^1,\mathbb{G}} \colon [\Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+,\mathcal{M}^{\mathsf{fr}}] \to \mathbf{Sp}_{S^1,\mathbb{G}}^{\mathsf{fr}}(k), \\ \mathcal X \mapsto \mathcal X_{S^1,\mathbb G} = \mathsf{ev}_{S^1,\mathbb{G}}(\mathcal X). \end{gathered}
\end{equation}
\tag{15}
$$
Example 2.3. For every $X\in\mathrm{Sm}/k$, we can form the motivic $\Gamma$-space with sections There is a natural morphism of motivic $\Gamma$-spaces
$$
\begin{equation}
\mathrm{Sm}/k_+\bigl(-,-\otimes (X\times-)\bigr)\to C_{\ast}\mathrm{Fr}\bigl(-,-\otimes (X\times-)\bigr).
\end{equation}
\tag{16}
$$
By Theorem 11.1 in [8], the evaluation functor in (14) takes the morphism in (16) to a stable motivic equivalence. In particular, the special framed motivic $\Gamma$-space $\underline{\mathrm{Hom}}(\mathsf{pt},C_{\ast}\mathrm{Fr})$ is a model for the motivic sphere $\mathbf {1}$. By linearization, we obtain the special framed motivic $\Gamma$-space $\underline{\mathrm{Hom}}(X,C_{\ast}\mathbb Z\mathrm{F})$ with sections
$$
\begin{equation*}
(K,U) \mapsto C_{\ast}\mathbb Z\mathrm{F}\bigl(-,K\otimes (X\times U)\bigr).
\end{equation*}
\notag
$$
The underlying motivic $S^1$-spectrum $\underline{\mathrm{Hom}} (X,C_{\ast}\mathbb Z\mathrm{F})(\mathbb S,\mathsf{pt})$ is the linear framed motive of $X$ (see [8]). Example 2.4. Let $\mathcal E$ be a motivic symmetric Thom $T$- or $T^{2}$-spectrum with bounding constant $d\leqslant 1$ and contractible alternating group action in the sense of § 1 in [23]. The main examples are algebraic cobordism $\mathbf{MGL}$ [6] and the $T^2$-spectra $\mathbf {MSL}$, $\mathbf {MSp}$ in [24] (in all of these cases $d=1$). Under these assumption, there exists a special framed motivic $\Gamma$-space $\underline{\mathrm{Hom}}(X,C_{\ast}\mathrm{Fr}^{\mathcal E})$ with sections Likewise, we obtain the special framed motivic $\Gamma$-space $\underline{\mathrm{Hom}}(X,C_{\ast}\mathbb Z\mathrm{F}^{\mathcal E})$, whose underlying motivic $S^1$-spectrum is the linear $\mathcal E$-framed motive of $X$, as defined in [23], § 9. Example 2.5. Suppose that $\mathbf{A}$ is a strict category of Voevodsky correspondences in the sense of Definition 2.3 in [25], and there exists a functor $\mathrm{Fr}_+(k)\to\mathbf{A}$ which is the identity map on objects. Examples include finite Milnor–Witt correspondences $\widetilde{\mathrm{Cor}}$ [26], finite Voevodsky correspondences $\mathrm{Cor}$ [27], and $K_0^{\oplus}$-correspondences [28]. We define $C_{\ast}\mathbf{A}$ to be the very special framed motivic $\Gamma$-space whose sections are the Suslin complex of the Nisnevich sheaf $\mathbf{A}(-,K\otimes U)^{\mathsf{nis}}$, that is, the motivic $\Gamma$-space of the form Remark 2.3. The motivic $\Gamma$-spaces in Examples 2.3–2.5 share the common trait of factoring through the functor $\otimes\colon \Gamma^{\mathrm{op}}\boxtimes \mathrm{Sm}/k_+\to\mathrm{Sm}/k_+$. § 3. Special framed motivic $\Gamma$-spaces and infinite motivic loop spacesLet $\mathcal E$ be a motivic $(S^1,\mathbb{G})$-bispectrum. Using the $n$th weight motivic $S^1$-spectrum, that is, the motivic $S^1$-spectra $\mathcal E(n)$ of the bispectrum $\mathcal E$, as defined by $\mathcal E(n)_{i}=\mathcal E_{i,n}$, we write $\mathcal E=(\mathcal E(0),\mathcal E(1),\dots)$. For integers $p,n\in\mathbb Z$, let $\pi^{\mathbb A^1}_{p,n}\mathcal E$ be the Nisnevich sheaf on $\mathrm{Sm}/k$, associated with the presheaf
$$
\begin{equation*}
U \mapsto \mathbf{SH}(k)(U_+\wedge S^{p-n}\wedge \mathbb{G}^{\wedge n},\mathcal E).
\end{equation*}
\notag
$$
Recall that $\mathcal E$ is connective if $\pi^{\mathbb A^1}_{p,n}\mathcal E=0$ for all $p<n$. Similarly, a motivic $S^1$-spectrum $\mathcal E\in \mathbf{Sp}_{S^1}(k)$ is connective if $\pi^{\mathbb A^1}_n\mathcal E=0$ for all $n<0$. For a Nisnevich sheaf $F$ of abelian groups on $\mathrm{Sm}/k$, let $F_{-1}$ denote the Nisnevich sheaf given by $U\mapsto\ker(1^{\ast}\colon F(U\times\mathbb G_m)\to F(U))$. Lemma 3.1. A framed motivic $(S^1,\mathbb{G})$-bispectrum $\mathcal E=(\mathcal E(0),\mathcal E(1),\dots)$ in the sense of § 2 in [13] is connective if and only if $\mathcal E(n)$ is a connective motivic $S^1$-spectrum for every $n\geqslant 0$. Proof. Without loss of generality, we may assume that the underlying motivic bispectrum $\mathcal E$ is fibrant (we use here Lemma 2.6] of [13]). Writing $|\,{-}\,|$ for the absolute value, we have $\pi_{p,n}^{\mathbb A^1}\mathcal E=\pi_{p-n}^{\mathsf{nis}}\mathcal E(|n|)$ if $n\leqslant 0$. At the same time, $\pi_{p,n}^{\mathbb A^1}\mathcal E= \pi_{p-n}^{\mathsf{nis}}\Omega_{\mathbb G^{\wedge n}}\mathcal E(0)$ if $n>0$. Here, $\pi_{\ast}^{\mathsf{nis}}$ denotes the Nisnevich sheaf associated with $\pi_{\ast}$. The proof of the sublemma in § 12 of [8] shows that
$$
\begin{equation*}
\pi_{p-n}^{\mathsf{nis}}\Omega_{\mathbb G^{\wedge n}}\mathcal E(0) = \pi_{p-n}^{\mathsf{nis}}\mathcal E(0)_{-n}.
\end{equation*}
\notag
$$
If $\mathcal E$ is connective, then $\pi_{p-n}^{\mathsf{nis}}\mathcal E(|n|)=0$ for all $n\leqslant 0$ and $p<n$. In particular, for all $s>0$ and $n\leqslant 0$ the sheaf $\pi_{-s}^{\mathsf{nis}}\mathcal E(|n|)$ is trivial. The converse implication is evident. This proves Lemma 3.1. Recall that $\mathbf{Sp}_{S^1}(k)$ is naturally enriched in $\mathcal M$ (see the proof of Theorem 6.3 in [29]). In fact, for $\mathcal E,\mathcal F\in\mathbf{Sp}_{S^1}(k)$ one defines $\mathcal M(\mathcal E,\mathcal F)$ as the equalizer of the diagram
$$
\begin{equation}
\prod_n\mathcal M(\mathcal E_n,\mathcal F_n)\quad \Longrightarrow \quad \prod_n\mathcal M\bigl(\mathcal E_n, \underline{\mathrm{Hom}}_{\,\mathcal M}(S^1,\mathcal F_{n+1})\bigr).
