January 01, 1970
In this paper, we develop "motivic derived algebraic geometry," an enhancement of derived algebraic geometry for the \(\mathbb{A}^1\)-homotopy theory of Morel and Voevodsky. We formulate motivic versions of \(\infty\)-categories, \(\infty\)-topoi, spectral schemes, and spectral Deligne-Mumford stacks based on the theories of Joyal, Lurie, Toën, and Vezzosi. As the main application, we establish the existence of the motivic stackification functor. This functor is constructed as the left adjoint of the pullback along the geometric morphism between classifying motivic \(\infty\)-topoi.
Derived algebraic geometry studies geometric objects by allowing sheaves of functions to take values in homotopical or higher-categorical objects rather than in ordinary sets. In the work of Lurie [1], [2] and Toën–Vezzosi [3], spectral schemes and spectral Deligne–Mumford stacks are formulated in terms of structured \(\infty\)-topoi and their classifying \(\infty\)-topoi. From this viewpoint, the passage from affine or algebraic objects to stacks is realized as a universal construction in the \(\infty\)-categorical setting.
The purpose of this paper is to develop a motivic analogue of this framework. We work over a Grothendieck site \(\mathcal{X}\) equipped with an interval object \(I\); the principal example is the Nisnevich site of smooth schemes over a base scheme, with an interval \(\mathbb{A}^1\). Given a left proper combinatorial simplicial model category \(\mathbf{M}\), we construct a motivic model category \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\) by imposing both descent for the Grothendieck topology and invariance with respect to the interval object. This construction recovers the usual model category of motivic spaces of Morel–Voevodsky [4] when \(\mathbf{M}\) is the model category of simplicial sets, and it also applies to models of \(\infty\)-categories and \(\infty\)-bicategories.
Using these motivic model categories, we define motivic \(\infty\)-categories, motivic \(\infty\)-topoi, and motivic classifying \(\infty\)-topoi. The bicategorical part of this construction relies on Lurie’s model of \(\infty\)-bicategories, which utilizes scaled simplicial sets and the scaled straightening and unstraightening theorem [5, p. 128], Theorem 3.8.1. This theorem facilitates the transition between motivic functors valued in \(\infty\)-categories and motivic locally coCartesian fibrations over the corresponding base. Since motivic \(\infty\)-topoi are defined fiberwise by ordinary \(\infty\)-topoi, they inherit the universal-colimit property: colimits in each fiber are stable under pullback and compatible with motivic restriction functors. This observation also motivates the treatment of geometric objects defined by torsors. The stability of torsors under base change is a formal consequence of universality in an \(\infty\)-topos, while the \(\mathbb{A}^1\)-invariance of their classification is a motivic condition that requires additional hypotheses on the acting group object.
The main result of the paper is Theorem 10. It states that a geometric morphism \(f:\mathcal{K}\to\mathcal{K}'\) of motivic classifying \(\infty\)-topoi, compatible with the relevant geometric structures, induces a pullback functor \[f^{-1}:{}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}')\to {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K})\] between the motivic \(\infty\)-categories of structured topoi, and that this functor admits a left adjoint relative to the underlying motivic \(\infty\)-topos. Thus, the stackification procedure familiar from derived algebraic geometry has a motivic counterpart.
As an application, we introduce motivic spectral schemes and motivic spectral Deligne–Mumford stacks. The motivic Zariski and motivic étale classifying \(\infty\)-topoi are defined by imposing motivic analogues of the usual local and strictly Henselian local conditions on commutative algebra objects. Applying Theorem 10 to the geometric morphism from the motivic discrete classifying topos to the motivic étale classifying topos yields a functor from motivic algebraic spaces to motivic stacks with the expected universal property. In particular, any motivic algebraic space \(\mathbb{X}\) is associated with a motivic stack \(\mathbb{X}^{\text{\'et}}\).
The paper is organized as follows. In Section 2, we define motivic model categories and the associated motivic \(\infty\)-categories. In Section 3, we recall the scaled straightening and unstraightening formalism, constructing motivic classifying \(\infty\)-topoi; the proof of Theorem 10 provided therein. In Section 4, we apply the theorem to motivic spectral schemes and motivic spectral Deligne–Mumford stacks, concluding with the existence of associated motivic stacks.
In this section, we define motivic model categories associated with a Grothendieck site \(\mathcal{X}\) equipped with an interval object. An interval object \(I\) of \(\mathcal{X}\) is a triple \((\mu : I \times I \to I,\, i_0,\,i_1: * \to I)\) satisfying \[\mu \circ( i_0 \times \mathrm{id}) = \mu \circ( \mathrm{id} \times i_0 ) = i_0 \circ \mathbf{1}\] and \[\mu \circ( i_1 \times \mathrm{id}) = \mu \circ( \mathrm{id} \times i_1 ) = \mathrm{id}_I,\] where \(\mathbf{1}: I \to *\) denotes the canonical map.
Let \(\mathcal{X}\) be a Grothendieck site with an interval object \(I\). We assume that \(\mathcal{X}\) has enough points such that a morphism \(f:X \to Y\) in \(\mathcal{X}\) is an isomorphism if \(f_x:X_x \to Y_x\) is an isomorphism of sets for any point \(x:* \to \mathcal{X}\), where the functor \((-)_x: \mathcal{X} \to \mathit{Sets}\) denotes the right adjoint of the induced functor \(x_*: \mathit{Sets} \to \mathcal{X}\). A simplicial object \(U_\bullet : \Delta^{\rm op} \to \mathcal{X}\) with an augmentation \(\pi:U_\bullet \to X\) is a hypercover of \(X\) if it satisfies the following conditions:
For any \(n \ge 0\), \(U_\bullet([n])\) is a coproduct of compact objects represented by small objects of \(\mathcal{X}\).
The augmentation \(\pi:U_\bullet \to X\) is a stalkwise trivial Kan fibration; that is, \(\pi_x:U_{x,\,\bullet} \to *\) is a trivial Kan fibration for any point \(x:* \to X\).
Let \(\mathbf{M}\) be a left proper combinatorial simplicial model category, and let \(\mathbf{M}^{\mathcal{X}^{\rm op}}\) denote the category of \(\mathbf{M}\)-valued presheaves on \(\mathcal{X}\), equipped with the projective model structure. This model structure is again left proper, combinatorial, and simplicial. Let \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\) denote the Bousfield localization of \(\mathbf{M}^{\mathcal{X}^{\rm op}}\) defined as follows. The cofibrations of \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\) are the cofibrations of \(\mathbf{M}^{\mathcal{X}^{\rm op}}\). An \(\mathbf{M}\)-valued presheaf \(F\) on \(\mathcal{X}\) is \(\mathcal{X}\)-local if \(F\) is fibrant in \(\mathbf{M}^{\mathcal{X}^{\rm op}}\) and the induced map \(F(\pi):F(X) \to F(|U_\bullet|)\) is a weak equivalence in \(\mathbf{M}\) for any hypercover \(\pi:U_\bullet \to X\) with \(X \in \mathcal{X}\). Here, the functor \(|-|:\mathrm{Fun}(\Delta^{\rm op},\,\mathcal{X}) \to \mathcal{X}\) denotes the geometric realization of simplicial objects. Furthermore, \(F\) is \(I\)-local if the canonical map \(\mathbf{1}:I \to *\) induces a weak equivalence \(F(U) \to F(U \times I)\) in \(\mathbf{M}\) for any \(U \in \mathcal{X}\). We say that \(F\) is motivic \(\mathbf{M}\)-local if it is both \(\mathcal{X}\)-local and \(I\)-local. A map \(f:F \to G\) of \(\mathbf{M}\)-valued presheaves on \(\mathcal{X}\) is a motivic \(\mathbf{M}\)-equivalence if the induced map \[f^*:\mathrm{Hom}(G,\,Z) \to \mathrm{Hom}(F,\,Z)\] is a weak homotopy equivalence of simplicial sets for any motivic local \(\mathbf{M}\)-valued presheaf \(Z\). We call the resulting model structure on \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\) the motivic \(\mathcal{X}\)-model structure of \(\mathbf{M}\).
