2. Lax natural transformations

In this section, we recall the $\infty$ -categories of functors and (op)lax natural transformations, and discuss (op)lax slice categories and the mate equivalence.

Definition 2.1. Let ${\mathcal{I}}$ be an $\infty$ -category and $\mathbf{C}$ an $(\infty,2)$ -category. We define the $(\infty,2)$ -categories

$$ \mathbf{Fun}^{\mathrm{lax}}({\mathcal{I}},\mathbf{C})\quad \mathrm{and}\quad\mathbf{Fun}^{\mathrm{oplax}}({\mathcal{I}},\mathbf{C}) $$

as the right-adjoint objects for the left and right Gray tensor product by ${\mathcal{I}}$ , respectively, in the sense of [Reference Gagna, Harpaz and LanariGHL21], so that

\begin{align*} \operatorname{Map}_{\mathrm{Cat}_2}(\mathbf{A} \boxtimes {\mathcal{I}}, \mathbf{C})&\simeq \operatorname{Map}_{\mathrm{Cat}_2}(\mathbf{A}, \mathbf{Fun}^{\mathrm{lax}}({\mathcal{I}},\mathbf{C})),\\ \operatorname{Map}_{\mathrm{Cat}_2}({\mathcal{I}}\boxtimes \mathbf{A} , \mathbf{C})&\simeq \operatorname{Map}_{\mathrm{Cat}_2}(\mathbf{A}, \mathbf{Fun}^{\mathrm{oplax}}({\mathcal{I}},\mathbf{C})), \end{align*}

for every $(\infty,2)$ -category $\mathbf{A}$ . The underlying $\infty$ -categories will be denoted $\operatorname{Fun}^{\mathrm{lax}}({\mathcal{I}}, \mathbf{C})$ and $\operatorname{Fun}^{\mathrm{oplax}}({\mathcal{I}},\mathbf{C})$ , respectively.

Example 2.2. The objects of $\operatorname{Fun}^{\mathrm{lax}}({\mathcal{I}},\mathbf{C})$ agree with the objects of $\operatorname{Fun}({\mathcal{I}},\mathbf{C})$ . A morphism in $\operatorname{Fun}^{\mathrm{lax}}({\mathcal{I}},\mathbf{C})$ is precisely the data of a lax natural transformation. Informally, a lax natural transformation $\eta \colon F \Rightarrow G$ between functors $F,G\colon {\mathcal{I}} \to \mathbf{C}$ is a coherent collection of squares as follows, together with 2-morphisms $\epsilon\colon G(f) \circ \eta_i \Rightarrow \eta_j\circ F(f) $ .

If the 2-morphisms $\epsilon$ are all equivalences, then $\eta$ defines a natural transformation. Dually, an oplax natural transformation $\eta \colon F \Rightarrow G$ is informally a coherent collection of oplaxly commuting squares.

We now give a definition of partially (op)lax limits, first introduced in [Reference BermanBer20].

Definition 2.4. Let $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ be a functor. We define the lax limit of F,

$$\operatorname{laxlim}_{{\mathcal{I}}} F := \operatorname{Nat}^{\mathrm{lax}}(\Delta(\ast),F), $$

to be the $\infty$ -category of lax cones: lax natural transformations from the constant functor on the terminal $\infty$ -category to F. Here, $\Delta\colon\mathrm{Cat}\to\operatorname{\mathbf{Fun}}^{\mathrm{lax}}({\mathcal{I}},\mathrm{Cat})$ denotes the diagonal which, under the identification $\mathrm{Cat}\simeq\operatorname{\mathbf{Fun}}^{\mathrm{lax}}(\ast,\mathrm{Cat})$ , is precomposition with the projection ${\mathcal{I}}\to\ast$ .

Suppose ${\mathcal{I}}$ is marked by a subcategory ${\mathcal{W}}$ . Then, we define the partially lax limit of F,

$$\operatorname{laxlim}^{\dagger}_{{\mathcal{I}},{\mathcal{W}}} F \subset \operatorname{laxlim}_{{\mathcal{I}}} F, $$

to be the full subcategory of $\operatorname{laxlim} F$ spanned by those lax cones whose restriction to ${\mathcal{W}}\subset {\mathcal{I}}$ is strictly natural.

Dually, we define the oplax limit $\operatorname{oplaxlim} F$ of F to be the $\infty$ -category $\operatorname{Nat}^{\mathrm{oplax}}(\Delta(\ast),F)$ of oplax cones and $\operatorname{oplaxlim^\dagger} F\subset \operatorname{oplaxlim} F$ to be the full subcategory spanned by those oplax cones whose restriction to ${\mathcal{W}}$ is strictly natural.

We now introduce an (un)straightening equivalence for the $\infty$ -category of functors and lax natural transformations. This will be a crucial tool for working with partially lax limits, and will also allow us to connect the definition given above to the definitions of [Reference BermanBer20] and [Reference Linskens, Nardin and PolLNP25].

Theorem 2.6 [Reference Haugseng, Hebestreit, Linskens and NuitenHHL⁺23a, Theorem E]. The (un)straightening equivalences of Lurie extend to natural equivalences

(2.1) \begin{equation}\mathbf{Cocart}^{\mathrm{lax}}({\mathcal{I}}) \simeq \mathbf{Fun}^{\mathrm{lax}}({\mathcal{I}},{\mathbf{Cat}}) \quad \text{and} \quad\mathbf{Cart}^{\mathrm{oplax}}({\mathcal{I}}) \simeq \mathbf{Fun}^{\mathrm{oplax}}({\mathcal{I}}^{\operatorname{op}},{\mathbf{Cat}}).\end{equation}

Remark 2.7. The previous equivalence gives an equivalence

$$\operatorname{Nat}^{\mathrm{lax}}(F,G)\simeq \operatorname{Fun}_{{\mathcal{I}}}(\mathrm{Un}^{\mathrm{co}}(F),\mathrm{Un}^{\mathrm{co}}(G)) $$

when restricted to the mapping $\infty$ -category from F to G. In particular, because the identity functor on ${\mathcal{I}}$ is the unstraightening of $\Delta(\ast)$ , we obtain an equivalence

$$\operatorname{laxlim} F \simeq \operatorname{Fun}_{{\mathcal{I}}}({\mathcal{I}},\mathrm{Un}^{\mathrm{co}}(F)). $$

That is, the lax limit of F is equivalent to the $\infty$ -category of sections of the cocartesian unstraightening of F. Furthermore, if ${\mathcal{I}}$ is marked by ${\mathcal{W}}$ , then a lax cone is in $\operatorname{laxlim}^{\dagger} F$ if and only if the associated section ${\mathcal{I}}\rightarrow \mathrm{Un}^{\mathrm{co}}(F)$ sends maps in ${\mathcal{W}}$ to cocartesian edges in $\mathrm{Un}^{\mathrm{co}}(F)$ . This follows immediately from the fact that Equation (2.1) restricts to the usual (un)straightening equivalence.

We conclude, by [Reference BermanBer20, Theorem 4.4], that our definition agrees with the definition of partially lax limits given there, and used in [Reference Linskens, Nardin and PolLNP25]. Finally, we note that all of the observations above dualize and imply analogous statements for oplax limits.

Example 2.8. A particularly degenerate consequence of the previous observations is already interesting. Recall that the cocartesian unstraightening of the constant functor $\Delta({\mathcal{C}})\colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ is given by the projection ${\mathcal{C}}\times {\mathcal{I}}\rightarrow {\mathcal{I}}$ . Therefore, we may compute that

$$\operatorname{laxlim} \Delta({\mathcal{C}}) \simeq \operatorname{Fun}_{{\mathcal{I}}}({\mathcal{I}},{\mathcal{I}}\times {\mathcal{C}}) \simeq \operatorname{Fun}({\mathcal{I}},{\mathcal{C}}). $$

We conclude that functors $F\colon {\mathcal{I}}\rightarrow {\mathcal{C}}$ are equivalent to lax cones over the constant functor on ${\mathcal{C}}$ . Dually, functors $F\colon {\mathcal{I}}^{\operatorname{op}}\rightarrow {\mathcal{C}}$ are equivalent to oplax cones over the constant functor on ${\mathcal{C}}$ .

While not immediately clear from the description above, we note that if $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ lifts to a diagram in $\mathrm{Cat}^{\otimes}$ , the $\infty$ -category of symmetric monoidal $\infty$ -categories and symmetric monoidal functors, then the partially lax limit $\operatorname{laxlim}^{\dagger} F$ is canonically symmetric monoidal.

Proposition 2.9. Let $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}^{\otimes}$ be a functor. Then $\operatorname{laxlim}^{\dagger} F$ admits a symmetric monoidal structure such that, for any symmetric monoidal $\infty$ -category ${\mathcal{C}}$ , we have a natural equivalence

$$\operatorname{Fun}^{\otimes\mathrm{-{lax}}}({\mathcal{C}},\operatorname{laxlim}^{\dagger} F) \simeq \operatorname{laxlim}^{\dagger} \operatorname{Fun}^{\otimes\mathrm{-{lax}}}({\mathcal{C}},F(i)) $$

between lax symmetric monoidal functors. In particular,

$$\operatorname{CAlg}(\operatorname{laxlim}^{\dagger} F) \simeq \operatorname{laxlim}^\dagger \operatorname{CAlg}(F(i)). $$

In the remainder of this section, we recall some technical material which we will use in later sections.