\end{equation}
\tag{17}
$$
Here, we employ the morphism $\mathcal M(\mathcal E_n,\mathcal F_n)\to \mathcal M(\mathcal E_n,\underline{\mathrm{Hom}}_{\,\mathcal M} (S^1,\mathcal F_{n+1}))$ induced by the adjoint of the structure maps of $\mathcal F$, and the canonically induced morphism
$$
\begin{equation*}
\mathcal M(\mathcal E_{n+1},\mathcal F_{n+1}) \to \mathcal M(\mathcal E_n\wedge S^1,\mathcal F_{n+1}) \cong \mathcal M\bigl(\mathcal E_n, \underline{\mathrm{Hom}}_{\,\mathcal M}(S^1,\mathcal F_{n+1})\bigr).
\end{equation*}
\notag
$$
We shall refer to $\mathbf{Sp}_{S^1}([\mathrm{Sm}/k_+,\mathcal M])$ as the category of spectral functors (see [13], § 5). The objects are $S^1$-spectra in the closed symmetric monoidal $\mathcal M$-category $[\mathrm{Sm}/k_+,\mathcal M]$, introduced in Example 2.2. Similarly to (13) (see [13], § 5, (3)) there exists an evaluation functor
$$
\begin{equation*}
\mathsf{ev}_\mathbb{G}\colon \mathbf{Sp}_{S^1}([\mathrm{Sm}/k_+,\mathcal M]) \to \mathbf{Sp}_{S^1,\mathbb G}(k).
\end{equation*}
\notag
$$
We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. For $\mathcal X\in \mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)$ and $n\geqslant 0$, by Definition 1.1 the geometric realization functor furnishes the associated $\mathcal M$-enriched functor
$$
\begin{equation*}
\mathcal X(\mathbb{G}^{\wedge n})\colon =\bigl|l\mapsto \mathcal X\bigl(-,(\mathbb G^{\wedge n})_l\bigr)\bigr| \in [\Gamma^{\mathrm{op}},\mathcal M^{\mathsf{fr}}].
\end{equation*}
\notag
$$
By Example 2.1, this is a pointed functor from $\Gamma^{\mathrm{op}}$ to $\mathcal M^{\mathsf{fr}}$. Applying the functor $\mathsf{ev}_{S^1}$ in (9), yields the motivic $S^1$-spectrum $\mathsf{ev}_{S^1}(\mathcal X(\mathbb{G}^{\wedge n}))= \mathcal X(\mathbb S,\mathbb{G}^{\wedge n})$. By Lemma 2.5 in [13], the $S^1$-spectrum $\mathcal X(\mathbb S,\mathbb{G}^{\wedge n})$ is $\mathbb A^1$-local. Moreover, $\mathcal X(\mathbb S,\mathbb{G}^{\wedge n})$ is sectionwise connective because on every section it is the $S^{1}$-spectrum associated with a $\Gamma$-space. It follows that $\mathcal X(\mathbb S,\mathbb{G}^{\wedge n})$ is a connective motivic $S^1$-spectrum for every $n\geqslant 0$. For the evaluation functor $\mathsf{ev}_{S^1,\mathbb{G}}$ in (14) we have
$$
\begin{equation*}
\mathsf{ev}_{S^1,\mathbb{G}}(\mathcal X)(n)= \mathcal X(\mathbb S,\mathbb{G}^{\wedge n}).
\end{equation*}
\notag
$$
Combined with Lemma 3.1 we conclude that $\mathsf{ev}_{S^1,\mathbb{G}}(\mathcal X)\in \mathbf{SH}(k)_{\geqslant 0}$. Hence the evaluation functor (15) assumes value in $\mathbf{SH}(k)_{\geqslant 0}$:
$$
\begin{equation}
\mathsf{ev}_{S^1,\mathbb{G}}\colon \mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)\to \mathbf{SH}(k)_{\geqslant 0}.
\end{equation}
\tag{18}
$$
By the construction of $\mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)$ (see Definition 1.1) the functor $\mathsf{ev}_{S^1,\mathbb{G}}$ in (18) is fully faithful. It remains to show essential surjectivity — this is the most interesting part of the proof.
Suppose $\mathcal E$ is a cofibrant and fibrant symmetric motivic $(S^1,\mathbb{G})$-bispectrum. Then there exists a framed spectral functor $\mathcal M^{\mathcal E}_{\mathsf{fr}}$ in the sense of Definition 6.1 in [13] such that $\mathsf{ev}_\mathbb{G}(\mathcal M_{\mathsf{fr}}^{\mathcal E})$ is naturally isomorphic to $\mathcal E$ in $\mathbf{SH}(k)$ (see [13], § 6). In fact, $\mathcal M^{\mathcal E}_{\mathsf{fr}}$ enables the equivalence between $\mathbf{SH}(k)$ and framed spectral functors in (see [13], Theorem 6.3, Definition 6.5). We briefly recall the construction of $\mathcal M_{\mathsf{fr}}^{\mathcal E}$, since it is important for the details of this proof. The motivic spaces $C_{\ast}\mathrm{Fr}(\mathcal E_{i,j})$ conspire into a motivic $(S^1,\mathbb{G})$-bispectrum $C_{\ast}\mathrm{Fr}(\mathcal E)$. For $n\geqslant 0$, we let $R^n_{\mathbb{G}} C_{\ast}\mathrm{Fr}(\mathcal E)$ denote $\underline{\mathrm{Hom}}(\mathbb{G}^{\wedge n}, C_{\ast}\mathrm{Fr}(\mathcal E[n]))$, where $\mathcal E[n]$ is the $n$th shift of $\mathcal E$ in the $\mathbb{G}$-direction. In each weight $i\geqslant 0$, we have the motivic $S^1$-spectrum
$$
\begin{equation*}
R^n_{\mathbb{G}} C_{\ast}\mathrm{Fr}(\mathcal E)(i)= \underline{\mathrm{Hom}}\bigl(\mathbb{G}^{\wedge n}, C_{\ast}\mathrm{Fr}(\mathcal E(n+i))\bigr).
\end{equation*}
\notag
$$
Next, there is a canonical morphism of motivic $(S^1,\mathbb{G})$-bispectra
$$
\begin{equation*}
R^n_{\mathbb{G}} C_{\ast}\mathrm{Fr}(\mathcal E)\to R^{n+1}_{\mathbb{G}} C_{\ast}\mathrm{Fr}(\mathcal E).
\end{equation*}
\notag
$$
We also set
$$
\begin{equation*}
R^\infty_{\mathbb{G}} C_{\ast}\mathrm{Fr}(\mathcal E):= \operatorname{colim}\bigl(C_{\ast}\mathrm{Fr}(\mathcal E)\to R^1_{\mathbb{G}} C_{\ast}\mathrm{Fr}(\mathcal E)\to R^2_{\mathbb{G}} C_{\ast}\mathrm{Fr}(\mathcal E)\to\cdots\bigr).
\end{equation*}
\notag
$$
According to [13], § 6, Claim 2, there are stable motivic equivalences
$$
\begin{equation*}
\mathcal E\to C_{\ast}\mathrm{Fr}(\mathcal E)\to R^\infty_{\mathbb{G}} C_{\ast}\mathrm{Fr}(\mathcal E).
\end{equation*}
\notag
$$
For $n\geqslant 0$, we define the spectral functor $\mathbb GC_{\ast}\mathrm{Fr}^{\mathcal E}[n]$ sectionwise by
$$
\begin{equation*}
U\mapsto \underline{\mathrm{Hom}}\bigl(\mathbb{G}^{\wedge n}, C_{\ast}\mathrm{Fr}(\mathcal E(n)\wedge U_+)\bigr).