By [6], the projective model structure on \(\mathbf{M}^{\mathcal{X}^{\rm op}}\) is left proper, combinatorial, and symmetric monoidal. Therefore, the iterated Bousfield localization \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\) of \(\mathbf{M}^{\mathcal{X}^{\rm op}}\) is also left proper. Moreover, by [6], the Bousfield localization \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\) is a symmetric monoidal localization of \(\mathbf{M}^{\mathcal{X}^{\rm op}}\). Hence, \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\) is also a symmetric monoidal model category:
Theorem 1. Let \(\mathbf{M}\) be a left proper combinatorial simplicial model category, and let \(\mathcal{X}\) be a Grothendieck site with an interval object \(I\). Assume that \(\mathcal{X}\) has enough points. Then there is a left proper combinatorial simplicial model structure on \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\) determined by the following data:
Cofibrations are pointwise cofibrations.
Weak equivalences are motivic \(\mathbf{M}\)-weak equivalences.
Fibrations are morphisms having the right lifting property with respect to all morphisms that are both cofibrations and motivic \(\mathbf{M}\)-weak equivalences.
Furthermore, if \(\mathbf{M}\) is a symmetric monoidal model category, then \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\) is also a symmetric monoidal model category. 0◻
Write \(\mathbf{MS}_\infty^I=\mathrm{Mot}^{I}_{\mathcal{X}}(\mathrm{Set}_{\Delta})_\infty\) and \(\mathrm{MCat}_{\infty}^I=\mathrm{Mot}^{I}_{\mathcal{X}}(\mathrm{Set}_{\Delta}^+)_\infty\). We refer to \(\mathbf{MS}_\infty^I\) as the \(\infty\)-category of motivic spaces and to \(\mathrm{MCat}_{\infty}^I\) as the \(\infty\)-category of motivic \(\infty\)-categories. We now give a more explicit description of motivic spaces and motivic \(\infty\)-categories. By the straightening and unstraightening theorem [7], we have a Quillen adjunction \[\mathrm{St}_\mathcal{X}: (\mathrm{Set}_{\Delta})_{/ N(\mathcal{X})} \rightleftarrows \mathrm{Set}_{\Delta}^{\mathcal{X}^{\rm op}} : \mathrm{Un}_{\mathcal{X}},\] where the model structure on the left-hand side is the contravariant model structure [7], and the model structure on the right-hand side is the projective model structure associated with the Kan–Quillen model structure on \(\mathrm{Set}_{\Delta}\). If \(X\) is a motivic space, then there exists a right fibration \(p_X:\overline{X} \to N(\mathcal{X})\) such that, for any \(U \in \mathcal{X}\), the fiber \(\overline{X} \times_{N(\mathcal{X})} U\) is homotopy equivalent to the Kan complex \(X(U)\) and satisfies \(X(U \times I)\simeq X(U)\). Similarly, a motivic \(\infty\)-category \(\mathcal{C}\) is represented by an \(\mathrm{Set}_{\Delta}^+\)-valued presheaf on \(\mathcal{X}\) satisfying \(\mathcal{C}(U \times I) \simeq \mathcal{C}(U)\) for any \(U \in \mathcal{X}\). In addition, \(\mathrm{MCat}_{\infty}^I\) is the full subcategory of \(\mathrm{Fun}(\mathcal{X}^{\rm op},\,\mathrm{Cat}_{\infty})\) spanned by \(I\)-local objects. By the straightening and unstraightening theorem for marked simplicial sets with the Cartesian model structure [7, p. 169], Theorem 3.2.0.1, we have a Quillen adjunction \[\mathrm{St}^+_\mathcal{X}: (\widehat{\mathrm{Set}_{\Delta}^+})_{/ N(\mathcal{X})} \rightleftarrows (\widehat{\mathrm{Set}_{\Delta}^+})^{\mathcal{X}^{\rm op}} : \mathrm{Un}^+_{\mathcal{X}}\] between left proper combinatorial simplicial model categories. Thus there exists a Cartesian fibration \(p_\mathcal{C}:\overline{\mathcal{C}} \to N(\mathcal{X})\) such that, for any \(U \in \mathcal{X}\), the fiber \(\overline{\mathcal{C}} \times_{N(\mathcal{X})} U\) is weakly equivalent to the \(\infty\)-category \(\mathcal{C}(U)\) and satisfies \(\mathcal{C}(U \times I) \simeq \mathcal{C}(U)\).
For any left proper combinatorial Cartesian closed simplicial monoidal model category \(\mathbf{M}\), the constant functor \[\mathbf{M} \to \mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\] is a left Quillen functor. Its right adjoint is the global section functor, namely the limit over \(\mathcal{X}\). Writing the constant functor as \(-\otimes \mathbf{1}\), we obtain the Quillen adjunction \[- \otimes \mathbf{1} :\mathbf{M} \rightleftarrows \mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M}) : \mathrm{Map}_{\widehat{\mathrm{MCat}_{\infty}}}( \mathbf{1},\, - ).\] Consider the case \(\mathbf{M}= (\mathrm{Set}_{\Delta}^+)_{/\Delta^0}\). Let \(\mathrm{MCat}_{\infty}\) denote the underlying \(\infty\)-category of \(\mathrm{Mot}^{I}_{\mathcal{X}}(\mathbf{M})\).
Definition 1 ([7] p.20, Definition \(1.1.5.3\)). Let \(n\) be a non-negative integer. For the standard \(n\)-simplex \(\Delta^n\), we define the simplicial category \(\mathfrak{C}[\Delta^n]\) as follows:
The objects of \(\mathfrak{C}[\Delta^n]\) are the objects of the category \([n]=\{0 \to 1 \to \cdots \to n \}\).
For \(0 \le i,\,j \le n\), the set of morphisms is defined by \[\mathrm{Hom}_{\mathfrak{C}[\Delta^n]}(i,\,j)=\begin{cases} \emptyset &(j<i), \\ N(P_{i,\,j}) &(i \leq j), \end{cases}\] where \(P_{i,\,j}\) denotes the partially ordered set \(\{I \subset [n] \mid i, j \in I, I \subset \{i,\,\ldots , j \} \}\), ordered by inclusion.
The functor \[\Delta \times \mathcal{X} \ni ([n],\,U) \mapsto (i_U)_*( \mathfrak{C}[\Delta^n]) \in (\mathrm{Cat}_{\Delta})^{\mathcal{X}^{\rm op}},\] where \((i_U)_*( \mathfrak{C}[\Delta^n]) : \mathcal{X}^{\rm op} \to \mathrm{Cat}_{\Delta}\) is the left Kan extension of \(\mathfrak{C}[\Delta^n]:\{U\} \to \mathrm{Cat}_{\Delta}\) along the inclusion \(i_U:\{U\} \to \mathcal{X}\), induces an adjunction \[\mathfrak{C}_\mathcal{X}: \mathrm{Set}_{\Delta_\mathcal{X}} \rightleftarrows (\mathrm{Cat}_{\Delta})^{\mathcal{X}^{\rm op}} :N_{\Delta_\mathcal{X}}.\] The right adjoint \(N_\Delta(\mathcal{C})\) of a functor \(\mathcal{C}:\mathcal{X}^{\rm op} \to \mathrm{Cat}_{\Delta}\) is defined by \[\mathrm{Hom}_{ \mathrm{Set}_{\Delta_\mathcal{X}}}(\Delta^n \times U , N_\Delta (\mathcal{C})) = \mathrm{Hom}_{ (\mathrm{Cat}_{\Delta})^{\mathcal{X}^{\rm op}}}( (i_U)_*( \mathfrak{C}[\Delta^n]),\, \mathcal{C}) = \mathrm{Hom}_{ (\mathrm{Cat}_{\Delta})^{\mathcal{X}^{\rm op}}}( \mathfrak{C}[\Delta^n], \mathcal{C}(U))\] for each \(n \ge 0\).