2.1 Lax slice categories

In this subsection, we introduce the definition of (op)lax slice categories, and use this to make some identifications of lax limits which will be useful later.

Definition 2.10. Given an $(\infty,2)$ -category $\mathbf{C}$ with underlying $\infty$ -category ${\mathcal{C}}$ and an object X in $\mathbf{C}$ , we define $\mathbf{C} \mathbin{\downarrow^{\operatorname{opl}}} X $ , the oplax slice category over X, via the following pullback.

Analogously, we can define the notion of lax slice category ${\mathbf{C}} \mathbin{\downarrow^{\mathrm{lax}}} X$ .

Our interest in oplax slice categories comes from the following theorem.

Theorem 2.11. Let ${\mathbf{C}}$ be an $(\infty,2)$ -category with underlying $\infty$ -category ${\mathcal{C}}$ , and fix $X\in {\mathbf{C}}$ .

1. (1) The forgetful functor $\mathrm{fgt}\colon {\mathbf{C}} \mathbin{\downarrow^{\operatorname{opl}}} X\rightarrow {\mathcal{C}}$ is a cartesian fibration and classifies the mapping functor $\operatorname{Map}_{{\mathbf{C}}}(-,X)\colon {\mathcal{C}}^{\operatorname{op}}\rightarrow \mathrm{Cat}$ .

2. (2) The forgetful functor $\mathrm{fgt} \colon ({\mathbf{C}} \mathbin{\downarrow^{\mathrm{lax}}} X)^{\operatorname{op}} \to {\mathcal{C}}^{\operatorname{op}}$ is a cocartesian fibration and classifies the same functor as above.

Proof. Part (1) is proved in [Reference Haugseng, Hebestreit, Linskens and NuitenHHL⁺23b, Theorem 7.21]. Part (2) follows by applying part (1) to ${\mathbf{C}}^{2\text{-op}}$ , the $(\infty,2)$ -category obtained from ${\mathbf{C}}$ by passing to opposites on hom-categories.

Applying this to ${\mathbf{C}} = {\mathbf{Cat}}$ , we obtain another formulation of the (un)straightening equivalence. We write $\operatorname{Cart}$ for the subcategory of $\operatorname{Ar}(\mathrm{Cat})$ spanned by cartesian fibrations and maps of cartesian fibrations (see Definition 7.1).

Corollary 2.12. There exists an equivalence

$$ \mathrm{Un} \colon {\mathbf{Cat}}\mathbin{\downarrow^{\operatorname{opl}}} \mathrm{Cat} \xrightarrow{\sim} \operatorname{Cart}. $$

Proof. The equivalence is induced by unstraightening the natural equivalence

$$\operatorname{Fun}(-,\mathrm{Cat}) \simeq \operatorname{Cart}((-)^{\operatorname{op}}).$$

Construction 2.13. Suppose we have a functor

$$ F\colon {\mathcal{I}}\rightarrow {{\mathbf{Cat}}\mathbin{\downarrow^{\operatorname{opl}}} {{\mathcal{C}}}},\quad i\mapsto [\phi_i\colon F(i)\rightarrow {\mathcal{C}}]. $$

We abuse notation and write F also for the functor $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ obtained by forgetting the functor to ${\mathcal{C}}$ . By Theorem 2.11(1), the forgetful functor ${{\mathbf{Cat}}\mathbin{\downarrow^{\operatorname{opl}}} {{\mathcal{C}}}} \to \mathrm{Cat}$ is the cartesian unstraightening of the functor $\operatorname{Fun}(-,{\mathcal{C}})\colon \mathrm{Cat}^{\operatorname{op}} \to \mathrm{Cat}$ . By pulling back along $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ , this implies that we can interpret F as a section of the cartesian fibration

$$ q\colon \mathrm{Un}^{\mathrm{ct}}(\operatorname{Fun}(F(-), {\mathcal{C}}))\rightarrow {\mathcal{I}}^{\operatorname{op}}. $$

By the dual of Remark 2.7, this equivalently defines an object of $\operatorname{oplaxlim}_{{\mathcal{I}}^{\operatorname{op}}} \operatorname{Fun}(F(-),{\mathcal{C}})$ . Summarizing, we have exhibited, for any $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ , an equivalence between $\operatorname{oplaxlim} \operatorname{Fun}(F(-),{\mathcal{C}})$ and $\operatorname{Fun}_{\mathrm{Cat}}({\mathcal{I}}, {\mathbf{Cat}}\mathbin{\downarrow^{\operatorname{opl}}} {{\mathcal{C}}})$ . The previous argument in fact applies to any $(\infty,2)$ -category ${\mathbf{C}}$ and an object $X\in {\mathbf{C}}$ , and gives an identification of

$$F\colon {\mathcal{I}}\rightarrow {\mathbf{C}} \mathbin{\downarrow^{\operatorname{opl}}} X $$

and objects of $\operatorname{oplaxlim} {\mathbf{C}}(F(-),X)$ . We later apply this also to ${\mathbf{Cat}}^{\otimes,\mathrm{lax}}$ .

2.2 The mate equivalence

We now introduce the mate equivalence, which in fact takes two forms. To state them, we introduce some notation.

Definition 2.14. Let ${\mathcal{I}}$ be an $\infty$ -category and let Cat be the (very large) $(\infty,2)$ -category of large $\infty$ -categories. We write

$$ \operatorname{\mathbf{Fun}}^{\mathrm{lax}}_{\mathrm{L}}({\mathcal{I}}, {\mathbf{Cat}}) \quad \text{and} \quad \operatorname{\mathbf{Fun}}^{\mathrm{oplax}}_{\mathrm{R}}({\mathcal{I}}, {\mathbf{Cat}}) $$

for the full subcategory of $\operatorname{\mathbf{Fun}}^{\mathrm{lax}}({\mathcal{I}}, {\mathbf{Cat}})$ and $\operatorname{\mathbf{Fun}}^{\mathrm{oplax}}({\mathcal{I}}, {\mathbf{Cat}})$ , respectively, spanned by those functors ${\mathcal{I}} \to \mathrm{Cat}$ which send each morphism in ${\mathcal{I}}$ to a left and right adjoint, respectively.

Theorem 2.15 [Reference Haugseng, Hebestreit, Linskens and NuitenHHL⁺23a, Theorem 5.3.6]. Let ${\mathcal{I}}$ be an $\infty$ -category. Then, there is an equivalence

$$ \operatorname{\mathbf{Fun}}^{\mathrm{lax}}_{\mathrm{L}}({\mathcal{I}}, {\mathbf{Cat}})\simeq \operatorname{\mathbf{Fun}}^{\mathrm{oplax}}_{\mathrm{R}}({\mathcal{I}}^{\operatorname{op}},{\mathbf{Cat}}) $$

which sends a diagram $F\colon {\mathcal{I}} \to \mathrm{Cat}^L$ to the corresponding diagram $G\colon {\mathcal{I}}^{\mathrm{op}}\to \mathrm{Cat}^R$ of right adjoints.

Example 2.16. Suppose $F,F'\colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ are two objects of $\operatorname{Fun}_{\mathrm{L}}^{\mathrm{lax}}({\mathcal{I}},\mathrm{Cat})$ , and write G and G’, respectively, for the associated ${\mathcal{I}}^{\operatorname{op}}$ -shaped diagrams of right adjoints. Consider a lax natural transformation $\gamma \colon F \Rightarrow F'$ , given at a map $f\colon i\to j$ in ${\mathcal{I}}$ by the following laxly commuting square.

(2.2)

By [Reference Haugseng, Hebestreit, Linskens and NuitenHHL⁺23a, Proposition 3.2.7], the first mate equivalence sends $\gamma$ to the oplax natural transformation $\gamma \colon G\Rightarrow G'$ with squares of the following form.

(2.3)

The oplax square above is filled by the Beck–Chevalley transformation $\beta$ of $\alpha$ , defined to be the composite

$$ \gamma_j G(f) \Rightarrow{\eta} G'(f)F'(f) \gamma_i G(f) \Rightarrow{\alpha} G'(f) \gamma_j F(f) G(f) \Rightarrow{\epsilon} G'(f)\gamma_i. $$

For the second mate equivalence, we again introduce some notation.

Definition 2.17. Let ${\mathcal{I}}$ be an $\infty$ -category. We write

$$ \operatorname{Fun}^{\mathrm{R},\mathrm{lax}}({\mathcal{I}}, {\mathbf{Cat}}) \quad \text{and} \quad \operatorname{Fun}^{\mathrm{L},\mathrm{oplax}}({\mathcal{I}}, {\mathbf{Cat}}) $$

for the wide subcategories of $\operatorname{Fun}^{\mathrm{lax}}({\mathcal{I}}, {\mathbf{Cat}})$ and $\operatorname{Fun}^{\mathrm{oplax}}({\mathcal{I}}, {\mathbf{Cat}})$ , respectively, spanned on morphisms by those (op)lax natural transformations $\eta\colon F \Rightarrow G$ such that each component $\eta_i$ is a right/left adjoint, respectively.