\end{equation*}
\notag
$$
By construction, there is a natural morphism of spectral functors
$$
\begin{equation*}
\mathbb GC_{\ast}\mathrm{Fr}^{\mathcal E}[n]\to \mathbb GC_{\ast}\mathrm{Fr}^{\mathcal E}[n+1],
\end{equation*}
\notag
$$
and we set
$$
\begin{equation*}
\mathcal M_{\mathsf{fr}}^{\mathcal E}:= \operatorname{colim} (\mathbb GC_{\ast}\mathrm{Fr}^{\mathcal E}[0]\to \mathbb GC_{\ast}\mathrm{Fr}^{\mathcal E}[1]\to\cdots).
\end{equation*}
\notag
$$
By [13], Lemma 6.6, there is a morphism of motivic $(S^1,\mathbb{G})$-bispectra
$$
\begin{equation}
\mathsf{ev}_\mathbb{G}(\mathcal M_{\mathsf{fr}}^{\mathcal E})\to R^\infty_{\mathbb{G}}C_{\ast}\mathrm{Fr}(E^c).
\end{equation}
\tag{19}
$$
In every weight, (19) is a stable local equivalence of motivic $S^1$-spectra due to Lemma 6.7 in [13], this implies the zigzag of stable motivic equivalences
$$
\begin{equation*}
\mathcal E\to R^\infty_{\mathbb{G}} C_{\ast}\mathrm{Fr}(\mathcal E) \leftarrow \mathsf{ev}_\mathbb{G}(\mathcal M_{\mathsf{fr}}^{\mathcal E}),
\end{equation*}
\notag
$$
and therefore, an isomorphism in $\mathbf{SH}(k)$
$$
\begin{equation}
\mathsf{ev}_\mathbb{G}(\mathcal M_{\mathsf{fr}}^{\mathcal E}) \cong \mathcal E.
\end{equation}
\tag{20}
$$
For $U\in\mathrm{Sm}/k_+$, the motivic $S^1$-spectrum $\mathcal M^{\mathcal E}_{\mathsf{fr}}(U)$ is not necessarily a sectionwise $\Omega$-spectrum. However, the above property holds for the framed spectral functor ${\mathbb M}^{\mathcal E}_{\mathsf{fr}}$, with sections
$$
\begin{equation*}
U\mapsto \Theta^{\infty}_{S^1}\mathcal M^{\mathcal E}_{\mathsf{fr}}(U).
\end{equation*}
\notag
$$
Here, $\Theta^{\infty}_{S^1}$ is the motivic $S^1$-stabilization functor defined in [29], Definition 4.2. By construction, there is a canonical morphism
$$
\begin{equation}
{\mathcal M}^{\mathcal E}_{\mathsf{fr}}(U)\to {\mathbb M}^{\mathcal E}_{\mathsf{fr}}(U).
\end{equation}
\tag{21}
$$
We note that (21) is a sectionwise stable equivalence of motivic $S^1$-spectra.
Next, we use (17) to define the motivic $\Gamma$-space $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}$ by setting
$$
\begin{equation*}
\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(n_+,U) := \mathcal M(\mathbb S^{\times n}, \mathbb M_{\mathsf{fr}}^{\mathcal E}(U)),\qquad n\geqslant 0,\quad U\in\mathrm{Sm}/k.
\end{equation*}
\notag
$$
Here, the $S^1$-spectrum $\mathbb S^{\times n}:=\mathbb S\times \overset{n}{\cdots}\times\mathbb S$ is regarded as a constant motivic $S^1$-spectrum. For all $U,V\in\mathrm{Sm}/k_+$ and the adjunction $(\mathsf{ev}_{S^1},\Phi)$ between $\Gamma$-spaces and spectra in [3], § 5, we have
$$
\begin{equation}
\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(n_+,U)(V)= \Phi\bigl(\mathbb M_{\mathsf{fr}}^{\mathcal E}(U)(V)\bigr)(n_+)= \mathbf{S}_{\bullet}\bigl(\mathbb S^{\times n}, \mathbb M_{\mathsf{fr}}^{\mathcal E}(U)(V)\bigr).
\end{equation}
\tag{22}
$$
This expression determines the values of $\Phi$ at the $S^1$-spectrum $\mathbb M_{\mathsf{fr}}^{\mathcal E}(U)(V)$. Moreover, the counit $\mathsf{ev}_{S^1}\circ\Phi\to\operatorname{id}$ induces a morphism of spectral functors
$$
\begin{equation}
\mathsf{ev}_{S^1}(\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E})\to \mathbb M_{\mathsf{fr}}^{\mathcal E}.
\end{equation}
\tag{23}
$$
By construction, $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}$ is a framed motivic $\Gamma$-space in the sense of Definition 2.1. Moreover, in each weight $n\geqslant 0$ morphism (21) induces a sectionwise stable equivalence of motivic $S^1$-spectra
$$
\begin{equation*}
\mathsf{ev}_{\mathbb{G}}(\mathcal M^{\mathcal E}_{\mathsf{fr}})(n) \to \mathsf{ev}_{\mathbb{G}}(\mathbb M^{\mathcal E}_{\mathsf{fr}})(n).
\end{equation*}
\notag
$$
In combination with (20) we deduce an isomorphism in $\mathbf{SH}(k)$
$$
\begin{equation}
\mathsf{ev}_{\mathbb{G}}(\mathbb M^{\mathcal E}_{\mathsf{fr}})\cong \mathcal E.
\end{equation}
\tag{24}
$$
We will show that $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}$ satisfies axioms 1–4, and also (5) provided that $\mathcal E\in \mathbf{SH}(k)_{\geqslant 0}$.
Clearly we have $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(0_+,U)=\ast= \Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(n_+,\varnothing)$ for all $U\in\mathrm{Sm}/k_+$ and $n\geqslant 0$. Moreover, the canonical sectionwise stable equivalence of cofibrant motivic $S^1$-spectra
$$
\begin{equation*}
\mathbb S\vee\overset{n}{\cdots}\vee\mathbb S \to \mathbb S\times\overset{n}{\cdots}\times\mathbb S
\end{equation*}
\notag
$$
induces, via (17) and (22), the sectionwise equivalence of motivic spaces
$$
\begin{equation*}
\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(n_+,U) = \mathcal M\bigl(\mathbb S^{\times n}, \mathbb M_{\mathsf{fr}}^{\mathcal E}(U)\bigr)\to \mathcal M\bigl(\mathbb S^{\vee n}, \mathbb M_{\mathsf{fr}}^{\mathcal E}(U)\bigr) \cong \Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(1_+,U)^{\times n}.
\end{equation*}
\notag
$$
This establishes Axiom 1.