The category \(\mathrm{Cat}_{\Delta_\mathcal{X}}\) carries the model structure induced by the motivic model structure on \(\mathrm{Set}_{\Delta_\mathcal{X}}\). The motivic model structures on \(\mathrm{Set}_{\Delta_\mathcal{X}}\) and \((\mathrm{Cat}_{\Delta})^{\mathcal{X}^{\rm op}}\) are chosen so that the functors \[(\mathrm{Cat}_{\Delta})^{\mathcal{X}^{\rm op}} \overset{N_\Delta}{\leftarrow} \mathrm{Set}_{\Delta_\mathcal{X}} \overset{(-)^\flat}{\rightarrow} (\mathrm{Set}_{\Delta}^+)^{\mathcal{X}^{\rm op}}\] are left Quillen functors.
Definition 2. Let \((-)^\flat: \mathrm{Set}_{\Delta}\to \mathrm{Set}_{\Delta}^+\) denote the functor defined by \(X^\flat=(X,\,s_0(X))\), where \(s_0(X)\) is the collection of degenerate edges of \(X\). Let \(f:F \to G\) be a morphism of simplicial sheaves on \(\mathcal{X}\).
We say that \(f\) is a motivic cofibration if \(f^\flat:F^\flat \to G^\flat\) is a motivic cofibration in \((\mathrm{Set}_{\Delta}^+)^{\mathcal{X}^{\rm op}}\).
We say that \(f\) is a motivic equivalence if \(f^\flat:F^\flat \to G^\flat\) is a motivic equivalence in \((\mathrm{Set}_{\Delta}^+)^{\mathcal{X}^{\rm op}}\).
Definition 3. Let \(\mathfrak{C}_\mathcal{X}: \mathrm{Set}_{\Delta_\mathcal{X}} \to (\mathrm{Cat}_{\Delta})^{\mathcal{X}^{\rm op}}\) denote the functor defined above. Let \(f:\mathcal{C} \to \mathcal{D}\) be a natural transformation between \(\mathrm{Cat}_{\Delta}\)-valued presheaves on \(\mathcal{X}\).
We say that \(f\) is a motivic equivalence if \(N_{\Delta_\mathcal{X}}(f)\) is a motivic equivalence in \(\mathrm{Set}_{\Delta_\mathcal{X}}\).
We say that \(f\) is a motivic fibration if \(N_{\Delta_\mathcal{X}}(f)\) is a motivic fibration in \(\mathrm{Set}_{\Delta_\mathcal{X}}\).
Theorem 2. The displayed left Quillen functors \[(\mathrm{Cat}_{\Delta})^{\mathcal{X}^{\rm op}} \overset{N_\Delta}{\leftarrow} \mathrm{Set}_{\Delta_\mathcal{X}} \overset{(-)^\flat}{\rightarrow} (\mathrm{Set}_{\Delta}^+)^{\mathcal{X}^{\rm op}}\] are Quillen equivalences.
This follows from results of Bergner and Lurie [[7] p.89, Theorem 2.2.5.1; p.164, Theorem 3.1.5.1; and p.826, Remark A.2.8.6]. 0◻
Let \(\mathcal{C}\) be a motivic \(\infty\)-category, and let \(p:\overline{\mathcal{C}} \to N(\mathcal{X})\) be the Cartesian fibration associated with \(\mathcal{C}\). For each \(n \ge 0\), an \(N(\mathcal{X})\)-morphism \(\Delta^n \times N(\mathcal{X}) \to \overline{\mathcal{C}}\) determines an \(n\)-simplex in the fiber \(\mathcal{C}(U)\) for any object \(U \in \mathcal{X}\). Let \(\mathrm{Ob}(\mathcal{C})\) denote the set of \(N(\mathcal{X})\)-morphisms \(\Delta^0 \times N(\mathcal{X}) \to \overline{\mathcal{C}}\), and let \(\mathrm{Ar}(\mathcal{C})\) denote the set of morphisms \(\Delta^1 \times N(\mathcal{X}) \to \overline{\mathcal{C}}\). We call their elements the motivic objects and motivic morphisms of \(\mathcal{C}\), respectively. The constant \(\mathrm{Cat}_{\infty}\)-valued sheaf \(\mathcal{S}_\infty\) on \(\mathcal{X}\) determines the constant Cartesian fibration \(\mathcal{S}_\infty \times N(\mathcal{X}) \to N(\mathcal{X})\). We regard \(N(\mathcal{X}) \times \mathcal{S}_\infty\) as the motivic \(\infty\)-category of spaces.
Definition 4. Let \(\mathcal{C}\) be a motivic \(\infty\)-category. For two objects \(x,\,y: \Delta^0\times N(\mathcal{X}) \to \overline{\mathcal{C}}\), the mapping space \(\mathrm{Map}_\mathcal{C}(x,\,y)\) is defined to be the homotopy limit \[\{x\} \times N(\mathcal{X}) \times_{ \overline{\mathcal{C}} } \mathrm{Ar}(\mathcal{C}) \times_{ \overline{\mathcal{C}} } \{y\} \times N(\mathcal{X}).\]
We now define the Yoneda functor for motivic \(\infty\)-categories. A motivic \(\infty\)-category is said to be small if it is termwise small. Let \(p:\overline{\mathcal{C}} \to N(\mathcal{X})\) be the Cartesian fibration determined by a small motivic \(\infty\)-category \(\mathcal{C}\). For any \(N(\mathcal{X})\)-morphism \(q:K \to \overline{\mathcal{C}}\), the overcategory \(r:\overline{\mathcal{C}}_{/q} \to \overline{\mathcal{C}}\) is a Cartesian fibration. For any point \(y \times N(\mathcal{X}):\Delta^0 \times N(\mathcal{X}) \to \overline{\mathcal{C}}\), the fiber of \(r:\overline{\mathcal{C}}_{/y \times N(\mathcal{X})} \to \overline{\mathcal{C}}\) is a Cartesian fibration and, therefore, a motivic space. Therefore, by the unstraightening \[\mathrm{Un}_{ \overline{\mathcal{C}}}: ({\mathrm{Set}_{\Delta}^+}_{/\mathcal{X}})_{/\overline{\mathcal{C}}} \to ((\mathrm{Set}_{\Delta}^+)_{/\mathcal{X}})^{\mathcal{C}^{\rm op}},\] we obtain a functor \[\mathrm{Un}_{ \overline{\mathcal{C}}} (\overline{\mathcal{C}}_{/y \times N(\mathcal{X})}) : \mathcal{C}^{\rm op} \to {\mathrm{Set}_{\Delta}^+}_{/N(\mathcal{X})},\] whose values at points \(\Delta^0 \times N(\mathcal{X}) \to \overline{\mathcal{C}}\) are motivic spaces. We define \(\mathbf{Y}(y)=\mathrm{Un}_{ \overline{\mathcal{C}}} (\overline{\mathcal{C}}_{/y \times N(\mathcal{X})})\) as the (motivic) Yoneda functor of \(y \times N(\mathcal{X}): \Delta^0 \times N(\mathcal{X}) \to \overline{\mathcal{C}}\).