Theorem 2.18 [Reference Haugseng, Hebestreit, Linskens and NuitenHHL⁺23a, Theorem 5.3.5]. Let ${\mathcal{I}}$ be an $\infty$ -category. Then, there is an equivalence

$$ \operatorname{\mathbf{Fun}}^{\mathrm{R},\mathrm{lax}}({\mathcal{I}}, {\mathbf{Cat}})\simeq \operatorname{\mathbf{Fun}}^{\mathrm{L},\mathrm{oplax}}({\mathcal{I}},{\mathbf{Cat}})^{(1,2)\text{-op}}, $$

where $(-)^{(1,2)\text{-op}}$ denotes the functor obtained by passing to opposites on the level of both 1- and 2-morphisms.

Example 2.19. Given the square (2.2), let $\gamma_i^{\mathrm{L}}$ and $\gamma_j^{\mathrm{L}}$ denote the left adjoints of $\gamma_i$ and $\gamma_j$ , respectively. Then, the second mate equivalence sends the square (2.2) to the following oplaxly commuting square.

Here $\beta$ is the Beck–Chevalley transformation of $\alpha$ , defined to be the composite

\begin{align*} \gamma_j^{\mathrm{L}} F'(f) \Rightarrow{\eta\,} \gamma_j^{\mathrm{L}} F'(f)\gamma_i \gamma_i^{\mathrm{L}} \Rightarrow{\alpha\,} \gamma_j^{\mathrm{L}}\gamma_j F(f)\gamma_i^{\mathrm{L}} \Rightarrow{\epsilon\,} F(f)\gamma_i^{\mathrm{L}}\\[-20pt] \end{align*}

Example 2.20. Specializing the second mate equivalence to the case ${\mathcal{I}} \simeq \ast$ , we obtain an equivalence ${\mathbf{Cat}}^{\mathrm{R}}\simeq ({\mathbf{Cat}}^{\mathrm{L}})^{(1,2)\text{-op}}$ .

As a first example of the mate equivalence, we observe that one can identify certain lax and oplax limits taken over adjoint diagrams.

Proposition 2.21. Let $G\colon {\mathcal{I}} \rightarrow \mathrm{Cat}^{\mathrm{R}}$ be a diagram of right adjoints. Write $F\colon {\mathcal{I}}^{\operatorname{op}}\rightarrow \mathrm{Cat}^{\mathrm{L}}$ for the diagram of left adjoints. Then, there exists an equivalence $\operatorname{oplaxlim}_{{\mathcal{I}}} G\simeq \operatorname{laxlim}_{{\mathcal{I}}^{\operatorname{op}}} F$ .

Proof. On categories, this follows immediately from the following chain of equivalences:

$$ \operatorname{oplaxlim} G = \operatorname{Nat}^{\mathrm{oplax}}(\Delta(\ast),G) \simeq \operatorname{Nat}^{\mathrm{lax}}(\Delta(\ast),F) = \operatorname{laxlim} F, $$

where the first and third are by definition, and the middle one follows from the definition of adjoint diagrams in higher category theory.

3. Unstable global homotopy theory

In this section, we recall the necessary background on global homotopy theory following [Reference Gepner and HenriquesGH07, Reference RezkRez14, Reference Linskens, Nardin and PolLNP25].

To begin, recall that a global family is a collection of compact Lie groups ${\mathcal{E}}$ which is closed under isomorphisms, passage to subgroups, and quotients. A global family is said to be multiplicative if, in addition, it is closed under finite products.

1. (1) We let $\operatorname{Glo}_{{\mathcal{E}}}$ denote the global orbit $\infty$ -category of [Reference Gepner and HenriquesGH07] with isotropy in ${\mathcal{E}}$ , whose objects are given by ${\mathbf{B}} G$ for $G\in{\mathcal{E}}$ . Up to homotopy, morphisms ${\mathbf{B}} H\rightarrow {\mathbf{B}} G$ in $\operatorname{Glo}_{{\mathcal{E}}}$ are given by conjugacy classes of continuous group homomorphisms $\alpha\colon H\rightarrow G$ . More precisely, by [Reference Linskens, Nardin and PolLNP25, Proposition 6.3], we have

   (3.1) \begin{equation} \operatorname{Glo}_{{\mathcal{E}}}({\mathbf{B}} H, {\mathbf{B}} G) \simeq \coprod\limits_{[\alpha]} BC(\alpha), \end{equation}
   where $[\alpha]$ runs through the set of conjugacy classes of continuous group homomorphisms from H to G, and $C(\alpha)$ denotes the centralizer of the image of $\alpha$ .

2. (2) We denote by $\mathrm{Orb}_{{\mathcal{E}}}\subseteq \operatorname{Glo}_{{\mathcal{E}}}$ the wide subcategory spanned by those morphisms which are represented by an injective group homomorphism.

3. (3) The $\infty$ -category of ${\mathcal{E}}$ -global spaces $\mathcal{S}_{\mathcal{E}-\mathrm{gl}} := \operatorname{Fun}(\operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}},\mathcal{S})$ is the presheaf $\infty$ -category on $\operatorname{Glo}_{{\mathcal{E}}}$ . We identify ${\mathbf{B}} G$ with its image under the Yoneda embedding.

Crucial to the story of global homotopy theory is its close relationship to equivariant homotopy theory. It follows from [Reference Linskens, Nardin and PolLNP25, Lemma 6.13] that, for any $G\in {\mathcal{E}}$ , the slice $\infty$ -category $(\mathrm{Orb}_{{\mathcal{E}}})_{/{{\mathbf{B}} G}}$ can be identified with $\mathrm{Orb}_G$ , the orbit $\infty$ -category of the compact Lie group G. Restricting along the functor $(\mathrm{Orb}_{{\mathcal{E}}})_{/{{\mathbf{B}} G}} \to \operatorname{Glo}_{{\mathcal{E}}}$ , $({\mathbf{B}} H \to {\mathbf{B}} G)\mapsto{\mathbf{B}} H$ , and using Elmendorf’s theorem yields a restriction functor

$$\mathrm{res}_G \colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}} \to \mathcal{S}_G $$

into the $\infty$ -category of G-spaces. The right Kan extension along ${\mathrm{Orb}_{{\mathcal{E}}}}_{/{{\mathbf{B}} G}} \to {\operatorname{Glo}_{{\mathcal{E}}}}$ defines a right adjoint to $\mathrm{res}_G$ . We record the following special case.

The restriction functor $\mathrm{res}_G$ also admits a left adjoint $-{/{\kern1pt}\!\!/} G\colon \mathcal{S}_G \to {\mathcal{S}_{\mathcal{E}-\mathrm{gl}}}$ , which is given by left Kan extension. It is important to record the following special case.

One verifies that $\mathrm{pt} /{\kern1pt}\!\!/ G \simeq {\mathbf{B}} G$ , so we obtain an induced functor $-/{\kern1pt}\!\!/ G \colon \mathcal{S}_G \to {\mathcal{S}_{\mathcal{E}-\mathrm{gl}}}_{/{\mathbf{B}} G}$ . To describe the essential image of this functor, we introduce the following definition.

Definition 3.9. A morphism of global spaces $f\colon \mathcal{X} \to \mathcal{Y}$ is faithful (or representable) if, for every ${\mathbf{B}} G\in\operatorname{Glo}_{{\mathcal{E}}}$ and normal subgroup $N \triangleleft G$ , the following diagram is a pullback.

We distinguish faithful maps by writing $\mathcal{X} \hookrightarrow \mathcal{Y} $ .

Theorem 3.15. Let ${\mathcal{E}}$ be a global family and consider $G\in {\mathcal{E}}$ . The functor

$$ -{/{\kern1pt}\!\!/} G\colon \mathcal{S}_G\rightarrow {\mathcal{S}_{\mathcal{E}-\mathrm{gl}}}_{/{\mathbf{B}} G} $$

is fully faithful and an object $f\colon \mathcal{X}\rightarrow {\mathbf{B}} G$ is in its essential image if and only if f is faithful.

Proof. The claim for the global family of all compact Lie groups is proved in [Reference Gepner and MeierGM23, Proposition 2.19]. We can obtain the general case from this one by noting that there is a factorization

$$ -{/{\kern1pt}\!\!/} G\colon \mathcal{S}_G \to (\mathcal{S}_{\mathcal{E}-\mathrm{gl}})_{/{\mathbf{B}} G} \to (\mathcal{S}_{\mathrm{gl}})_{/{\mathbf{B}} G}, $$

where both functors are fully faithful.

Remark 3.17. We can also give an interpretation of the functoriality of equivariant spaces using the previous result. Given a group homomorphism $\alpha\colon H\rightarrow G$ , pullback and postcomposition by $B\alpha\colon {\mathbf{B}} H\rightarrow {\mathbf{B}} G$ induces an adjunction

$${\mathbf{B}} \alpha_!\colon {\mathcal{S}_{\mathcal{E}-\mathrm{gl}}}_{/{\mathbf{B}} H} \rightleftarrows {\mathcal{S}_{\mathcal{E}-\mathrm{gl}}}_{/{\mathbf{B}} G}{:}{\mathbf{B}} \alpha^*. $$

As observed in Remark 3.12, pulling back preserves faithful maps. Therefore, the right adjoint above restricts to a functor

$$({\mathbf{B}} \alpha)^*\colon (\mathcal{S}_{\mathcal{E}-\mathrm{gl}})_{/{\mathbf{B}} G}^{\mathrm{fth}} \rightarrow (\mathcal{S}_{\mathcal{E}-\mathrm{gl}})_{/{\mathbf{B}} H}^{\mathrm{fth}} $$

between faithful morphisms. By [Reference Linskens, Nardin and PolLNP25, Proposition 6.17], this functor agrees under the equivalences of Theorem 3.15 with the standard restriction functoriality $\alpha^*\colon \mathcal{S}_G\rightarrow \mathcal{S}_H$ of equivariant spaces.