Next, for $U\in\mathrm{Sm}/k_+$ the presheaf of stable homotopy groups $\pi_n\mathsf{ev}_{S^1}({\Gamma\mathbb M}^{\mathcal E}_{\mathsf{fr}}(U))$ is isomorphic to $\pi_n({\mathbb M}^{\mathcal E}_{\mathsf{fr}}(U))$ if $n\geqslant 0$, and trivial if $n<0$ (this follows as in Theorem 5.1 of [3]). By (21), there is an isomorphism of presheaves between $\pi_{\ast}({\mathcal M}^{\mathcal E}_{\mathsf{fr}}(U))$ and $\pi_{\ast}({\mathbb M}^{\mathcal E}_{\mathsf{fr}}(U))$. Since the former is framed in addition to being $\mathbb A^1$-invariant and $\sigma$-invariant, the same holds for $\pi_{\ast}\mathsf{ev}_{S^1} ({\Gamma\mathbb M}^{\mathcal E}_{\mathsf{fr}}(U))$. This shows that Axiom 2 holds. Axioms 3 and 4 hold because ${\mathbb M}^{\mathcal E}_{\mathsf{fr}}$ is a framed spectral functor and the presheaves of stable homotopy groups $\pi_n\mathsf{ev}_{S^1}({\Gamma\mathbb M}^{\mathcal E}_{\mathsf{fr}}(U))$ of the connective $\mathbb A^1$-local motivic $S^1$-spectrum $\mathsf{ev}_{S^1}({\Gamma\mathbb M}^{\mathcal E}_{\mathsf{fr}}(U))$ are isomorphic to $\pi_n({\mathbb M}^{\mathcal E}_{\mathsf{fr}}(U))$ for all $n\geqslant 0$ and $U\in\mathrm{Sm}/k_+$. Let us verify Axiom 5 assuming that $\mathcal E\in \mathbf{SH}(k)_{\geqslant 0}$. Indeed, the proof of Theorem 6.3 in [13] shows that $\mathcal E\wedge U_+\in \mathbf{SH}(k)_{\geqslant 0}$ is isomorphic to $\mathsf{ev}_\mathbb{G}(\mathbb M^{\mathcal E}_{\mathsf{fr}}(-\times U))$ for all $U\in\mathrm{Sm}/k_+$. Here, $\mathbb M^{\mathcal E}_{\mathsf{fr}}(-\times U) $ is the framed spectral functor with sections
$$
\begin{equation*}
X \mapsto \mathbb M^{\mathcal E}_{\mathsf{fr}}(X\times U).
\end{equation*}
\notag
$$
By Lemma 3.1, the $\mathbb A^1$-local motivic $S^1$-spectrum $\mathbb M^{\mathcal E}_{\mathsf{fr}}(U)$ is connective. Indeed, $\mathbb M^{\mathcal E}_{\mathsf{fr}}(U)$ is the zeroth weight of the framed bispectrum $\mathsf{ev}_\mathbb{G}(\mathbb M^{\mathcal E}_{\mathsf{fr}}(-\times U))$, whose weights are $\mathbb A^1$-local by Lemma 2.6 in [13]. Thus, for all $U\in\mathrm{Sm}/k_+$, the morphism (23) yields a stable local equivalence of connective motivic $S^1$-spectra
$$
\begin{equation*}
\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}}(\mathbb S,U) \to \mathbb M^{\mathcal E}_{\mathsf{fr}}(U).
\end{equation*}
\notag
$$
As a result, $\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}}(\mathbb S,-)$ is a framed spectral functor and the framed motivic $\Gamma$-space $\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}}$ satisfies Nisnevich excision as in Axiom 5. This completes the proof of Theorem 1.1. Remark 3.1. The proof of Theorem 1.1 shows that a quasi-inverse functor $\Gamma\mathbb M_{\mathsf{fr}}$ to the equivalence $\mathsf{ev}_{S^1,\mathbb{G}}\colon \mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k){\to}\mathbf{SH}(k)_{\geqslant 0}$ is given as follows: for $\mathcal E\in\mathbf{SH}(k)_{\geqslant 0}$ take a functorial cofibrant and fibrant replacement $\mathcal E'$ in the stable model structure on symmetric motivic $(S^1,\mathbb G)$-bispectra. Then map $\mathcal E$ to the framed motivic $\Gamma$-space $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E'}$. With Theorem 1.1 in hand we can prove Theorem 1.2. Proof of Theorem 1.2. Following § 3, p. 1131 in [30], and § 5 in [14], we have
$$
\begin{equation*}
\mathbf{SH}^{\mathsf{veff}}(k)=\mathbf{SH}(k)_{\geqslant 0}\cap \mathbf{SH}^{\mathsf{eff}}(k),
\end{equation*}
\notag
$$
where $\mathbf{SH}^{\mathsf{eff}}(k)$ is the full subcategory of $\mathbf{SH}(k)$ comprised of effective bispectra. For $\mathcal X\in \mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{veffr}}(k)$, the evaluation $\mathcal X_{S^1,\mathbb G}$ is contained in $\mathbf{SH}(k)_{\geqslant 0}$ by Theorem 1.1. By Axiom 6, the $S^1$-spectrum
$$
\begin{equation*}
\bigl|\mathcal X(\mathbb S,\mathbb{G}\times U) \bigl(\widehat{\Delta}^\bullet_{K/k}\bigr)\bigr|
\end{equation*}
\notag
$$
is stably contractible for any finitely generated field extension $K/k$ and $U\in\mathrm{Sm}/k$. It follows that
$$
\begin{equation*}
\bigl|\mathcal X(\mathbb S,\mathbb{G}^{\wedge n}) \bigl(\widehat{\Delta}^\bullet_{K/k}\bigr)\bigr|
\end{equation*}
\notag
$$
is stably contractible for every $n>0$. We deduce that $\mathcal X_{S^1,\mathbb G}\in\mathbf{SH}^{\mathsf{eff}}(k)$ and thus $\mathcal X_{S^1,\mathbb G}\in\mathbf{SH}^{\mathsf{veff}}(k)$ thanks to [12], Theorem 4.4, and [13], Definition 3.5, Theorem 3.6.
We have shown the restriction of the equivalence $\mathsf{ev}_{S^1,\mathbb{G}} \colon \mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)\xrightarrow{\simeq} \mathbf{SH}(k)_{\geqslant 0}$ in Theorem 1.1 to the full subcategory $\mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{veffr}}(k)$ takes values in $\mathbf{SH}^{\mathsf{veff}}(k)$. It remains to show that it is essentially surjective. Suppose that $\mathcal E$ is a very effective cofibrant and fibrant symmetric motivic $(S^1,\mathbb G)$-bispectrum. By Theorem 1.1, there exists a framed motivic $\Gamma$-space $\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}}$ and an isomorphism between $\mathsf{ev}_{S^1,\mathbb{G}}(\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}})$ and $\mathcal E$ in $\mathbf{SH}(k)_{\geqslant 0}$. Moreover, the proof of Theorem 1.1 shows that, for every $U\in\mathrm{Sm}/k_+$, there is an isomorphism in $\mathbf{SH}(k)_{\geqslant 0}$ between $\mathcal E\wedge U_+$ and $\mathsf{ev}_{S^1,\mathbb{G}}(\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}} (-\times U))$. Here, $\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}}(-\times U)$ is the framed motivic $\Gamma$-space with sections
$$
\begin{equation*}
(n_+,X)\mapsto \Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}}(n_+,X\times U).