Lemma 1. Let \(\mathcal{C}\) be a small motivic \(\infty\)-category. Then, for any functor \(f: \mathcal{C}^{\rm op} \to (\mathrm{Set}_{\Delta})_{/\mathcal{X}}\), the following square \[\xymatrix@1{ \mathcal{C} \ar[rrr]^f \ar[d]_{\mathbf{Y}} && & {\mathrm{Set}_{\Delta}}_{/N(\mathcal{X})} \ar[d]^{i} \\ ( {\mathrm{Set}_{\Delta}}_{/N(\mathcal{X})} )^{\mathcal{C}^{\rm op}} \ar[rrr]_{\mathrm{Map}_{({\mathrm{Set}_{\Delta}}_{/N(\mathcal{X})} )^{\mathcal{C}^{\rm op}} }(-,\,f)} & & & {\widehat{{\mathrm{Set}_{\Delta}}}}_{/N(\mathcal{X})} }\] is homotopically commutative, where \(i\) denotes the canonical inclusion from small simplicial sets to large simplicial sets over \(N(\mathcal{X})\).
For any object \(U \in \mathcal{X}\), the fiber of the square is homotopically commutative by Lurie’s strong Yoneda lemma [7, p. 461], Proposition 5.5.2.1. Hence the square itself is homotopically commutative. 0◻
Corollary 1. For any pair of points \(x,\,y: \Delta^0 \times N(\mathcal{X}) \to \mathcal{C}\), the mapping spaces \(\mathrm{Map}_{\mathcal{C}}(x,\,y)\) and \[\mathrm{Map}_{({\mathrm{Set}_{\Delta}}_{/N(\mathcal{X})} )^{\mathcal{C}^{\rm op}} } (\mathbf{Y}(x),\,\mathbf{Y}(y))\] are weakly equivalent motivic spaces.
Applying Lemma 1 to the case \(f=\mathbf{Y}(y)\), for any \(x\) one obtains a chain of equivalences \[\mathrm{Map}_{({\mathrm{Set}_{\Delta}}_{/N(\mathcal{X})} )^{\mathcal{C}^{\rm op}} } (\mathbf{Y}(x),\,\mathbf{Y}(y)) \simeq \mathbf{Y}(y)(x) \simeq \mathrm{Map}_{\mathcal{C}}(x,\,y).\] 0◻
Let \(\widehat{\mathrm{Cat}_{\infty}}\) denote the very large \(\infty\)-category of large \(\infty\)-categories, and let \(\widehat{\mathrm{MCat}_{\infty}}\) denote the very large \(\infty\)-category of large motivic \(\infty\)-categories. Let \({}^{\rm L}\mathrm{Pr}\) denote the subcategory of \(\widehat{\mathrm{Cat}_{\infty}}\) spanned by presentable \(\infty\)-categories and colimit-preserving functors. We define the very large \(\infty\)-category \({}^{\rm L}\mathrm{MPr}\) of presentable motivic \(\infty\)-categories by the pullback \[\xymatrix@1{ {}^{\rm L}\mathrm{MPr} \ar[d] \ar[r] & \mathrm{Fun}(\mathcal{X}^{\rm op},\,{}^{\rm L}\mathrm{Pr}) \ar[d] \\ \widehat{\mathrm{MCat}_{\infty}} \ar[r] & \mathrm{Fun}(\mathcal{X}^{\rm op},\,\widehat{\mathrm{Cat}_{\infty}}). }\]
Proposition 3. Let \(F:\mathcal{C} \to \mathcal{D}\) be a functor of motivic \(\infty\)-categories. Assume that \(\mathcal{C}\) is small and that \(\mathcal{D}\) is presentable. Let \(y: \mathcal{C} \to \mathrm{Fun}(\mathcal{C}^{\rm op},\, \mathfrak{MS}_\infty)\) denote the motivic Yoneda embedding. Then the induced functor \[y_*(F): \mathcal{D} \ni d \mapsto \mathrm{Map}_{\mathcal{D}}(F(-),\, d) \in \mathrm{Fun}(\mathcal{C}^{\rm op},\, \mathfrak{MS}_\infty)\] admits a left adjoint.
For any \(U \in \mathcal{X}\), the presentability of \(\mathcal{D}(U)\) implies that the restriction functor \[y_*(F(U)): \mathcal{D}(U) \to \mathrm{Fun}(\mathcal{C}(U)^{\rm op},\,\mathcal{S}_\infty)\] admits a left adjoint, namely left Kan extension along \(F(U)\). These fiberwise left Kan extensions are functorial in \(U\) by the universal property of left Kan extensions, and hence assemble to a left adjoint of \(y_*(F)\) in the motivic functor category. 0◻
Let \({}^{\rm L}\mathrm{Top}\) denote the very large \(\infty\)-category of \(\infty\)-topoi and left-exact colimit-preserving functors. We define the very large \(\infty\)-category \({}^{\rm L}\mathrm{MTop}\) of motivic \(\infty\)-topoi by the pullback \[\xymatrix@1{ {}^{\rm L}\mathrm{MTop} \ar[d] \ar[r] & \mathrm{Fun}(\mathcal{X}^{\rm op},\,{}^{\rm L}\mathrm{Top}) \ar[d] \\ \widehat{\mathrm{MCat}_{\infty}} \ar[r] & \mathrm{Fun}(\mathcal{X}^{\rm op},\,\widehat{\mathrm{Cat}_{\infty}}). }\]
Proposition 4. Let \(\mathcal{T}\) be a motivic \(\infty\)-topos. Then colimits in \(\mathcal{T}\) are universal fiberwise. More precisely, for any \(U\in\mathcal{X}\), any morphism \(x\to y\) in \(\mathcal{T}(U)\), and any small diagram \(D:K\to\mathcal{T}(U)_{/y}\), the canonical map \[\left(\operatorname*{colim}_{k\in K}D_k\right)\times_y x \longrightarrow \operatorname*{colim}_{k\in K}(D_k\times_y x)\] is an equivalence. Moreover, for any morphism \(f:U\to V\) in \(\mathcal{X}\), the restriction functor \(\mathcal{T}(f):\mathcal{T}(V)\to\mathcal{T}(U)\) preserves these universal colimits.
By definition, \(\mathcal{T}(U)\) is an \(\infty\)-topos for any \(U\in\mathcal{X}\). Hence colimits in \(\mathcal{T}(U)\) are universal by the \(\infty\)-categorical Giraud theorem [7]. The transition functors of a motivic \(\infty\)-topos are left exact and colimit-preserving, so they preserve both pullbacks and colimits. This proves the compatibility with motivic restriction functors. 0◻
Remark 5. The preceding proposition separates two notions that are often related in motivic examples. Let \(G\) be a group object of \(\mathcal{T}(U)\) and let \(P\to X\) be a \(G\)-torsor. For any morphism \(Y\to X\), the pullback \(P\times_XY\to Y\) is again a torsor, now under the pulled-back group \(G_Y=G\times Y\). This is a formal base-change property: the torsor condition is expressed by an effective epimorphism together with the equivalence \(G_X\times_XP\simeq P\times_XP\), where \(G_X=G\times X\), and these conditions are preserved by pullbacks in an \(\infty\)-topos. This should not be confused with \(\mathbb{A}^1\)-invariance of the classification of \(G\)-torsors. The latter is a separate motivic representability condition, familiar from Morel–Voevodsky theory and from affine representability results for vector bundles and principal bundles [4], [8], [9].
Proposition 6. Let \(\mathcal{C}\) be a small motivic \(\infty\)-category. Assume that, for any morphism \(f:U \to V\) in \(\mathcal{X}\), the induced functor \(\mathcal{C}(f):\mathcal{C}(V) \to \mathcal{C}(U)\) is left exact. Then the motivic \(\infty\)-category \[\mathrm{Fun}_{\widehat{\mathrm{MCat}_{\infty}}}(\mathcal{C}^{\rm op},\, \mathfrak{MS}_\infty)\] is a motivic \(\infty\)-topos.