It follows, by standard arguments, that the left adjoint of ${\mathbf{B}}\alpha^*$ , restricted to global spaces over ${\mathbf{B}} G$ and ${\mathbf{B}} H$ with faithful structure map, is computed by first applying ${\mathbf{B}} \alpha_!$ and then reflecting back into $(\mathcal{S}_{\mathcal{E}-\mathrm{gl}})_{/{\mathbf{B}}G}^{\mathrm{fth}}$ . This agrees, by uniqueness of adjoints, with the induction functor on equivariant spaces. In particular, the counit of the adjunction $\alpha_!\dashv \alpha^*$ is given by factoring the counit $\alpha^* X/{\kern1pt}\!\!/ H \simeq ({\mathbf{B}} \alpha)^* (X/{\kern1pt}\!\!/G)\rightarrow X/{\kern1pt}\!\!/ G$ of the pullback–postcomposition adjunction into a quotient map followed by a faithful map, as in the following diagram.

We also briefly discuss the identification of free G-spaces under the equivalence of Theorem 3.15. To establish notation, we recall that $\mathrm{End}_{\mathrm{Orb}_{G}} (G/e) \simeq G$ , and so we obtain a restriction functor

$$\mathcal{S}_G := \operatorname{Fun}(\mathrm{Orb}_{G},\mathcal{S})\xrightarrow{\mathrm{ev}_{G/e}} \operatorname{Fun}(BG,\mathcal{S})\simeq \mathcal{S}_{/BG}. $$

We call G-spaces in the image of the fully faithful left adjoint, free G-spaces.

Proposition 3.18. A map $\mathcal{X}\rightarrow BG$ of global spaces is faithful if and only if it is in the image of the constant functor. Now, consider the map $\nu\colon BG\rightarrow {\mathbf{B}} G$ from Example 3.11. Both functors in the adjunction

$$ \nu_!\colon {\mathcal{S}_{\mathcal{E}-\mathrm{gl}}}_{/BG} \rightleftarrows {\mathcal{S}_{\mathcal{E}-\mathrm{gl}}}_{/{\mathbf{B}} G} \, {:} \nu^* $$

preserve objects with a faithful structure map, and the following square commutes.

In particular, passing to left adjoints, we conclude that an object $\mathcal{X}\rightarrow {\mathbf{B}} G$ of ${\mathcal{S}_{\mathcal{E}-\mathrm{gl}}}_{/{\mathbf{B}} G}$ corresponds to a free G-space if and only if $\mathcal{X}$ is a constant global space and the map to ${\mathbf{B}} G$ is faithful.

Proof. Suppose $f\colon \mathcal{X}\rightarrow BG$ is a faithful map of global spaces. For any ${\mathbf{B}} H$ , the following square is a pullback square.

Thus, we conclude that the map $\mathcal{X}(\mathrm{pt})\rightarrow \mathcal{X}({\mathbf{B}} H)$ is an equivalence, and so $\mathcal{X}$ is constant. For the second statement, we first note that faithful maps are closed under pullback, and so $\nu^*$ clearly preserves faithful maps. Since composition of faithful maps is again faithful, and $\nu_!$ is given by postcomposing with $\nu$ , it follows that $\nu_!$ preserves faithful maps. The commutativity of the diagram follows from the observation that $\mathrm{res}_e$ (which agrees with evaluation at pt) preserves pullback squares, so we obtain the first equivalence in the following sequence:

$$ \mathrm{res}_e \nu^*\mathcal{X}\simeq \mathrm{res}_e X \simeq \mathrm{ev}_{G/e}\circ \mathrm{res}_G \mathcal{X}.$$

The connection between global and equivariant homotopy theory, expressed by Theorem 3.15, can be extended to give a different perspective on global homotopy theory. Recall from [Reference Linskens, Nardin and PolLNP25, § 6] that there exists a functor $\mathcal{S}_{\bullet}\colon \operatorname{Glo}_{{\mathcal{E}}}^{\mathrm{op}} \to \mathrm{Cat} $ , which sends ${\mathbf{B}} G$ to $\mathcal{S}_G$ , and sends a morphism ${\mathbf{B}}\alpha \colon {\mathbf{B}} H\to {\mathbf{B}} G$ to the restriction–inflation functor $\alpha^* \colon \mathcal{S}_G \to \mathcal{S}_H$ . We recall the following result.

Theorem 3.19 [Reference Linskens, Nardin and PolLNP25, Theorem 6.18]. Let ${\mathcal{E}}$ be a global family of compact Lie groups. Then, the restriction functors induce an equivalence of $\infty$ -categories

$$ \mathcal{S}_{\mathcal{E}-\mathrm{gl}}\simeq \operatorname{laxlim}^{\dagger}_{\operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}},\mathrm{Orb}_{{\mathcal{E}}}^{\operatorname{op}}} \mathcal{S}_\bullet, $$

where we view $\operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}$ as a marked $\infty$ -category via the inclusion $\mathrm{Orb}_{{\mathcal{E}}}^{\operatorname{op}}\subset \operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}$ .

• – a G-space $\mathrm{res}_G \mathcal{X}$ for each group $G\in {\mathcal{E}}$ ;

• – an H-equivariant map $f_\alpha\colon \alpha^*\mathrm{res}_G \mathcal{X}\to \mathrm{res}_H \mathcal{X}$ for each continuous group homomorphism $\alpha\colon H \to G$ ;

• – the maps $f_\alpha$ are functorial in the sense that there are given homotopies $f_{\beta\circ \alpha}\simeq f_\beta\circ \beta^*(f_{\alpha})$ for all composable maps $\alpha$ and $\beta$ , and $f_{\mathrm{id}}\simeq\mathrm{id}$ ;

• – the map $f_\alpha$ is an equivalence for every continuous injective homomorphism $\alpha$ ;

• – a homotopy between the map $f_{c_g}$ induced by the conjugation isomorphism and the map $l_g \colon c_g^* \mathrm{res}_G X \to \mathrm{res}_G X$ given by left multiplication by g;

• – higher coherences for the homotopies.

Here, we record a consequence of this description for constructing functors out of global spaces.

Proposition 3.22. Let $\mathcal{C}$ be an $\infty$ -category. Restriction along the functors $-{/{\kern1pt}\!\!/}G\colon\mathcal{S}_G\rightarrow \mathcal{S}_{\mathcal{E}-\mathrm{gl}}$ induces a functor

$$\Phi\colon \operatorname{Fun}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}},\mathcal{C})\to \operatorname{oplaxlim^\dagger}_{\operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}} \operatorname{Fun}(\mathcal{S}_\bullet,\mathcal{C}), $$

where the diagram $\operatorname{Fun}(\mathcal{S}_\bullet,\mathcal{C})$ is functorial in $\operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}$ as follows: a morphism ${\mathbf{B}} \alpha \colon {\mathbf{B}} H \to {\mathbf{B}} G$ in $\operatorname{Glo}_{{\mathcal{E}}}$ is sent to restriction along the induction functor $\alpha_!\colon \mathcal{S}_H \rightarrow \mathcal{S}_G$ . Furthermore, suppose $\mathcal{C}$ admits all small colimits. Then $\Phi$ restricts to an equivalence

$$\operatorname{Fun}^{\mathrm{L}}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}},\mathcal{C})\simeq \operatorname{oplaxlim^\dagger}\limits_{\operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}} \operatorname{Fun}^{\mathrm{L}}(\mathcal{S}_\bullet,\mathcal{C}). $$

Proof. Recall that $\mathcal{S}_{\mathcal{E}-\mathrm{gl}}\simeq \operatorname{laxlim}^{\dagger} \mathcal{S}_G$ . In particular, we obtain a universal partially lax cone as follows.

Passing to left adjoints (by which we mean using the equivalence ${\mathbf{Cat}}^{\mathrm{R}} \simeq ({\mathbf{Cat}}^{\mathrm{L}})^{(1,2)\text{-}\operatorname{op}}$ , see Example 2.20), we obtain a partially oplax cocone as follows.

(3.2)

Given a functor $F\colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}}\to \mathcal{C}$ , we may precompose by this cocone to obtain a partially oplax cocone with target $\mathcal{C}$ , encoded by a diagram $\operatorname{Glo}_{{\mathcal{E}}}\rightarrow {\mathbf{Cat}} \mathbin{\downarrow^{\operatorname{opl}}} \mathcal{C}$ . This, in turn, gives an object of $\operatorname{oplaxlim^\dagger} \operatorname{Fun}(\mathcal{S}_G,\mathcal{C})$ (see Construction 2.13).