\end{equation*}
\notag
$$
Recall that $\mathbf{SH}^{\mathsf{veff}}(k)$ is closed under the smash products in $\mathbf{SH}(k)$ by Lemma 5.6 in [14]. In particular, $\mathcal E\wedge U_+\in \mathbf{SH}^{\mathsf{veff}}(k)$. To verify that the $S^1$-spectrum
$$
\begin{equation*}
\bigl|\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}}(\mathbb S,\mathbb{G}\times U) \bigl(\widehat{\Delta}^\bullet_{K/k}\bigr)\bigr|
\end{equation*}
\notag
$$
is stably contractible we appeal to Theorem 3.6 in [13]. Hence the framed motivic $\Gamma$-space $\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}}$ is effective, and so $\mathcal E$ is isomorphic to $\mathsf{ev}_{S^1,\mathbb{G}}(\Gamma\mathbb M^{\mathcal E}_{\mathsf{fr}})$ in $\mathbf{SH}^{\mathsf{veff}}(k)$. Theorem 1.2 is proved. Suppose that $\mathcal E$ is a motivic $(S^1,\mathbb G)$-bispectrum with motivic fibrant replacement $\mathcal E^{f}$. We will write $\Omega_{S^1}^{\infty}\Omega_\mathbb{G}^{\infty} \mathcal E$ for the pointed motivic space $\mathcal E^{f}_{0,0}$. Definition 3.1. A pointed motivic space $A$ is an infinite motivic loop space if there exists a motivic $(S^1,\mathbb G)$-bispectrum $\mathcal E$ and local equivalence $A\simeq\Omega_{S^1}^{\infty}\Omega_\mathbb{G}^{\infty} \mathcal E$. Lemma 3.2. Suppose that $\mathcal X$ is a very special framed motivic $\Gamma$-space. Then the bispectrum ${\mathcal X}^{f}_{S^1,\mathbb{G}}$, as obtained from ${\mathcal X}_{S^1,\mathbb{G}}$ by taking levelwise local fibrant replacements is motivically fibrant. Proof. This follows from Lemma 2.6 in [13], since the $S^1$-spectrum, associated with a very special $\Gamma$-space is an $\Omega$-spectrum after taking levelwise fibrant replacements (see Corollary 2.2.1.7 in [4]). The above brings us to the proof of Theorem 1.3. Proof of Theorem 1.3. Without loss of generality we may assume that $\mathcal E\in\mathbf{SH}(k)_{\geqslant 0}$. Indeed, it follows from p. 374 in [31] that, for any $\mathcal E$, the connective cover $\tau_{\geqslant 0}\mathcal E\to\mathcal E$ yields a sectionwise equivalence
$$
\begin{equation*}
\Omega^\infty_{S^1}\Omega^\infty_\mathbb{G}(\tau_{\geqslant 0}\mathcal E) \to \Omega^\infty_{S^1}\Omega^\infty_\mathbb{G}(\mathcal E).
\end{equation*}
\notag
$$
Now every $\mathcal E\in \mathbf{SH}(k)_{\geqslant 0}$ is isomorphic to the bispectrum $\mathsf{ev}_{S^1,\mathbb{G}}(\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E})$ for some special framed motivic $\Gamma$-space $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}$ (see the proof of Theorem 1.1). For $n\geqslant 0$ and $U,V\in\mathrm{Sm}/k_+$, from (22) we have
$$
\begin{equation*}
\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(n_+,U)(V)= \Phi\bigl(\mathbb M_{\mathsf{fr}}^{\mathcal E}(U)(V)\bigr)(n_+)= \mathbf{S}_{\bullet}\bigl(\mathbb S^{\times n}, \mathbb M_{\mathsf{fr}}^{\mathcal E}(U)(V)\bigr).
\end{equation*}
\notag
$$
Here, $\mathbb M_{\mathsf{fr}}^{\mathcal E}(U)(V)$ is the $\Omega$-spectrum $\Theta^{\infty}_{S^1}\mathcal M_{\mathsf{fr}}^{\mathcal E}(U)(V)$, which was introduced in the proof of Theorem 1.1. It follows that $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(1_+,U)(V)$ is the zeroth space ${\mathbb M}_{\mathsf{fr}}^{\mathcal E}(U)(V)_0$ of the $\Omega$-spectrum ${\mathbb M}_{\mathsf{fr}}^{\mathcal E}(U)(V)$. Thus $\pi_0(\mathbb M_{\mathsf{fr}}^{\mathcal E}(U)(V)_0)$ is an abelian group, and $\pi_0^{\mathsf{nis}}\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(1_+,U)$ is a Nisnevich sheaf of abelian groups. This shows that $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}$ is a very special framed motivic $\Gamma$-space (see Axiom 7).
An appeal to Lemma 3.2 shows that the bispectrum $\mathsf{ev}_{S^1,\mathbb{G}}(\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E})^f$, as obtained by taking levelwise local fibrant replacements from $\mathsf{ev}_{S^1,\mathbb{G}} (\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E})$, is motivically fibrant. Hence there exists a sectionwise equivalence of pointed motivic spaces
$$
\begin{equation*}
\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(1_+,\mathsf{pt})^f= \mathsf{ev}_{S^1,\mathbb{G}} (\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E})^f_{0,0} \simeq \Omega^{\infty}_{S^1}\Omega^{\infty}_\mathbb{G}\mathcal E.
\end{equation*}
\notag
$$
We conclude that $\Gamma\mathbb M_{\mathsf{fr}}^{\mathcal E}(1_+,\mathsf{pt})$ is locally equivalent to $\Omega^{\infty}_{S^1}\Omega^{\infty}_\mathbb{G}\mathcal E$.
Now suppose $\mathcal X$ is a very special framed motivic $\Gamma$-space. By Lemma 3.2, ${\mathcal X}^{f}_{S^1,\mathbb{G}}$ is motivically fibrant and we deduce
$$
\begin{equation*}
\mathcal X(1_+,\mathsf{pt})^f = \mathsf{ev}_{S^1,\mathbb{G}}(\mathcal X)^f_{0,0} \simeq \Omega^{\infty}_{S^1}\Omega^{\infty}_\mathbb{G} {\mathcal X}^{f}_{S^1,\mathbb{G}}.
\end{equation*}
\notag
$$
Since $\mathcal X(1_+,\mathsf{pt})$ is locally equivalent to $\mathcal X(1_+,\mathsf{pt})^f$ it follows that $\mathcal X(1_+,\mathsf{pt})$ is an infinite motivic loop space in the sense of Definition 3.1. Theorem 1.3 is proved. Remark 3.2. Every special framed motivic $\Gamma$-space $\mathcal X\colon \Gamma^{\mathrm{op}}\boxtimes\mathrm{Sm}/k_+\to\mathcal M$ has a canonically associated very special framed motivic $\Gamma$-space with sections In this expression, the Kan fibrant replacement functor $\mathrm{Ex}^{\infty}$ is applied sectionwise in the category $\mathbf{S}_{\bullet}$. We complete this section by discussing the diagram (4) of adjoint functors from the introduction: The functor $u\colon \mathbf{H}_{\Gamma\mathcal{M}}^{\mathsf{fr}}(k)\to\mathbf{H}(k)$ sends a framed motivic $\Gamma$-space $\mathcal X$ to its underlying motivic space $\mathcal X(1_+,\mathsf{pt})$. Moreover, $C_{\ast}\mathrm{Fr}$ sends a motivic space $A$ to $C_{\ast}\mathrm{Fr}(A^c\otimes-)$, where $A^c $ is the projective cofibrant replacement of $A$. Recall that $A^c$ is a directed colimit in $\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$ of simplicial smooth schemes.The composite functor $\mathsf{ev}_{S^1,\mathbb{G}}\circ C_{\ast}\mathrm{Fr}$ is equivalent to $\Sigma^{\infty}_{S^1,\mathbb G}$ due to § 11 in [8]. Theorem 1.3 implies that $u\circ\Gamma\mathbb M_{\mathsf{fr}}$ is equivalent to $\Omega^{\infty}_{S^1,\mathbb G}$. Thus the adjoint pair $(\Sigma^{\infty}_{S^1,\mathbb G},\Omega^{\infty}_{S^1,\mathbb G})$ is equivalent to $(\mathsf{ev}_{S^1,\mathbb{G}}\circ C_{\ast}\mathrm{Fr},u\circ\Gamma\mathbb M_{\mathsf{fr}})$. Since $(\mathsf{ev}_{S^1,\mathbb{G}},\Gamma\mathbb M_{\mathsf{fr}})$ is an adjoint equivalence by Theorem 1.1, $(C_{\ast}\mathrm{Fr},u)$ is a pair of adjoint functors. Corollary 3.1. The diagram of adjoint functors (4) commutes up to equivalence of functors. § 4. Further properties of motivic $\Gamma$-spacesLet $\mathcal X\colon \Gamma^{\mathrm{op}}\boxtimes\mathrm{Sm}/k_+ \to \mathcal M^{\mathsf{fr}}$ be a framed motivic $\Gamma$-space. One has an enriched functor
$$
\begin{equation*}
\mathcal X(1_+,-)\colon \mathrm{Sm}/k_+ \to \mathcal M^{\mathsf{fr}},\qquad U \mapsto \mathcal X(1_+,U).