For any \(U \in \mathcal{X}\), the fiber \[\mathrm{Fun}_{\widehat{\mathrm{Cat}_{\infty}}}(\mathcal{C}(U)^{\rm op},\, \mathfrak{MS}_\infty(U)) \simeq \mathrm{Fun}_{\widehat{\mathrm{Cat}_{\infty}}}(\mathcal{C}(U)^{\rm op},\, \mathcal{S}_\infty)\] is an \(\infty\)-topos. The left exactness assumption on the transition functors \(\mathcal{C}(f)\) implies, by [7], that the induced transition functors between these presheaf \(\infty\)-topoi are left exact and preserve colimits. Hence the displayed motivic \(\infty\)-category is an object of \({}^{\rm L}\mathrm{MTop}\). 0◻
We use Lurie’s model of \(\infty\)-bicategories by scaled simplicial sets [5]. A scaled simplicial set is a pair \(\overline{X}=(X,\,T)\) consisting of a simplicial set \(X\) and a collection of thin \(2\)-simplices containing all degenerate \(2\)-simplices; we write \(\mathrm{Set}_{\Delta}^{\rm sc}\) for the category of scaled simplicial sets. Let \(\mathrm{Set}_{\Delta}^+\) denote the category of marked simplicial sets with the Cartesian model structure, and let \(\mathrm{Cat}_{\Delta}^+\) denote the category of \(\mathrm{Set}_{\Delta}^+\)-enriched categories with the model structure of [7]. We use the scaled nerve adjunction \[\mathfrak{C}^{\rm sc}: \mathrm{Set}_{\Delta}^{\rm sc} \rightleftarrows \mathrm{Cat}_{\Delta}^+ :N^{\rm sc}\] constructed in [5]. We also use the scaled model structure on \(\mathrm{Set}_{\Delta}^{\rm sc}\) and Lurie’s scaled straightening and unstraightening theorem, recalled below only in the forms needed for our motivic constructions.
We recall the definition of the scaled straightening and unstraightening functors from [5]. Let \(\overline{S}=(S,\,T)\) be a scaled simplicial set, and let \(\mathcal{C}\) be a \(\mathrm{Set}_{\Delta}^+\)-enriched category. Given a functor \(\phi: \mathfrak{C}^{\rm sc}[\overline{S}] \to \mathcal{C}\), we define the scaled straightening functor \(\mathrm{St}_\phi^{\rm sc}:\mathrm{Set}_{\Delta/\overline{S}}^+ \to (\mathrm{Set}_{\Delta}^+)^\mathcal{C}\) as follows.
Definition 5 ([5] p.114, Definition 3.5.1). Let \(\overline{X}=(X,\,M)\) be a marked simplicial set. Let \(T \subset X \times \Delta^1\) be the collection of all \(2\)-simplices \(\sigma\) with the following properties:
Under the projection \(X \times \Delta^1 \to X\), the image of \(\sigma\) is a degenerate \(2\)-simplex of \(X\).
For any \(2\)-simplex \(\pi:\Delta^2 \overset{\sigma}\to X \times \Delta^1 \to \Delta^1\) satisfying \(\pi^{-1}(\{0\}) = \Delta^{0,\,1}\), the restriction \(\sigma|_{\Delta^{0,\,1}}\) determines a marked edge of \(X\).
We define a scaled simplicial set \(C(\overline{X})\) by \[C(\overline{X}) = (X \times \Delta^1) \coprod_{(X \times \{0\})_\flat} \{ v \}_\flat.\] We call \(C(\overline{X})\) the scaled cone of \(\overline{X}\). More generally, for any scaled simplicial set \(\overline{S}\) and any \(\overline{X} \in (\mathrm{Set}_{\Delta}^{\rm sc})_{/\overline{S}}\), we set \(C_{\overline{S}}(\overline{X}) = C(\overline{X}) \coprod_{(X \times \{1\})_\flat} \overline{S}\). We call \(C_{\overline{S}}(\overline{X})\) the scaled cone of \(\overline{X}\) over \(\overline{S}\).
Definition 6 ([5] p.115, Definition 3.5.4). Let \(\overline{S}\) be a scaled simplicial set, let \(\mathcal{C}\) be a \(\mathrm{Set}_{\Delta}^+\)-enriched category, and let \(\phi: \mathfrak{C}^{\rm sc}[\overline{S}] \to \mathcal{C}\) be a functor of \(\mathrm{Set}_{\Delta}^+\)-enriched categories. The scaled straightening functor associated with \(\phi\) is the functor \[\mathrm{St}_\phi^{\rm sc}: \mathrm{Set}_{\Delta/\overline{S}}^+ \to (\mathrm{Set}_{\Delta}^+)^\mathcal{C}\] defined on \(\overline{X}\) by \[(\mathrm{St}^{\rm sc}_\phi(\overline{X}))(C)= \mathrm{Map}_{ C_{\overline{S}}[\overline{X}] \coprod_{\mathfrak{C}^{\rm sc}[\overline{S}]} \mathcal{C} } (v,\, C),\] for any \(C \in \mathcal{C}\).
Remark 7. The straightening functor \(\mathrm{St}^{\rm sc}_\phi:\mathrm{Set}_{\Delta/\overline{S}}^+ \to (\mathrm{Set}_{\Delta}^+)^\mathcal{C}\) is defined as a \(\mathrm{Set}_{\Delta}^+\)-enriched categorical colimit. Let \(f:X \to S\) be a marked simplicial set over \(S\). Then the straightening \(\mathrm{St}^{\rm sc}_\phi(\overline{X})\) is equivalent to the colimit of the diagram \[j^{\rm op}\circ \phi \circ \mathfrak{C}^{\rm sc}[F]: \mathfrak{C}^{\rm sc} [ (X \times \Delta^1,\,T )\coprod_{X \times \{1\} }\overline{S} ] \to \mathfrak{C}^{\rm sc}[\overline{S}] \to \mathcal{C} \to (\mathrm{Set}_{\Delta}^+)^\mathcal{C},\] where \(F\) is induced by \(f\) and \(j: \mathcal{C} \to (\mathrm{Set}_{\Delta}^{+})^{\mathcal{C}^{\rm op}}\) denotes the enriched Yoneda embedding.
Since the scaled straightening functor \(\mathrm{St}^{\rm sc}_\phi\) preserves all small colimits, it has a right adjoint by the adjoint functor theorem; we denote this right adjoint by \(\mathrm{Un}^{\rm sc}_\phi\). We call \(\mathrm{Un}_\phi^{\rm sc}\) the scaled unstraightening functor associated with \(\phi\). It is known that the adjunction \((\mathrm{St}^{\rm sc}_\phi,\, \mathrm{Un}_\phi^{\rm sc})\) is a Quillen adjunction. Here the model structure on \(\mathrm{Set}_{\Delta/\overline{S}}^+\) is the locally coCartesian model structure [5], and the model structure on \((\mathrm{Set}_{\Delta}^+)^{\mathcal{C}}\) is the projective model structure [7, pp. 823–824], Definition A.2.8.1 and Proposition A.2.8.2.
We recall the scaled straightening and unstraightening theorem.
Theorem 8 ([5] p.128, Theorem 3.8.1). Let \(\overline{S}\) be a scaled simplicial set, let \(\mathcal{C}\) be a \(\mathrm{Set}_{\Delta}^+\)-enriched category, and let \(\phi: \mathfrak{C}^{\rm sc}[\overline{S}] \to \mathcal{C}\) be a weak equivalence of \(\mathrm{Set}_{\Delta}^+\)-enriched categories. Then the Quillen adjunction \[\mathrm{St}_\phi^{\rm sc}: \mathrm{Set}_{\Delta/\overline{S}}^+ \rightleftarrows (\mathrm{Set}_{\Delta}^+)^{\mathcal{C}}:\mathrm{Un}_\phi^{\rm sc}\] is a Quillen equivalence. 0◻
We recall the model structure on \(\mathrm{Set}_{\Delta}^{\rm sc}\) using the model structure on \(\mathrm{Cat}_{\Delta}^+\).