For the final statement we first observe that, because each functor $-/{\kern1pt}\!\!/ G$ preserves colimits, the functor constructed above restricts appropriately. To see that it is an equivalence, we compute

\begin{align*} \operatorname{Fun}^{\mathrm{L}}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}},\mathcal{C}) \simeq \operatorname{Fun}(\operatorname{Glo}_{{\mathcal{E}}},\mathcal{C}) &\simeq \operatorname{Fun}(\operatorname{oplaxcolim^\dagger} \mathrm{Orb}_\bullet, \mathcal{C}) \\ &\simeq \operatorname{oplaxlim^\dagger} \operatorname{Fun}(\mathrm{Orb}_\bullet,\mathcal{C}) \simeq \operatorname{oplaxlim^\dagger} \operatorname{Fun}^{\mathrm{L}}(\mathcal{S}_\bullet,\mathcal{C}),\end{align*}

where the second and third equivalences are justified by the proof of Theorem 6.18 and Proposition 4.15 of [Reference Linskens, Nardin and PolLNP25], respectively.

Remark 3.23. Unwinding the proof of the previous proposition, we find that the image of $F\colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}} \rightarrow {\mathcal{C}}$ under the functor of Proposition 3.22 and the equivalence of Theorem 3.15 is given by the family of functors $\{F_G(X ) = F(X/{\kern1pt}\!\!/ G)\}$ . The oplax structure map $F_G(\alpha_!(-))\rightarrow F_H(-)$ is given at an object X by applying F to the unique quotient map $X/{\kern1pt}\!\!/ G\twoheadrightarrow \alpha_! X/{\kern1pt}\!\!/ H$ , for which the following square commutes.

4. Stable global homotopy theory

Global spectra were defined in [Reference SchwedeSch18] as an $\infty$ -category representing objects for genuine cohomology theories on global spaces. Just as in the unstable setting, one can describe global spectra as a laxly compatible collection of genuine equivariant spectra.

Theorem 4.1 [Reference Linskens, Nardin and PolLNP25, Theorem 11.10]. Let ${\mathcal{E}}$ be a multiplicative global family of compact Lie groups. Then, there exists a functor

$$\mathrm{Sp}_\bullet\colon \operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}\rightarrow \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}), \quad {\mathbf{B}} G\mapsto \mathrm{Sp}_G $$

which sends ${\mathbf{B}} G$ to the $\infty$ -category of genuine G-spectra and an equivalence of symmetric monoidal $\infty$ -categories

$$\mathrm{Sp}_{\mathcal{E}-\mathrm{gl}} \simeq \operatorname{laxlim}^{\dagger} \mathrm{Sp}_\bullet. $$

In this section, we refine this result and show that, in certain cases, a simpler description of global spectra is possible.

1. (1) ${\mathcal{T}}$ is closed under products in $\operatorname{Glo}_{{\mathcal{E}}}$ ; and

2. (2) given any object ${\mathbf{B}} G$ , there exists a faithful map ${\mathbf{B}} G\hookrightarrow {\mathbf{B}} K$ for some ${\mathbf{B}} K\in {\mathcal{T}}$ .

We view ${\mathcal{T}}$ as a marked $\infty$ -category by once again marking the subcategory of faithful maps.

The variant of Theorem 4.1 which we prove is as follows.

Theorem 4.7. Let ${\mathcal{E}}$ be a multiplicative global family of compact Lie groups, and suppose ${\mathcal{T}}\subset \operatorname{Glo}_{{\mathcal{E}}}$ is a family of enough injective objects. Then, there exists an equivalence of symmetric monoidal $\infty$ -categories

$$ \mathrm{Sp}_{\mathcal{E}-\mathrm{gl}} \simeq \operatorname{laxlim}^{\dagger}_{{\mathcal{T}}^{\operatorname{op}}} \mathrm{Sp}_\bullet. $$

These equivalences follow from the fact that the inclusion ${\mathcal{T}}^{\operatorname{op}}\subset \operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}$ is marked final, a concept from [Reference Abellán GarcaAbe22] which we recall now.

1. (1) Given any functor $G\colon {\mathcal{J}}\rightarrow \mathrm{Cat}$ , $\operatorname{laxlim}^{\dagger} G\simeq \operatorname{laxlim}^{\dagger} GF$ whenever either exist.

2. (2) Given any functor $G\colon {\mathcal{J}}\rightarrow \mathrm{Cat}$ , $\operatorname{oplaxlim^\dagger} G\simeq \operatorname{oplaxlim^\dagger} GF$ whenever either exist.

Given a marked $\infty$ -category ${\mathcal{C}}$ , we write $\mathcal{L}({\mathcal{C}})$ for the localization of ${\mathcal{C}}$ at the marked edges in ${\mathcal{C}}$ .

Proof. The product functor $-\times-\colon {\mathcal{C}}\times{\mathcal{C}}\rightarrow {\mathcal{C}}$ is determined by being right adjoint to the diagonal functor $\Delta_{{\mathcal{C}}}\colon {\mathcal{C}}\rightarrow {\mathcal{C}}\times{\mathcal{C}}$ . Our assumptions imply that the product derives to a functor $\mathcal{L}(-\times-)\colon \mathcal{L}({\mathcal{C}}\times {\mathcal{C}})\rightarrow \mathcal{L}({\mathcal{C}})$ , which is right adjoint to $\mathcal{L}(\Delta_{{\mathcal{C}}})$ . Here, we consider ${\mathcal{C}}\times {\mathcal{C}}$ as a marked $\infty$ -category by taking the product in marked $\infty$ -categories; explicitly, this is given by marking an edge (f,g) whenever both f and g are marked. One can compute that $\mathcal{L}({\mathcal{C}}\times {\mathcal{C}})\simeq \mathcal{L}({\mathcal{C}})\times \mathcal{L}({\mathcal{C}})$ ; that is, $\mathcal{L}$ preserves products. Therefore, we have the following commutative diagram.

The dotted arrow defines a product in $\mathcal{L}({\mathcal{C}})$ . The commutativity of the above diagram also shows that the localization functor preserves products.

It will be helpful to keep the following example in mind.

Example 4.13. Let ${\mathcal{T}} \subset \operatorname{Glo}_{{\mathcal{E}}}$ be a subcategory of enough injectives. For any ${\mathbf{B}} G\in \operatorname{Glo}_{{\mathcal{E}}}$ , we can form the comma category ${\mathcal{T}}_{{\mathbf{B}} G/}$ . A morphism as follows in this comma category is marked precisely when f is represented by an injective group homomorphism.

Note that ${\mathcal{T}}_{{\mathbf{B}} G/}$ has finite products, since ${\mathcal{T}}$ is closed in $\operatorname{Glo}_{{\mathcal{E}}}$ under finite products. Furthermore, taking products with a fixed object of ${\mathcal{T}}_{{\mathbf{B}} G/}$ preserves faithful maps, and therefore Lemma 4.12 implies that $\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})$ has finite products and that the localization ${\mathcal{T}}_{{\mathbf{B}} G/}\rightarrow \mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})$ preserves them.

The key ingredient for the proof of Theorem 4.7 is the following criterion.

1. (1) For every $j\in J$ , there exists an object $g\colon j\rightarrow F(i)$ of $I_{j/}$ which is marked when viewed as a morphism in J.

2. (2) Any object of $I_{j/}$ of the form above is an initial object in $\mathcal{L}(I_{j/})$ .

3. (3) Given a marked edge $j\rightarrow j'$ of J, the induced map $\mathcal{L}(I_{j'/})\rightarrow \mathcal{L}(I_{j/})$ preserves initial objects.

In order to verify condition (2) of the previous criterion, we need the following consequence of the axioms of a subcategory of enough injectives.

Lemma 4.15. Let ${\mathcal{T}}\subset \operatorname{Glo}_{{\mathcal{E}}}$ be a subcategory of enough injective objects. Given any map $f\colon {\mathbf{B}} H\rightarrow {\mathbf{B}} G$ in $\operatorname{Glo}_{{\mathcal{E}}}$ , there is a commutative square as follows in $\operatorname{Glo}_{{\mathcal{E}}}$ , such that the vertical maps are faithful and ${\mathbf{B}} K,{\mathbf{B}} J \in{\mathcal{T}}$ .

Proof. By condition (2) for ${\mathcal{T}}$ , there are faithful maps $i \colon {\mathbf{B}} H \hookrightarrow {\mathbf{B}} H'$ and $j \colon {\mathbf{B}} G \hookrightarrow {\mathbf{B}} G'$ with ${\mathbf{B}} H',{\mathbf{B}} G' \in {\mathcal{T}} $ . We can then form the following commutative square.

It is only left to note that ${\mathbf{B}} H' \times {\mathbf{B}} G'\in {\mathcal{T}}$ by condition (1), and that the map $i \times (j \circ f)$ is faithful since i is so.

Proof. We apply Theorem 4.14 to the inclusion ${\mathcal{T}}\rightarrow \operatorname{Glo}_{{\mathcal{E}}}$ . Condition (1) follows from Definition 4.2(2). Note that, if we assume condition (2), then condition (3) is immediate from the fact that a composite of faithful maps is faithful. Therefore, all that remains is condition (2). Fix an object ${\mathbf{B}} G\in \operatorname{Glo}_{{\mathcal{E}}}$ , and consider the $\infty$ -category ${\mathcal{T}}_{{\mathbf{B}} G/}$ . As discussed in Example 4.13, the comma category $\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})$ has finite products and the localization ${\mathcal{T}}_{{\mathbf{B}} G/}\rightarrow \mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})$ preserves them. However, given an object $\alpha\colon {\mathbf{B}} G\rightarrow X$ in ${\mathcal{T}}_{{\mathbf{B}} G/}$ , the following composite in ${\mathcal{T}}_{{\mathbf{B}} G/}$ exhibits pr_1 as an equivalence in $\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})$ .