\end{equation*}
\notag
$$
For all $U,V\in\mathrm{Sm}/k_+$ we have the elementary Nisnevich square: If $\mathcal X$ is (very) special in the sense of Axioms, then axioms 1 and 5 imply the stable local equivalence
$$
\begin{equation}
\mathcal X(\mathbb S,U)\vee\mathcal X(\mathbb S,V)\to \mathcal X(\mathbb S,U\sqcup V).
\end{equation}
\tag{25}
$$
On the other hand, the sectionwise stable equivalence
$$
\begin{equation*}
\mathcal X(\mathbb S,U)\vee\mathcal X(\mathbb S,V)\to \mathcal X(\mathbb S,U)\times\mathcal X(\mathbb S,V)
\end{equation*}
\notag
$$
factors as
$$
\begin{equation*}
\mathcal X(\mathbb S,U)\vee\mathcal X(\mathbb S,V)\to \mathcal X(\mathbb S,U\sqcup V)\to \mathcal X(\mathbb S,U)\times\mathcal X(\mathbb S,V).
\end{equation*}
\notag
$$
It follows that the rightmost morphism is a local stable equivalence. This shows that the morphism of motivic spaces
$$
\begin{equation*}
\mathcal X(1_+,U\sqcup V)\to\mathcal X(1_+,U)\times\mathcal X(1_+,V)
\end{equation*}
\notag
$$
is a local equivalence, and similarly, so is
$$
\begin{equation*}
\mathcal X(1_+,n_+\otimes U)\to\mathcal X(1_+,U)\times \overset{n}{\cdots}\times\mathcal X(1_+,U),\qquad n\geqslant 1.
\end{equation*}
\notag
$$
Here, we write $n_+\otimes U:= U\sqcup\,{\overset{n}{\cdots}}\,\sqcup U\in\mathrm{Sm}/k_+$. Axiom 1 ensures that $\mathcal X(1_+, 0_+\otimes U)=\ast$, since by definition $0_+\otimes U:=\varnothing$. Moreover, if $\mathcal X$ is very special then the Nisnevich sheaf $\pi^{\mathsf{nis}}_{0}\mathcal X(1_+,U)$ takes values in abelian groups due to Axiom 7. We record these observations in the next lemma. Lemma 4.1. For any very special framed motivic $\Gamma$-space $\mathcal X$ and $U\in\mathrm{Sm}/k_+$, the functor is locally a very special $\Gamma$-space. Let us fix a cofibrant replacement functor $A\to A^{\mathrm{c}}$ in the projective motivic model structure on $\mathcal M$ in the sense of § 3 in [19], [20]. Then $A^{\mathrm{c}}$ is a sequential colimit of simplicial $k$-smooth schemes in $\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$. For a motivic $\Gamma$-space $\mathcal X$, we define the functor $\mathcal X(1_+,-)\colon \mathcal M\to\mathcal M$ by setting
$$
\begin{equation*}
\mathcal X(1_+,A):=\operatorname{colim}_{(\Delta[n]\times U)_+\to A} \mathcal X(1_+,\Delta[n]_+\otimes U),\qquad A\in\mathcal M.
\end{equation*}
\notag
$$
Here, we identify the pointed motivic space $A$ with $\operatorname{colim}_{(\Delta[n]\times U)_+\to A}(\Delta[n]\times U)_+$. A key property of $\Gamma$-spaces says that if $f\colon K\to L$ is an equivalence in $\mathbf{S}_\bullet$, then so is $F(f)\colon F(K)\to F(L)$, that is, an equivalence for each $\Gamma$-space $F\colon \Gamma^\mathrm{op}\to\mathbf{S}_\bullet$ (see Proposition 4.9 in [3] and Lemma 2.2.1.3 in [4]). The following result is a motivic counterpart of this property. Theorem 4.1. For any very special framed motivic $\Gamma$-space $\mathcal X$, the functor takes motivic equivalences to local equivalences of motivic spaces. Hence if $\mathcal X$ is a special framed motivic $\Gamma$-space, then the functor takes motivic equivalences to stable local equivalences of motivic $S^1$-spectra. Our proof of Theorem 4.1 is inspired by Voevodsky’s theory of left derived radditive functors as in Theorem 4.19 of [32]. The basic notions we will need in this paper are recalled below. In this context, we note that the category $\mathrm{Sm}/k_+$ has finite coproducts. Recall that a morphism $e\colon A\to X$ in a category $\mathcal C$ is a coprojection if it is isomorphic to the canonical morphism $A\to A\sqcup Y$ for some $Y$ (see § 2 in [32]). A morphism $f\colon A\to X$ in $\Delta^{\mathrm{op}}\mathcal C$ is a termwise coprojection if, for all $i\geqslant 0$, the morphism $f_i\colon A_i\to X_i$ is a coprojection. As observed in § 2 of [32], a morphism $f\colon B\to A$ and an object $X$ conspire into the pushout: It follows that there exist pushouts for all pairs of morphisms $(e,f)$ with $e$, a coprojection whenever $\mathcal C$ is a category with finite coproducts, and likewise for pairs of morphisms $(e,f)$ in $\Delta^{\mathrm{op}}\mathcal C$, where $e$ is a termwise coprojection. Following § 2 in [32], a square in $\Delta^{\mathrm{op}}\mathcal C$ is called an elementary pushout square if it is isomorphic to the pushout square for a pair of morphisms $(e,f)$, where $e$ is a termwise coprojection.If $\mathcal C$ has finite coproducts, then, for any commutative square $Q$ of the form we define the object $K_Q$ by the elementary pushout square: There is a canonically induced morphism $p_Q\colon K_Q\to X$. An important example is the cylinder $\operatorname{cyl}(f)$ of a morphism $f\colon X\to X'$. In terms of the construction above, this is the object associated with the square By Lemma 2.9 in [32], the natural morphisms $X'\to\operatorname{cyl}(f)$ and $\operatorname{cyl}(f)\to X'$ are mutually inverse homotopy equivalences.Lemma 4.2. Suppose $\mathcal X$ is a special framed motivic $\Gamma$-space. Then $\mathcal X(\mathbb S,-)$ takes elementary pushout squares in $\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$ to homotopy pushout squares in the stable local model structure on motivic $S^1$-spectra. Proof. Consider the pushout square in $\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$ with horizontal coprojections: The associated square of $S^1$-spectra is a homotopy pushout because by (25) it is stably locally equivalent to the pushout square By definition, an elementary pushout square is isomorphic to the pushout square of morphisms $(e,f)$, where $e$ is a termwise coprojection. It remains to observe that the geometric realization of a simplicial homotopy pushout square of spectra is a homotopy pushout. This proves Lemma 4.2. Corollary 4.1. Suppose $\mathcal X$ is a special framed motivic $\Gamma$-space, and is an elementary pushout square in $\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$ of morphisms $(e,f)$, where $e$ is a termwise coprojection. If $\mathcal X(\mathbb S,e)$ is a stable local equivalence of $S^1$-spectra, then so is $\mathcal X(\mathbb S,e')$. Proof of Theorem 4.1. Let $Q$ denote an elementary Nisnevich square in $\mathrm{Sm}/k$: By applying the cylinder construction and forming pushouts in $\mathcal M$, we obtain the commutative diagram Note that $U'_+\to\operatorname{cyl}(U'_+\to X'_+)$ is a termwise coprojection and a projective cofibration between projective cofibrant objects of $\mathcal M$. Thus $s(Q):=\operatorname{cyl}(U'_+\to X'_+)\bigsqcup_{U'_+}\!U_+$ is projective cofibrant (see Corollary 1.1.11 in [33]), and $U_+\to s(Q)$ is a termwise coprojection. Likewise, by applying the cylinder construction to $s(Q)\to X_+$ and setting $t(Q):=\operatorname{cyl}(s(Q)\to X_+)$, we get a projective cofibration Here, $\operatorname{cyl}(Q)$ is a termwise coprojection and a local equivalence in $\mathcal M$.