Definition 7 ([5] p.115, Definition 3.5.6). Let \(f:\overline{X} \to \overline{Y}\) be a morphism of scaled simplicial sets. We say that \(f\) is a bicategorical equivalence if the induced functor \(\mathfrak{C}^{\rm sc}[f]: \mathfrak{C}^{\rm sc}[\overline{X}] \to \mathfrak{C}^{\rm sc}[\overline{Y}]\) is a weak equivalence of \(\mathrm{Set}_{\Delta}^+\)-enriched categories.
Theorem 9 ([5] p.143, Theorem 4.2.7). Let \(\mathrm{Set}_{\Delta}^{\rm sc}\) denote the category of scaled simplicial sets. Then \(\mathrm{Set}_{\Delta}^{\rm sc}\) has a left proper combinatorial model structure determined by the following data:
The weak equivalences in \(\mathrm{Set}_{\Delta}^{\rm sc}\) are bicategorical equivalences.
The cofibrations in \(\mathrm{Set}_{\Delta}^{\rm sc}\) are monomorphisms.
The fibrations in \(\mathrm{Set}_{\Delta}^{\rm sc}\) are the morphisms having the right lifting property with respect to all morphisms satisfying (W) and (C).
Furthermore, the Quillen adjunction \[\mathfrak{C}^{\rm sc} : \mathrm{Set}_{\Delta}^{\rm sc} \rightleftarrows \mathrm{Cat}_{\Delta}^+ : N^{\rm sc}\] is a Quillen equivalence. 0◻
We call the model structure on \(\mathrm{Set}_{\Delta}^{\rm sc}\) in Theorem 9 the scaled model structure.
Definition 8 ([5] p.145, Definition 4.2.8). An \(\infty\)-bicategory is a fibrant object of \(\mathrm{Set}_{\Delta}^{\rm sc}\) with respect to the scaled model structure.
Definition 9. The scaled nerve of the \(\mathrm{Set}_{\Delta}^+\)-enriched category \(\mathrm{Set}_{\Delta}^+\) is an \(\infty\)-bicategory of \(\infty\)-categories, denoted \(\mathbf{Cat}_\infty\). Let \({}^{\rm L}\mathbf{Top}\) denote the subcategory of the \(\infty\)-bicategory \(\mathbf{Cat}_\infty\) spanned by \(\infty\)-topoi, whose morphisms are colimit-preserving functors.
Definition 10 (cf. [7] p.369, Definition 5.2.8.8 (Joyal)). Let \(\mathcal{C}\) be a motivic \(\infty\)-category. A factorization system \((S_L,\,S_R)\) is a pair of collections of morphisms of \(\mathcal{C}\) satisfying the following axioms:
The collections \(S_L\) and \(S_R\) are closed under retracts.
The collection \(S_L\) is left orthogonal to \(S_R\).
For any morphism \(h:X \to Z\) in \(\mathcal{C}\), there are an object \(Y\) of \(\mathcal{C}\) and morphisms \(f:X \to Y\) and \(g:Y \to Z\) such that \(h=g \circ f\), \(f \in S_L\), and \(g \in S_R\).
Let \(\mathcal{T}\) and \(\mathcal{T}'\) be motivic \(\infty\)-topoi. Let \(\mathrm{Fun}^*(\mathcal{T},\,\mathcal{T}')\) denote the full subcategory of \(\mathrm{Fun}(\mathcal{T},\,\mathcal{T}')\) spanned by functors \(f:\mathcal{T} \to \mathcal{T}'\) that admit geometric left adjoints. A motivic classifying \(\infty\)-topos is a motivic \(\infty\)-topos \(\mathcal{K}\) such that \(\mathcal{K}(U)\) is a classifying \(\infty\)-topos for any \(U \in \mathcal{X}\), and such that the transition morphisms \(\mathcal{K}(U) \to \mathcal{K}(V)\) are compatible with the geometric structures for any morphism \(V \to U\) in \(\mathcal{X}\).
Definition 11 ([1] p.27, Definition 1.4.3). Let \(\mathcal{K}\) be a motivic \(\infty\)-topos. A geometric structure on \(\mathcal{K}\) is a factorization system \((S_L^\mathcal{T},\, S_R^\mathcal{T})\) on \(\mathrm{Fun}^*(\mathcal{K},\,\mathcal{T})\) that depends functorially on \(\mathcal{T}\). We call \(\mathcal{K}\) a classifying motivic \(\infty\)-topos, and we call a morphism in \(S^{\mathcal{T}}_R\) a local morphism. For any classifying motivic \(\infty\)-topos \(\mathcal{K}\) and any motivic \(\infty\)-topos \(\mathcal{T}\), we let \(\mathrm{Str}^{\rm loc}_{\mathcal{K}}(\mathcal{T})\) denote the subcategory of \(\mathrm{Fun}^*(\mathcal{K},\,\mathcal{T})\) with the same objects and with morphisms given by local morphisms. We call an object of \(\mathrm{Str}^{\rm loc}_{\mathcal{K}}(\mathcal{T})\) a \(\mathcal{K}\)-structured sheaf on \(\mathcal{T}\). If a geometric morphism \(f:\mathcal{K} \to \mathcal{K}'\) of classifying motivic \(\infty\)-topoi induces, for any motivic \(\infty\)-topos \(\mathcal{T}\), a pullback functor that carries local morphisms in \(\mathrm{Fun}^*(\mathcal{K}',\,\mathcal{T})\) to local morphisms in \(\mathrm{Fun}^*(\mathcal{K},\,\mathcal{T})\), then we say that \(f\) is compatible with the geometric structures.
The scaled straightening and unstraightening adjunction \[\mathrm{St}^{\rm sc}: ( \mathrm{Set}_{\Delta}^+)_{/N(\mathcal{X})\times {}^{\rm L} \mathbf{Top}^{\rm op} } \rightleftarrows ( \mathrm{Set}_{\Delta}^+)^{ \mathcal{X}^{\rm op} \times \mathfrak{C}^{\rm sc}\left[ {}^{\rm L} \mathbf{Top} \right]} :\mathrm{Un}^{\rm sc}\] induces a Quillen equivalence \[\mathrm{Mot}^{I}_{\mathcal{X}}(\mathrm{St}^{\rm sc}): (\mathrm{Set}_{\Delta})^+_{/ N(\mathcal{X})\times {}^{\rm L} \mathbf{Top}^{\rm op} } \rightleftarrows \mathrm{Mot}^{I}_{\mathcal{X}}((\mathrm{Set}_{\Delta}^+)^{\mathfrak{C}^{\rm sc}\left[{}^{\rm L} \mathbf{Top} \right]}) :\mathrm{Mot}^{I}_{\mathcal{X}}(\mathrm{Un}^{\rm sc})\] of left proper combinatorial simplicial model categories. Let \[{}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}): \mathcal{X} \ni U \mapsto {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}(U)) \in N((\mathrm{Set}_{\Delta}^+)^\circ)_{/{}^{\rm L}\mathbf{Top}^{\rm op}}\] denote the scaled unstraightening of the \(\mathrm{Cat}_{\infty}\)-valued presheaf \[\mathrm{Str}^{\rm loc}_\mathcal{K}: \mathcal{X}^{\rm op} \ni U \mapsto \mathrm{Str}^{\rm loc}_{\mathcal{K}(U)}(-) \in \mathrm{Fun}_{\mathrm{Cat}_{(\infty,\,2)}} ({}^{\rm L}\mathbf{Top},\,\mathbf{Cat}_{\infty}).\]
Furthermore, the motivic Yoneda functor \[\mathrm{Fun}^*(\mathcal{K},\,- ): N(\mathcal{X}) \times {}^{\rm L}\mathbf{Top} \to \widehat{\mathbf{Cat}_{\infty}}\] classifies an \(\infty\)-category \({}^{\mathrm{L}}\mathfrak{MTop}_{\mathcal{K}/}\) and a motivic locally coCartesian fibration \(q:{}^{\mathrm{L}}\mathfrak{MTop}_{\mathcal{K}/} \to N(\mathcal{X})\times{}^{\rm L}\mathbf{Top}\). By an argument analogous to the proof of [7], the \(\infty\)-category \({}^{\mathrm{R}}\mathfrak{MTop}\) admits pullbacks. In other words, for any geometric morphism \(f:\mathcal{K} \to \mathcal{K}'\), the forgetful functor \(f_* : {}^{\mathrm{R}}\mathfrak{MTop}_{/\mathcal{K}} \to {}^{\mathrm{R}}\mathfrak{MTop}_{/\mathcal{K}'}\) admits a right adjoint. The opposite category of \({}^{\mathrm{L}}\mathfrak{MTop}_{\mathcal{K}/}\) is weakly equivalent to \(({}^{\mathrm{R}}\mathfrak{MTop}_{/\mathcal{K}})^{\rm op}\) as a motivic \(\infty\)-category. Hence we have a homotopically commutative diagram of \(\infty\)-categories: \[\xymatrix@1{ {}^{\mathrm{L}}\mathfrak{MTop}_{ \mathcal{K} / } \ar[r]<0.5mm>^{f_*} & {}^{\mathrm{L}}\mathfrak{MTop}_{\mathcal{K}'/ } \ar[l]<0.5mm>^{f^{-1}} \\ {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}) \ar[u] & {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}') \ar[l]^{f^{-1}} \ar[u] }.\] We now prove that the lower horizontal functor has a left adjoint.