That is, every object is equivalent to its product. On mapping spaces we obtain that

$$ \mathrm{pr}_1\colon \operatorname{Map}_{\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})}(\beta, \alpha)\times \operatorname{Map}_{\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})}(\beta, \alpha) \rightarrow \operatorname{Map}_{\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})}(\beta, \alpha) $$

is an equivalence for every two objects $\alpha,\beta$ of ${\mathcal{T}}_{{\mathbf{B}} G/}$ , which implies that $\operatorname{Map}_{\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})}(\beta, \alpha)$ is always either empty or contractible. Condition (2) of Theorem 4.14 requires that any faithful map $i\colon{\mathbf{B}} G\hookrightarrow {\mathbf{B}} J$ is initial in $\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})$ . First, we note that the following commutative diagram shows that any two faithful maps $i\colon{\mathbf{B}} G\hookrightarrow {\mathbf{B}} J$ and $j\colon{\mathbf{B}} G\hookrightarrow {\mathbf{B}} K$ are isomorphic in ${\mathcal{L}}({\mathcal{T}}_{/{\mathbf{B}} G})$ .

Combining this with the observation before, we see that it suffices to show that for any $\beta\in {\mathcal{T}}_{{\mathbf{B}} G/}$ , there exists a faithful map $i\colon {\mathbf{B}} G\hookrightarrow {\mathbf{B}} J$ such that $\operatorname{Map}_{\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})}(i, \beta)$ is non-empty. For this, we apply Lemma 4.15 to the map $\beta$ to obtain the following diagram.

Observe that we may interpret this as a zig–zag

in ${\mathcal{T}}_{{\mathbf{B}} G/}$ , where the second map is faithful. This proves that $\operatorname{Map}_{\mathcal{L}({\mathcal{T}}_{{\mathbf{B}} G/})}(i, \beta)$ is non-empty.

Proof of Theorem 4.7. Since the forgetful functor $\mathrm{Cat}^\otimes\to \mathrm{Cat}$ is conservative and preserves partially lax limits (see [Reference Linskens, Nardin and PolLNP25, Remark 5.1]), it suffices to verify that the induced functors on underlying $\infty$ -categories are equivalences. This now follows from combining Theorems 4.1 and 4.16.

PART II. Naive global 2-rings

In this part, we introduce naive global and equivariant ring spectra. We give a precise definition of a genuine refinement for such a ring, and introduce the notion of a naive global 2-ring, the central concept of this work. We then discuss how a naive global 2-ring decategorifies to give a naive global ring and list some examples. Finally, we spend the remainder of the part on the proof of Theorem A and some related material.

5. Naive global rings and genuine refinements

In this section, we introduce the naive analogs of global and equivariant ring spectra. We then discuss multiplicative cohomology theories and introduce the notion of genuine refinements.

Before diving into the main definitions of this section, we review some background on spectrum objects in an $\infty$ -category. Recall that, given any presentable $\infty$ -category $\mathcal{B}$ , one can define the category $\mathrm{Sp}(\mathcal{B})$ of spectrum objects in $\mathcal{B}$ . Combining [Reference LurieLur17, Proposition 4.8.1.17 and Example 4.8.1.23], we deduce that

$$\mathrm{Sp}(\mathcal{B}) \simeq \mathcal{B} \otimes \mathrm{Sp} \simeq \operatorname{Fun}^{\mathrm{R}}(\mathcal{B}^{\operatorname{op}},\mathrm{Sp}). $$

If $\mathcal{B}$ is presentably symmetric monoidal, then $\mathrm{Sp}(\mathcal{B})$ acquires a symmetric monoidal structure uniquely determined by the following universal property: for any stable and presentably symmetric monoidal $\infty$ -category ${\mathcal{C}}$ , precomposition with the suspension functor $\Sigma_+^\infty \colon \mathcal{B} \to \mathrm{Sp}(\mathcal{B})$ induces an equivalence

$$\operatorname{Fun}^{\mathrm{L},\otimes}(\mathrm{Sp}(\mathcal{B}), {\mathcal{C}})\simeq \operatorname{Fun}^{\mathrm{L},\otimes}(\mathcal{B}, {\mathcal{C}}) $$

(see [Reference Gepner, Groth and NikolausGGN15, Theorem 5.1]).

Remark 5.1. For a stable $\infty$ -category ${\mathcal{C}}$ and two objects $X,Y\in {\mathcal{C}}$ , we write $\operatorname{map}_{\mathcal{C}}(X,Y)$ for the spectrum of maps from X to Y. Moreover, we let

$$y \colon {\mathcal{C}} \rightarrow \operatorname{Fun}^{\mathrm{R}}({\mathcal{C}}^{\operatorname{op}}, \mathrm{Sp}), \quad X\mapsto \operatorname{map}_{{\mathcal{C}}}(-,X), $$

denote the spectral Yoneda embedding. For an arbitrary presentably symmetric monoidal $\infty$ -category $\mathcal{B}$ , the equivalence $\mathrm{Sp}(\mathcal{B}) \simeq \operatorname{Fun}^{\mathrm{R}}(\mathcal{B}^{\operatorname{op}},\mathrm{Sp})$ is concretely given by the assignment $X \mapsto \operatorname{map}_{\mathrm{Sp}(\mathcal{B})}(\Sigma_+^\infty(-),X)$ .

We now specialize to our cases of interest.

The terminology above is intended to distinguish such objects from their genuine analogs, which contain substantially more structure.

It has been crucial in equivariant homotopy theory to build and exploit the additional structure contained in a genuine G-spectrum, over and above that of a naive G-spectrum. To systematically consider this, we make the following definition.

Definition 5.7. Let X be a naive G-ring. We say a G-ring $\tilde{X}\in\operatorname{CAlg}(\mathrm{Sp}_G)$ is a genuine refinement of X if there is an equivalence $\mathbb{U}_G(\tilde{X}) \simeq X$ . The space of genuine refinements of X is the following pullback in Cat.

Because $\mathbb{U}_G\colon \operatorname{CAlg}(\mathrm{Sp}_G) \rightarrow \operatorname{CAlg}(\mathrm{Sp}(\mathcal{S}_G))$ is conservative, $\mathrm{GenRef}(X)$ is a space.

Remark 5.8. We emphasize again that the space $\mathrm{GenRef}(X)$ of genuine refinements of a naive G-ring is not necessarily contractible. For a concrete example we note that, from the discussion in Remark 5.6, it suffices to give two different Green functor structures on the same coefficient system of rings. Consider the following $C_2$ -coefficient system R.

Note that there are different choices of transfer maps $\mathrm{tr}\colon \mathbb{F}_2\to \mathbb{F}_2[x]/x^2$ such that R is a Green functor; for example, $a \mapsto 0$ and $a \mapsto ax$ .

We may also consider a global analog of the previous definition. To do so, we first note that the equivariant suspension spectrum functors assemble to give a natural transformation $\Sigma_\bullet^\infty \colon \mathcal{S}_\bullet \Rightarrow \mathrm{Sp}_\bullet$ of symmetric monoidal left-adjoint functors defined over $\operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}$ (see [Reference Linskens, Nardin and PolLNP25, Proposition 10.5]).

Definition 5.9. We define the global suspension functor

$$\Sigma_{\mathrm{gl}}^{\infty}\colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}} \to \mathrm{Sp}_{\mathcal{E}-\mathrm{gl}} $$

as the functor induced on partially lax limits by the natural transformation $\Sigma_\bullet^\infty\colon \mathcal{S}_\bullet\rightarrow \mathrm{Sp}_\bullet$ .

Proof. As a partially lax limit of strong monoidal functors, $\Sigma_{\mathrm{gl}}^\infty$ is automatically strong monoidal. To construct $\Omega_{\mathrm{gl}}^\infty$ and exhibit it as the right adjoint of $\Sigma_{\mathrm{gl}}^\infty$ , we apply the criteria of [Reference LinskensLin24, Proposition 3.22]. For this, it suffices to show that for any faithful morphism $f \colon {\mathbf{B}} H \to {\mathbf{B}} G$ , the Beck–Chevalley natural transformation

$$f^* \Omega^\infty_G \Rightarrow \Omega^\infty_H f^* $$

is an equivalence. In other words, we have to check that $\Omega^\infty_\bullet$ is a natural transformation when restricted to $\mathrm{Orb}_{{\mathcal{E}}}^{\operatorname{op}}\subset \operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}$ . To check this, we can pass to total mates and instead verify that taking suspension spectra commutes with induction. This is a well-known fact, which could, for instance, be checked in one’s preferred model of equivariant spectra.

Given this, we can discuss the notion of a genuine refinement of a naive global ring.

5.1 Multiplicative cohomology theories

We typically construct and compare naive global and equivariant rings via the cohomology theories they represent.

Definition 5.15. Let $\mathcal{B}$ be a presentably symmetric monoidal $\infty$ -category and let $\mathcal{C}$ be a presentably symmetric monoidal stable $\infty$ -category. Then, a multiplicative $\mathcal{C}$ -valued cohomology theory on $\mathcal{B}$ is a lax symmetric monoidal limit-preserving functor $F\colon \mathcal{B}^{\operatorname{op}}\rightarrow \mathcal{C}$ . We write $\operatorname{Fun}^{\mathrm{R},\otimes\mathrm{-lax}}(\mathcal{B}^{\operatorname{op}},\mathcal{C})$ for the category of multiplicative cohomology theories. We are most interested in the following two cases.