In the following we let $J_{\mathrm{mot}}=J_{\mathrm{proj}}\cup J_{\mathsf{nis}}\cup J_{\mathbb A^1}$, where
$$
\begin{equation*}
\begin{aligned} \, J_{\mathrm{proj}} &= \{\Lambda^{r}[n]_+\wedge U_+\to \Delta[n]_+\wedge U_+\mid U\in\mathrm{Sm}/k,\, n>0,\, 0\leqslant r\leqslant n\}, \\ J_{\mathsf{nis}} &= \biggl\{\Delta[n]_+\wedge s(Q) \bigsqcup_{\partial\Delta[n]_+\wedge s(Q)}\partial\Delta[n]_+\wedge t(Q) \\ &\qquad\qquad \to \Delta[n]_+\wedge t(Q)\biggm| Q \text{ is an elementary Nisnevich square}\biggr\}, \\ J_{\mathbb A^1} &= \biggl\{\Delta[n]_+\wedge U\times\mathbb A^1_+ \bigsqcup_{\partial\Delta[n]_+\wedge U\times\mathbb A^1_+} \partial\Delta[n]_+\wedge \operatorname{cyl}(U\times\mathbb A^1_+\to U_+) \\ &\qquad\qquad \to\Delta[n]_+\wedge\operatorname{cyl} (U\times\mathbb A^1_+\to U_+)\biggm| U\in\mathrm{Sm}/k\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
We note that every map in $J_{\mathrm{mot}}$ is a termwise coprojection. According to Lemma 2.15 in [20], a morphism is a fibration with fibrant codomain in the projective motivic model structure if and only if it has the right lifting property with respect to $J_{\mathrm{mot}}$.
Arguing as in Proposition 4.9 of [3], the functor $\mathcal X(1_+,-)$ maps members of $J_{\mathrm{proj}}$ to local equivalences. We note that $\mathcal X(1_+,-)$ preserves naive simplicial homotopies: if $A$ is a pointed motivic space then $\mathcal X(1_+,\Delta[1]_+\otimes A^{\mathrm{c}})$ is a cylinder object for $\mathcal X(1_+,A^{\mathrm{c}})$. Axiom 4 implies that there is a canonically induced local equivalence
$$
\begin{equation*}
\mathcal X(1_+,U\times\mathbb A^1)\to \mathcal X\bigl(1_+,\operatorname{cyl}(U\times\mathbb A^1\to U)\bigr).
\end{equation*}
\notag
$$
Axiom 5 implies the same holds for $\mathcal X(1_+,\operatorname{cyl}(Q))$.
To show that $\mathcal X(1_+,-)$ maps members of $J_{\mathsf{nis}}$ to local equivalences, let us start with a cofibration of simplicial sets $K\hookrightarrow L$ and the induced commutative diagram: An application of Lemma 4.1 to $a_0=K_+\wedge\operatorname{cyl}(Q)$, shows that the induced morphism $\mathcal X(1_+,a_0)$ is a local equivalence. The same applies to $a_2=L_+\wedge\operatorname{cyl}(Q)$ and $\mathcal X(1_+,a_2)$. Since $\mathcal X$ is very special, Corollary 4.1 shows $\mathcal X(1_+,a_1)$ is a local equivalence. Thus $\mathcal X(1_+,a_3)$ is a local equivalence and our claim for $J_{\mathsf{nis}}$ follows. Likewise, $\mathcal X(1_+,-)$ maps members of $J_{\mathbb A^1}$ to local equivalences.So far, we have established that $\mathcal X(1_+,-)$ takes members of $J_{\mathrm{mot}}$ to local equivalences. For every motivic equivalence $f\colon A \to B$, the induced morphism $f^{\mathrm{c}}\colon A^{\mathrm{c}} \to B^{\mathrm{c}}$ is also a motivic equivalence. It remains to show the canonical morphism
$$
\begin{equation*}
\mathcal X(1_+,f^{\mathrm{c}})\colon \mathcal X(1_+,A^{\mathrm{c}}) \to \mathcal X(1_+,B^{\mathrm{c}})
\end{equation*}
\notag
$$
is a local equivalence. To that end, we apply the “small object argument” (see Theorem 2.1.14 in [33]).
To begin, we note that all the morphisms in $J_{\mathrm{mot}}$ have finitely presentable (co)domains. For every pointed motivic space $A\in\mathcal M$, let $\alpha\colon A\to \mathcal L A$ be the transfinite composition of the $\aleph_{0}$-sequence:
$$
\begin{equation*}
A=E^0\xrightarrow{\alpha_0}E^1\xrightarrow{\alpha_1}E^2 \xrightarrow{\alpha_2}\cdots,
\end{equation*}
\notag
$$
constructed as follows. For $n\geqslant 0$ we let $S_n$ denote the set of all commutative squares where $g\in J_{\mathrm{mot}}$, and form the pushout This construction is plainly functorial in $A$. By definition, $\alpha$ is a trivial motivic cofibration in $\mathcal M$ belonging to $J_{\mathrm{mot}}$-cell (see Definition 2.1.9 in [33]).
We claim that the horizontal morphisms in the commutative diagram are local equivalences. Indeed, Corollary 4.1 shows that $\mathcal X(1_+,-)$ maps the cobase change of a member of $J_{\mathrm{mot}}$ to a local equivalence (here we use the assumption that $\mathcal X$ is very special). Local equivalences are closed under filtered colimits and $\mathcal X(1_+,-)$ preserves filtered colimits, so the same holds for members of $J_{\mathrm{mot}}$-cell. Since ${\mathcal L}(A^{\mathrm{c}})$ and ${\mathcal L}(B^{\mathrm{c}})$ are cofibrant and fibrant, ${\mathcal L}(f^{\mathrm{c}})$ is a homotopy equivalence. As noted above, $\mathcal X(1_+,-)$ preserves naive simplicial homotopies, and therefore, $\mathcal X(1_+,{\mathcal L}(f^{\mathrm{c}}))$ is a homotopy equivalence. Thus, $\mathcal X(1_+,f^{\mathrm{c}})$ is a local equivalence. Theorem 4.1 is proved.Let $M\mathbb Z$ be the motivic ring spectrum representing integral motivic cohomology in the sense of Voevodsky–Suslin [6]. Up to inversion of the exponential characteristic $e$ of the base field $k$, the highly structured category of $M\mathbb Z$-modules is equivalent to Voevodsky’s derived category of motives (see Theorem 58 in [34], and also Theorem 5.8 in [35]). A crucial part of the proof shows that, for every $U\in\mathrm{Sm}/k$, the natural assembly morphism is an isomorphism in $\mathbf{SH}(k)[1/e]$. For a $\Gamma$-space $F\colon \Gamma^{\mathrm{op}}\to\bf S_\bullet$, the corresponding statement says that the morphism
$$
\begin{equation*}
\mathsf{ev}_{S^1}(F)\wedge K \to \mathsf{ev}_{S^1}(F(-\wedge K))
\end{equation*}
\notag
$$
is a stable equivalence for every pointed simplicial set $K\in\mathbf S_\bullet$ (see Lemma 4.1 in [3]). We show a similar property for special framed motivic $\Gamma$-spaces. Theorem 4.2. Suppose $k$ is an infinite perfect field of exponential characteristic $e$. Let $U\in\mathrm{Sm}/k$ be such that $U_+$ is strongly dualizable in $\mathbf{SH}(k)$ (for example, $U$ is a smooth projective algebraic variety). Then, for every special framed motivic $\Gamma$-space $\mathcal X$, the natural morphism of bispectra Proof. Without loss of generality we may assume that $\mathcal X$ is very special motivic $\Gamma$-space (see Remark 3.2). We view $\mathcal X(1_+,-)$ as an $\mathcal M$-enriched functor from $\mathrm{Sm}/k_+$ to $\mathcal M$.