Theorem 10. Let \(f:\mathcal{K} \to \mathcal{K}'\) be a geometric morphism of motivic classifying \(\infty\)-topoi compatible with the geometric structures. Given the commutative diagram \[\xymatrix@1{ {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}) \ar[dr]_p & & \ar[ll]^{f^{-1}} {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}') \ar[dl]_q \\ & N(\mathcal{X})\times {}^{\rm L} \mathbf{Top} & }\] where \(f^{-1}\) is the functor induced by \(f\), and where \(p\) and \(q\) are motivic locally coCartesian fibrations. Then \(f^{-1}:{}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}') \to {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K})\) admits a left adjoint relative to \(N(\mathcal{X})\times {}^{\rm L}\mathbf{Top}\).
The unstraightening functor \[\mathrm{Mot}^{I}_{\mathcal{X}}(\mathrm{Un}^{\rm sc}) : \mathrm{Mot}^{I}_{\mathcal{X}}((\mathrm{Set}_{\Delta}^+)^{\mathfrak{C}^{\rm sc}\left[{}^{\rm L} \mathbf{Top} \right]}) \to (\mathrm{Set}_{\Delta})^+_{/ N(\mathcal{X}) \times {}^{\rm L} \mathbf{Top}^{\rm op} }\] carries fibrant objects to fibrant objects. Hence the induced morphism \(f^{-1}:{}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}') \to {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K})\) is a morphism between fibrant objects over \(N(\mathcal{X}) \times {}^{\rm L}\mathbf{Top}^{\rm op}\). In particular, \(f^{-1}\) carries motivic locally \(q\)-coCartesian edges to motivic locally \(p\)-coCartesian edges. By [10], it is sufficient to prove that the functor \[f^{-1}_{\mathcal{T}}: \mathrm{Str}^{\rm loc}_{\mathcal{K}'} (\mathcal{T}) \to \mathrm{Str}^{\rm loc}_{\mathcal{K}} (\mathcal{T})\] admits a left adjoint for any motivic \(\infty\)-topos \(\mathcal{T}\). Let \(\mathcal{O}_\mathcal{T}\) be an object of \(\mathrm{Fun}^*(\mathcal{K},\,\mathcal{T})\), and let \(\mathcal{O}'_\mathcal{T}\) be an object of \(\mathrm{Fun}^*(\mathcal{K}',\,\mathcal{T})\). Let \(\phi:\mathcal{O}_\mathcal{T} \to \mathcal{O}'_\mathcal{T} \circ f\) be a local morphism in \(\mathrm{Fun}^*(\mathcal{K},\,\mathcal{T})\). We obtain a left Kan extension \(f_*\mathcal{O}_\mathcal{T}: \mathcal{K}' \to \mathcal{T}\) along \(f\). The transformation \(\phi_*: f_*(\mathcal{O}_\mathcal{T}) \to \mathcal{O}'_\mathcal{T}\) induced by \(\phi\) gives a functorial factorization \[f_*(\mathcal{O}_\mathcal{T}) \to \mathrm{MSpc}^{\mathcal{K}}_{\mathcal{K}'} (\mathcal{O}_\mathcal{T}) \overset{\mathrm{MSpc}(\alpha)}\to \mathcal{O}'_\mathcal{T}\] where \(\mathrm{MSpc}(\alpha)\) is local. Hence we obtain a functor \[\mathrm{MSpc}^{\mathcal{K}'}_{\mathcal{K},\,\mathcal{T}}: f_*(\mathcal{O}_\mathcal{T}) \to \mathcal{O}'_\mathcal{T}\] which is a left Kan extension of \(f^{-1}_\mathcal{T}\). 0◻
Remark 11. In this paper, the motivic \(\infty\)-category \({}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K})\) is constructed following [1].
Fix a regular Noetherian separated scheme \(S\) of finite dimension, and let \(\mathbf{Sm}_S\) be the Nisnevich site of smooth schemes over \(S\) with interval object \(\mathbb{A}^1\). We write \(\mathfrak{MS}_\infty\) for the motivic \(\infty\)-category of motivic spaces and \(\mathfrak{MSp}_\infty\) for the stable motivic \(\infty\)-category of motivic spectra. Let \(\mathfrak{MSp}_\infty^\omega\) denote the full subcategory of compact motivic spectra.
Recall that Zariski topos is classified by the factorization system of local morphisms between local rings, and étale topos is classified by the factorization system of local morphisms between strict Henselian local rings. We introduce the definition of local ring objects, local morphisms and strict Henselian local ring objects of a motivic \(\infty\)-topos.
Let \(\mathcal{T}\) be a topos with a final object \(\boldsymbol{1}\) and \(\mathcal{O}\) a commutative algebra object of \(\mathcal{T}\). We say that \(\mathcal{O}\) is local [2] if the following conditions are satisfied:
Let \(0, \, 1 : \boldsymbol{1} \to \mathcal{O}\) denote the additive identity and multiplicative identity in \(\mathcal{O}\). Then \(\boldsymbol{1} \times_\mathcal{O} \boldsymbol{1}\) is an initial object of \(\mathcal{T}\).
Let \(\mathcal{O}^\times\) be the multiplicative group of \(\mathcal{O}\) which is given by the pullback square \[\xymatrix@1{ \mathcal{O}^\times \ar[d] \ar[r]^e & \mathcal{O} \times \mathcal{O} \ar[d] \\ \boldsymbol{1} \ar[r] & \mathcal{O}, }\] where \(m: \mathcal{O} \times \mathcal{O} \to \mathcal{O}\) is the multiplication equipped with \(\mathcal{O}\). Then the map \[(1-e)\coprod e : \mathcal{O}^\times \coprod \mathcal{O}^\times \to \mathcal{O}\] is an effective epimorphism (See [7]).
Let \(\alpha: \mathcal{O} \to \mathcal{O}'\) be a morphism of commutative local ring objects of \(\mathcal{T}\). We say that \(\alpha : \mathcal{O} \to \mathcal{O}'\) is a local morphism if the diagram \[\xymatrix@1{ \mathcal{O}^\times \ar[r]^\alpha \ar[d] & \mathcal{O}'^\times \ar[d] \\ \mathcal{O} \ar[r]_\alpha & \mathcal{O}' }\] is a pullback square.