1. (1) If ${\mathcal{E}}$ is a global family and $\mathcal{B} = \mathcal{S}_{\mathcal{E}-\mathrm{gl}}$ , we refer to the above functor as a multiplicative ${\mathcal{E}}$ -global ${\mathcal{C}}$ -valued cohomology theory.

2. (2) If G is a compact Lie group and $\mathcal{B}= \mathcal{S}_G$ , we refer to the above functor as a multiplicative G-equivariant ${\mathcal{C}}$ -valued cohomology theory.

Proposition 5.16. Let $\mathcal{B}$ be a presentably symmetric monoidal $\infty$ -category. The assignment $X \mapsto \operatorname{map}_{\mathrm{Sp}(\mathcal{B})}(\Sigma_+^\infty(-), X)$ defines an equivalence

$$\operatorname{CAlg}(\mathrm{Sp}(\mathcal{B})) \simeq \operatorname{Fun}^{\mathrm{R},\otimes\mathrm{-{lax}}}(\mathcal{B}^{\operatorname{op}},\mathrm{Sp}). $$

Proof. By Theorem A.3, there is a symmetric monoidal equivalence $\mathcal{B} \otimes \mathrm{Sp} \simeq \operatorname{Fun}^{\mathrm{R}}(\mathcal{B}^{\operatorname{op}},\mathrm{Sp})$ . Here, the right-hand side is symmetric monoidal by localizing the Day convolution symmetric monoidal structure, and the left-hand side is symmetric monoidal as it is the tensor product of two presentably symmetric monoidal $\infty$ -categories. As mentioned, there is an equivalence $\mathrm{Sp}(\mathcal{B})\simeq \mathcal{B} \otimes \mathrm{Sp}$ . We observe that, after precomposing with the suspension functor, it agrees with $\mathcal{B}\otimes \Sigma_+^\infty \colon \mathcal{B}\otimes \mathcal{S} \to\mathcal{B}\otimes\mathrm{Sp}$ , and so is canonically symmetric monoidal. Therefore, by the universal property of $\mathrm{Sp}(\mathcal{B})$ , the equivalence above is also symmetric monoidal. All in all, we have constructed a symmetric monoidal equivalence

$$\mathrm{Sp}(\mathcal{B}) \xrightarrow{\sim} \operatorname{Fun}^{\mathrm{R}}(\mathcal{B}^{\operatorname{op}}, \mathrm{Sp}), $$

which, by Remark 5.1, is given as in the proposition. In particular, we may compute

$$\operatorname{CAlg}(\mathrm{Sp}(\mathcal{B})) \simeq \operatorname{CAlg}(\operatorname{Fun}^{\mathrm{R}}(\mathcal{B}^{\operatorname{op}},\mathrm{Sp})) \simeq \operatorname{Fun}^{\mathrm{R},\otimes\mathrm{-lax}}(\mathcal{B}^{\operatorname{op}},\mathrm{Sp}), $$

where the second equivalence utilizes the universal property of Day convolution [Reference LurieLur17, Example 2.2.6.9], and the fact that the category of commutative algebras in a symmetric monoidal Bousfield localization is equivalent to commutative algebras whose underlying object is local.

Suppose $\mathcal{B}$ is equipped with the cartesian symmetric monoidal structure. By [Reference LurieLur17, Theorem 2.4.3.18], we obtain an equivalence

$$\operatorname{Fun}^{\otimes\mathrm{-lax}}(\mathcal{B}^{\operatorname{op}},{\mathcal{C}}) \simeq \operatorname{Fun}(\mathcal{B}^{\operatorname{op}},\operatorname{CAlg}({\mathcal{C}})) $$

between lax symmetric monoidal functors $F\colon \mathcal{B}^{\operatorname{op}}\rightarrow {\mathcal{C}}$ and ordinary functors $\tilde{F}\colon \mathcal{B}^{\operatorname{op}}\rightarrow \operatorname{CAlg}({\mathcal{C}})$ . Moreover, because the forgetful functor $\operatorname{CAlg}({\mathcal{C}})\rightarrow {\mathcal{C}}$ preserves and detects limits, F is limit-preserving if and only if $\tilde{F}$ is. Therefore, the equivalence above restricts to an equivalence

$$\operatorname{Fun}^{\mathrm{R},\otimes\mathrm{-lax}}(\mathcal{B}^{\operatorname{op}},{\mathcal{C}}) \simeq \operatorname{Fun}^{\mathrm{R}}(\mathcal{B}^{\operatorname{op}},\operatorname{CAlg}({\mathcal{C}})), $$

which gives an alternative definition of multiplicative $\mathcal{C}$ -valued cohomology theories on cartesian monoidal $\mathcal{B}$ . When $\mathcal{B}$ is the $\infty$ -category of G-spaces or of global spaces and ${\mathcal{C}}$ is the $\infty$ -category of spectra, we may combine this with the previous proposition to deduce the following result, where we write $\operatorname{CAlg}$ for $\operatorname{CAlg}(\mathrm{Sp})$ .

Proposition 5.17. For any compact Lie group G, the assignment $X \mapsto \operatorname{map}_{\mathrm{Sp}(\mathcal{S}_G)}(\Sigma_+^\infty(-), X)$ defines an equivalence

$$\operatorname{CAlg}(\mathrm{Sp}(\mathcal{S}_G))\simeq \operatorname{Fun}^{\mathrm{R}}(\mathcal{S}_G^{\operatorname{op}}, \operatorname{CAlg}). $$

Similarly, for any global family ${\mathcal{E}}$ , the assignment $X \mapsto \operatorname{map}_{\mathrm{Sp}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}})}(\Sigma_+^\infty(-), X)$ defines an equivalence

$$\operatorname{CAlg}(\mathrm{Sp}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}}))\simeq \operatorname{Fun}^{\mathrm{R}}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}}^{\operatorname{op}},\operatorname{CAlg}). $$

Now that we have discussed naive rings and multiplicative cohomology theories, we can further investigate the consequences of having a genuine refinement.

Proposition 5.18.

1. (1) The composite

   $$\mathbb{U}_G\colon \operatorname{CAlg}(\mathrm{Sp}_G)\rightarrow \operatorname{CAlg}(\mathrm{Sp}(\mathcal{S}_G)) \simeq \operatorname{Fun}^{\mathrm{R}}(\mathcal{S}_G^{\operatorname{op}},\operatorname{CAlg}) $$
   corresponds to the functor which sends a G-ring X to the functor $\operatorname{map}(\Sigma^\infty_{G}(-),X)$ . The analogous statement holds in the global case.

2. (2) Precomposition by the lax cocone $-/{\kern1pt}\!\!/ G\colon \mathcal{S}_G^{\operatorname{op}}\rightarrow \mathcal{S}_{\mathcal{E}-\mathrm{gl}}^{\operatorname{op}}$ defines an equivalence

   $$\Phi \colon \operatorname{Fun}^{\mathrm{R}}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}}^{\operatorname{op}}, \operatorname{CAlg}) \xrightarrow{\simeq}\operatorname{laxlim}^{\dagger} \operatorname{Fun}^{\mathrm{R}}(\mathcal{S}_G^{\operatorname{op}},\operatorname{CAlg}). $$

3. (3) The restriction functor induces an equivalence $F\colon \operatorname{CAlg}(\mathrm{Sp}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}}))\xrightarrow{\sim} \operatorname{laxlim}^{\dagger} \operatorname{CAlg}(\mathrm{Sp}(\mathcal{S}_G))$ which fits into the following commutative squares, where the top horizontal equivalence uses Proposition 2.9.

   

Proof. Part (1) follows from Proposition 5.17 combined with the adjunction equivalence

$$\operatorname{map}_{\mathrm{Sp}(\mathcal{S}_{G})}(\Sigma^\infty_+(-),\mathbb{U}_G(X)) \simeq \operatorname{map}_{\mathrm{Sp}_G}(\Sigma^\infty_G(-),X). $$

The proof of the global statement is completely analogous. The dual argument of Proposition 3.22 gives part (2). For part (3), we start by constructing a lax symmetric monoidal functor $F\colon \mathrm{Sp}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}}) \to \operatorname{laxlim}^{\dagger} \mathrm{Sp}(\mathcal{S}_G)$ . Using the universal property of the partially lax limits (see [Reference Linskens, Nardin and PolLNP25, Remark 5.1]), it suffices to define a partially lax family of lax symmetric monoidal functors $F_G \colon\mathrm{Sp}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}})\to \mathrm{Sp}(\mathcal{S}_G)$ . We can then invoke [Reference NikolausNik16] to see that there exists a unique lax symmetric monoidal functor $F_G$ making the following diagram commute.