Recall from § 2 the $\mathcal M$-category of finitely presentable motivic spaces $f\mathcal M$. Via an enriched left Kan extension functor the inclusion of $\mathcal M$-categories $\iota\colon \mathrm{Sm}/k_+\hookrightarrow f\mathcal M$ yields the functor
$$
\begin{equation*}
\Upsilon \colon [\mathrm{Sm}/k_+,\mathcal M]\to[f\mathcal M,\mathcal M].
\end{equation*}
\notag
$$
By expressing $\mathcal Y\in[\mathrm{Sm}/k_+,\mathcal M]$ as a coend
$$
\begin{equation*}
\mathcal Y=\int^{U\in\mathrm{Sm}/k_+}\mathcal Y(U)\wedge_{\mathcal M}[U,-],
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation*}
\Upsilon(\mathcal Y)= \int^{U\in\mathrm{Sm}/k_+}\mathcal Y(U) \wedge_{\mathcal M}[\iota(U),-].
\end{equation*}
\notag
$$
By construction $\Upsilon(\mathcal Y)(V)=\mathcal Y(V)$ for all $V\in\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$. More generally, we have $\Upsilon(\mathcal Y)(A^{\mathrm{c}})=\mathcal Y(A^{\mathrm{c}})$ for every pointed motivic space $A\in\mathcal M$.
Theorem 4.1 implies that $\Upsilon(\mathcal X(1_+,-))$ maps motivic weak equivalences of projective cofibrant motivic spaces to local equivalences. Owing to Corollary 56 in [34], the $\mathbb G$-evaluation of the assembly morphism
$$
\begin{equation*}
\Upsilon\bigl(\mathcal X(1_+,-\otimes\mathbb S)\bigr)\wedge U_+\to \Upsilon\bigl(\mathcal X(1_+,-\otimes\mathbb S\otimes U)\bigr)
\end{equation*}
\notag
$$
is a stable motivic equivalence between motivic $(S^1,\mathbb G)$-bispectra if $U_+$ is strongly dualizable in $\mathbf{SH}(k)$. Here, $\mathcal X(1_+,-\otimes\mathbb S\otimes U)$ is the evaluation at the sphere $\mathbb S$, of the $\Gamma$-space of Lemma 4.1. Since $\Upsilon(\mathcal X(1_+,V))=\mathcal X(1_+,V)$ for all $V\in\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$, the same holds for the $\mathbb G$-evaluation of the morphism
$$
\begin{equation*}
\mathcal X(1_+,-\otimes\mathbb S)\wedge U_+\to \mathcal X(1_+,-\otimes\mathbb S\otimes U).
\end{equation*}
\notag
$$
Given $n>0$, let $\mathcal X(S^n,-)$ be the very special framed motivic $\Gamma$-space with sections Replacing $\mathcal X$ with $\mathcal X(S^n,-)$, we deduce the stable motivic equivalence of motivic $(S^1,\mathbb G)$-bispectra
$$
\begin{equation}
\mathsf{ev}_{\mathbb{G}}\bigl(\mathcal X(S^n,-\otimes\mathbb S)\bigr) \wedge U_+ \to \mathsf{ev}_{\mathbb{G}} \bigl(\mathcal X(S^n,-\otimes\mathbb S\otimes U)\bigr).
\end{equation}
\tag{28}
$$
Combining (28) with Lemma 4.1 in [3], we obtain the stable motivic equivalences of motivic $(S^1,S^1,\mathbb G)$-trispectra
$$
\begin{equation*}
\begin{gathered} \, \mathsf{ev}_{\mathbb{G}}\bigl(\mathcal X(\mathbb S,-\otimes\mathbb S)\bigr) \wedge U_+\to \mathsf{ev}_{\mathbb{G}} \bigl(\mathcal X(\mathbb S,-\otimes\mathbb S\otimes U)\bigr), \\ \mathsf{ev}_{\mathbb{G}}\bigl(\mathcal X(\mathbb S,-)\bigr)\wedge U_+\wedge\mathbb S\to \mathsf{ev}_{\mathbb{G}} \bigl(\mathcal X(\mathbb S,-\otimes U)\bigr)\wedge\mathbb S. \end{gathered}
\end{equation*}
\notag
$$
For the cofibrant replacements of $\mathsf{ev}_{\mathbb{G}}(\mathcal X(\mathbb S,-))\wedge U_+$, $\mathsf{ev}_{\mathbb{G}}(\mathcal X(\mathbb S,-\otimes U))$ in $\mathbf{Sp}_{S^1,\mathbb G}(k)$ we find a stable motivic equivalence between cofibrant motivic $(S^1,S^1,\mathbb G)$-trispectra
$$
\begin{equation*}
\bigl(\mathsf{ev}_{\mathbb{G}}(\mathcal X(\mathbb S,-))\wedge U_+\bigr)^{\mathrm{c}}\wedge\mathbb S \to\mathsf{ev}_{\mathbb{G}} \bigl(\mathcal X(\mathbb S,-\otimes U)\bigr)^{\mathrm{c}}\wedge\mathbb S.
\end{equation*}
\notag
$$
Since $-\wedge S^1$ is a Quillen auto-equivalence on $\mathbf{Sp}_{S^1,\mathbb G}(k)$, we deduce the stable motivic equivalence
$$
\begin{equation*}
\bigl(\mathsf{ev}_{\mathbb{G}}(\mathcal X(\mathbb S,-))\wedge U_+\bigr)^{\mathrm{c}} \to \mathsf{ev}_{\mathbb{G}} \bigl(\mathcal X(\mathbb S,-\otimes U)\bigr)^{\mathrm{c}}
\end{equation*}
\notag
$$
between cofibrant motivic $(S^1,\mathbb G)$-bispectra (see also Theorem 5.1 in [29]). Therefore, (26) is a stable motivic equivalence.
Recall that $U_+$ is strongly dualizable in $\mathbf{SH}(k)[1/e]$ for every $U\in\mathrm{Sm}/k$ (see Appendix B in [36]). The previous arguments show that (26) is an $e^{-1}$-stable motivic equivalence Indeed, even though Corollary 56 in [34] is concerned with the stable motivic model structure on motivic functors, it readily extends to the $e^{-1}$-stable model structure. Finally, when $A\in\mathcal M$, recall that $A^{\mathrm{c}}$ is a sequential colimit of simplicial schemes from $\Delta^{\mathrm{op}}\mathrm{Sm}/k_+$. Since the geometric realization functor preserves $e^{-1}$-stable motivic equivalences, we conclude that (27) is an isomorphism in $\mathbf{SH}(k)[1/e]$. Theorem 4.2 is proved.
Citation in format AMSBIB
\Bibitem{GarPanOst23} |
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