Definition 12 (c.f. [2] p.13, Definition 2.5). Let \(\mathcal{O}\) be a commutative ring object of a motivic \(\infty\)-topos on \(\mathcal{T}\). We say that \(\mathcal{O}\) is a local if \(\pi_0 \mathcal{O}(U)\) is a local ring object of \(\mathrm{h} \mathcal{T} (U)\) for any \(U \in \mathcal{T}\).
Let \(\mathcal{K}^{M}_{disc}\) denote the motivic \(\infty\)-topos \(\mathrm{Fun}( \mathrm{CAlg}(\mathfrak{MSp}^\omega_\infty)^{\rm op},\,\mathfrak{MS}_\infty)\). We introduce the classifying Zariski topos on \(\mathcal{K}_{\rm disc}^M\). The motivic \(\infty\)-categorical Yoneda functor \(y: \mathfrak{MSp}^{\omega}_\infty \to \mathrm{Fun}( \mathfrak{MSp}_\infty^{\omega,\,\rm op} ,\,\mathfrak{MS}_\infty)\) induces \(\mathrm{CAlg}(y): \mathrm{CAlg}( \mathfrak{MSp}^{\omega}_\infty) \to \mathrm{CAlg}(\mathrm{Fun}( \mathfrak{MSp}_\infty^{\omega, \,\rm op} ,\,\mathfrak{MS}_\infty))\). Hence we obtain the canonical functor: \[\mathrm{Fun}(\mathrm{CAlg}(\mathfrak{MSp}^\omega_\infty)^{\rm op}, \, \mathfrak{MS}_\infty ) \to \mathrm{CAlg}( \mathrm{Fun}( \mathfrak{MSp}_\infty^{\omega,\, \rm op} ,\,\mathfrak{MS}_\infty) ).\] It is well-known that this canonical functor induces weak equivalences on each fiber on \(X \in \mathcal{X}\). Therefore the canonical functor is a weak equivalence of motivic \(\infty\)-categories. We let \(\mathcal{K}_{\rm Zar}^M\) denote the subcategory of \(\mathcal{K}_{\rm disc}^M\) whose morphisms are local morphisms. We say that \(\mathcal{K}_{\rm Zar}^M\) is the motivic \(\infty\)-Zariski topos. Write \(\mathrm{MSch}= {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}_{\rm Zar}^M)^{\rm op}\). A motivic scheme is an object of the motivic \(\infty\)-category \(\mathrm{MSch}\). Equivalently a motivic scheme is an \(\mathbb{A}^1\)-homotopy invariant spectral scheme-valued Nisnevich-local sheaf on \(\mathbf{Sm}_S\).
Write \(\mathrm{MAlgSp}= {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}_{\text{disc}}^M)^{\rm op}\). We refer to \(\mathrm{MAlgSp}\) as the motivic \(\infty\)-category of motivic algebraic spaces. Then the geometric morphism \(\mathcal{K}_{\text{disc}}^M \to \mathcal{K}_{\text{Zar}}^M\) induces the functor \[(-)^{\text{Zar}} : \mathrm{MAlgSp} \to \mathrm{MStk}\] which admits a left adjoint by Theorem 10.
Definition 13 ([2] p.68, Definition 8.1). Let \(\mathcal{T}\) be a topos and \(\mathcal{O}_\mathcal{X}\) a commutative ring object of \(\mathcal{T}\). For any finitely generated algebra \(R\), let \(\mathrm{Sol}_R(\mathcal{O}_\mathcal{T})\) be an object of \(\mathcal{T}\) defined by \[\mathrm{Sol}_R(\mathcal{O}_\mathcal{T}) : \mathcal{T} \ni U \mapsto \mathrm{Hom}_{\rm Ring}(R,\, \mathrm{Hom}_\mathcal{T}(U,\,\mathcal{O}_\mathcal{T}) ) \in \mathrm{Sets}.\] We say that \(\mathcal{O}_\mathcal{T}\) is strictly Henselian, if for any finite collection of étale maps \(R \to R_\alpha\) which induce a faithfully flat map \(R \to \prod_\alpha R_\alpha\), the induced map \[\coprod_\alpha \mathrm{Sol}_{R_\alpha}(\mathcal{O}_\mathcal{T}) \to \mathrm{Sol}_R (\mathcal{O}_\mathcal{T})\] is an effective epimorphism.
Definition 14 (c.f. [2] p.68, Definition 8.3). Let \(\mathcal{T}\) be a motivic \(\infty\)-topos and \(\mathcal{O}_\mathcal{T}\) a commutative algebra object of \(\mathcal{T}\). Then we say that \(\mathcal{O}_\mathcal{T}\) is strictly Henselian if \(\pi_0 \mathcal{O}_\mathcal{T}(U)\) is a strictly Henselian commutative ring object of the category \(\mathrm{h} \mathcal{T}(U)\) for each \(U \in \mathcal{T}\). Let \(\alpha:\mathcal{O}_\mathcal{T} \to \mathcal{O}'_\mathcal{T}\) be a local morphism of local commutative algebra objects of \(\mathcal{T}\). We say that \(\alpha\) is a strict Henselian local if \(\mathcal{O}_\mathcal{T}\) and \(\mathcal{O}'_\mathcal{T}\) are strict Henselian.
Let \(\mathcal{K}_{\text{\'et}}^M\) denote the subcategory of \(\mathcal{K}_{\rm disc}^M\) whose morphisms are strict Henselian local. We say that the motivic \(\infty\)-topos \(\mathcal{K}_{\text{\'et}}^M\) is the motivic \(\infty\)-étale topos. Write \(\mathrm{MStk}= {}^{\mathrm{L}}\mathfrak{MTop}(\mathcal{K}_{\text{\'et}}^M)^{\rm op}\). A motivic stack is an object of the motivic \(\infty\)-category \(\mathrm{MStk}\). By a similar argument to the case of motivic schemes, a motivic stack is a \(\mathbb{A}^1\)-homotopy invariant spectral Deligne–Mumford stack-valued Nisnevich-local sheaf. Similarly, the geometric morphism \(\mathcal{K}_{\text{disc}}^M \to \mathcal{K}_{\text{\'et}}^M\) induces the functor \[(-)^{\text{\'et}} : \mathrm{MAlgSp} \to \mathrm{MStk}\] which admits a left adjoint by Theorem 10. Let \(\mathbb{X}\) be a motivic algebraic space. Then we say that \(\mathbb{X}^{\text{\'et}}\) is the motivic stack associated to \(\mathbb{X}\).
Corollary 2. Let \(\mathrm{MAlgSp}\) and \(\mathrm{MStk}\) denote the motivic \(\infty\)-categories of motivic algebraic spaces and motivic stacks, respectively. The geometric morphism \(\mathcal{K}_{disc}^M \to \mathcal{K}_{\text{\'et}}^M\) induces a pullback functor \((-)^{\text{\'et}} : \mathrm{MAlgSp} \to \mathrm{MStk}\). This functor admits a left adjoint relative to the underlying \(\infty\)-topos. Consequently, for any motivic algebraic space \(\mathbb{X}\), there universally exists an associated motivic stack \(\mathbb{X}^{\text{\'et}}\).
This follows directly from Theorem 10 and the definition of the motivic \(\infty\)-étale topos \(\mathcal{K}_{\text{\'et}}^M\). The relative left adjoint provides the required universal property of the stackification.
The author used Google’s Gemini-pro 3.1 as a conversational research aid for brainstorming and refining mathematical formulations, and OpenAI’s Prism (GPT-5.2) for editorial and expository assistance, including structural refinement. These tools were used to improve the manuscript’s clarity and to indicate where abbreviated arguments warranted fuller exposition. All results, proofs, and final formulations were reviewed, verified, and approved by the author, who bears sole responsibility for the content of this work.