Passing to commutative algebras and using Proposition 2.9, we obtain a functor

$$F=\{F_G\}\colon\operatorname{CAlg}(\mathrm{Sp}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}}))\to\operatorname{laxlim}^{\dagger} \operatorname{CAlg}(\mathrm{Sp}(\mathcal{S}_G)).$$

We claim that F makes the bottom square in part (3) commute. This, in particular, shows that F is an equivalence. The commutativity follows from the series of equivalences

\begin{align*}\operatorname{Map}_{\mathrm{Sp}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}})}(\Sigma_+^\infty(-/{\kern1pt}\!\!/ G), X) & \simeq \operatorname{Map}_{\mathcal{S}_G}(-, \mathrm{res}_G \Omega_{-}^\infty X)\\& \simeq \operatorname{Map}_{\mathcal{S}_G}(-, \Omega_{-}^\infty F_G X) \\& \simeq \operatorname{Map}_{\mathrm{Sp}(\mathcal{S}_G)}(\Sigma_+^\infty(-), F_GX).\end{align*}

It remains to prove that the top square in part (3) commutes. Unraveling the definitions, we see that the commutativity of the square reduces to proving an equivalence $\mathbb{U}_G \mathrm{res}_G X\simeq F_G \mathbb{U}_{\mathrm{gl}}(X)$ . Consider the following diagram.

We note that the left and right triangles commute. We can check this after passing to left adjoints, where it holds by definition. Moreover, the other square also commutes by the definition of $\Omega_{\mathrm{gl}}^\infty$ . Then, the commutativity of the top square follows by the universal property of $\mathrm{Sp}(\mathcal{S}_G)$ ; we can check this after precomposition with $\Omega^\infty_{-}$ .

Corollary 5.19. Suppose $\tilde{X}\in \mathrm{Sp}_{\mathcal{E}-\mathrm{gl}}$ is a genuine refinement of a naive global ring X. Then, $\mathrm{res}_G X$ is a genuine refinement of the naive G-ring $\mathrm{res}_G X$ .

Proof. This follows from Proposition 5.18(3).

6. Naive global 2-rings

In this section, we introduce the notion of a naive global 2-ring, which is the central concept of this work, and list some key examples.

The starting point of our discussion is a functor ${\mathcal{R}}\colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}}^{\operatorname{op}}\rightarrow \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}})$ . We can now perform two different constructions to ${\mathcal{R}}$ .

Definition 6.1. The decategorification of ${\mathcal{R}}$ is the composite

$$ \text{H}_{{\mathrm{gl}}}(-,{\mathcal{R}}) \colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}}^{\operatorname{op}} \xrightarrow{{\mathcal{R}}}\operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}) \xrightarrow{\mathrm{End}_\bullet(\mathbb{1})} \operatorname{CAlg}, $$

where the global section functor is given by $\mathcal{C}\mapsto \operatorname{map}_{\mathcal{C}}(\mathbb{1},\mathbb{1})$ .

The second construction that we can perform is given by the following result, which we state here but defer proving until later sections, as its proof requires substantial additional work.

Theorem 6.2. Consider a functor ${\mathcal{R}} \colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}}^{\operatorname{op}} \to \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}})$ . Then, for all $G\in {\mathcal{E}}$ , there exists a functor as described in Definition 6.3,

$$ \mathbb{H}_{{G}}(-,{\mathcal{R}})\colon \mathcal{S}_G^{\operatorname{op}}\rightarrow \operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G}), $$

such that, for every group homomorphism $\alpha \colon H \to G$ , there exists a natural transformation filling the following square.

(6.1)

These data are coherently functorial: this is encoded in the existence of a functor

$$ \mathbb{H}_{{\bullet}}(-,{\mathcal{R}})\colon \operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}\rightarrow \operatorname{Fun}^{\mathrm{oplax}}([1],{\mathbf{Cat}}^{\otimes,\mathrm{lax}}) $$

extending the assignment above.

We call the output $\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})$ of the previous theorem the unraveling of ${\mathcal{R}}$ . Our consideration of such families is inspired by the perspective on equivariant elliptic cohomology given in [Reference Ginzburg, Kapranov and VasserotGKV95].

1. (1) Given a group $G\in {\mathcal{E}}$ and a G-space X, which we regard as a global space $f\colon X/{\kern1pt}\!\!/ G \rightarrow {\mathbf{B}} G$ equipped with a faithful map to ${\mathbf{B}} G$ , we define $\mathbb{H}_{{G}}(X,{\mathcal{R}}) := f_* \mathbb{1}_{X/{\kern1pt}\!\!/ G}$ to be the pushforward of the unit of ${\mathcal{R}}_{X/{\kern1pt}\!\!/ G}$ to ${\mathcal{R}}_{{\mathbf{B}} G}$ .

2. (2) Given a map of G-spaces $X\to Y$ , encoded by the following commutative diagram in $\mathcal{S}_{\mathcal{E}-\mathrm{gl}}$ , we define $\mathbb{H}_{{G}}(h,{\mathcal{R}})$ to be the map

   $$g_*\mathbb{1}_{{\mathcal{R}}_{Y/{\kern1pt}\!\!/ G}}\xrightarrow{g_*(\mathrm{lax}_{h})} g_* h_* \mathbb{1}_{{\mathcal{R}}_{X/{\kern1pt}\!\!/ G}} \simeq f_* \mathbb{1}_{{\mathcal{R}}_{X/{\kern1pt}\!\!/ G}}, $$
   where $\mathrm{lax}_{h}$ is the lax unit map of the lax monoidal functor $h_*$ .
   

3. (3) Finally, given a group homomorphism $\alpha\colon H\rightarrow G$ , we define the transformation

   $$Q_\alpha\colon \alpha^*\mathbb{H}_{{G}}(-,{\mathcal{R}}) \Rightarrow \mathbb{H}_{{H}}(\alpha^*(-),{\mathcal{R}}) $$
   to be the mate of a natural transformation
   $$T_\alpha\colon \mathbb{H}_{{G}}(\alpha_!(-),{\mathcal{R}})\Rightarrow \alpha_*\mathbb{H}_{{H}}(-,{\mathcal{R}}). $$
   We define $T_\alpha$ at an H-space Z, corresponding to a faithful map $f\colon Z/{\kern1pt}\!\!/ H\hookrightarrow {\mathbf{B}} H$ , by considering the factorization of $\alpha f$ into a quotient map h followed by an injection $\alpha_! f$ .
   
   We then define $T_\alpha$ as the map
   \begin{align*} \mathbb{H}_{{G}}(\alpha_! Z/{\kern1pt}\!\!/ H,{\mathcal{R}}) &\simeq (\alpha_! f)_*(\mathbb{1}_{{\mathcal{R}}(\alpha_! Z/{\kern1pt}\!\!/ H)}) \xrightarrow{(\alpha_! f)_* (\mathrm{lax}_{h})} (\alpha_! f)_* h_* \mathbb{1}_{{\mathcal{R}}_{Z/{\kern1pt}\!\!/ H})}\\& \simeq \alpha_* f_*\mathbb{1}_{{\mathcal{R}}_{Z/{\kern1pt}\!\!/ H}} \simeq \alpha_* \mathbb{H}_{{H}}(Z,{\mathcal{R}}). \end{align*}

It will not surprise the reader that carefully dealing with all of the coherences implicit in this construction is a nontrivial task, which we carry out in §§ 7 and 8.

We want to think of the functor ${\mathcal{R}}$ as the categorification of the naive global ring $\text{H}_{{\mathrm{gl}}}(-,{\mathcal{R}})$ , and we want to think of the unraveling $\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})$ as a family of equivariant cohomology theories equipped with change-of-group transformations which are canonically associated to ${\mathcal{R}}$ . However, in this generality, the functors $\mathbb{H}_{{G}}(-,{\mathcal{R}})$ and $\text{H}_{{\mathrm{gl}}}(-,{\mathcal{R}})$ will not be limit-preserving, so we can fix this with the following definition.

Example 6.6. By definition, the decategorication of a naive global 2-ring defines a functor

$$\mathrm{2CAlg}^{\mathrm{naive}}_{\mathcal{E}-\mathrm{gl}} \to \operatorname{Fun}^{\mathrm{R}}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}}^{\operatorname{op}},\operatorname{CAlg}) \simeq \operatorname{CAlg}(\mathrm{Sp}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}})), \quad {\mathcal{R}} \mapsto \text{H}_{{\mathrm{gl}}}(-,{\mathcal{R}}) $$

to naive global rings, where we used Proposition 5.17.

A good source of examples is given by the following result.

Proof. Since the global section functor preserves limits, it is clear that the decategorication $\text{H}_{{\mathrm{gl}}}(-,{\mathcal{R}})$ is limit-preserving. Let us now argue that $\mathbb{H}_{{G}}(-,{\mathcal{R}})$ preserves limits, which we can do after forgetting commutative algebra structures. Now, consider X in $\mathcal{S}_G$ and suppose X is equivalent to the colimit of G-spaces $X_i$ . Because $-/{\kern1pt}\!\!/ G\colon \mathcal{S}_G\rightarrow \mathcal{S}_{\mathcal{E}-\mathrm{gl}}$ preserves colimits, we find that $Y:=X/{\kern1pt}\!\!/ G$ is a colimit of the global spaces $Y_i:=X_i/{\kern1pt}\!\!/ G$ . By our assumption on ${\mathcal{R}}$ , we have an equivalence ${\mathcal{R}}_{Y}\simeq \lim {\mathcal{R}}_{Y_i}.$ In particular, writing $\pi_i\colon Y_i\rightarrow Y$ for the components of the universal cocone, given any object B in ${\mathcal{R}}_{X/{\kern1pt}\!\!/ G}$ , we obtain an equivalence

$$ B\simeq \lim_i (\pi_i)_*\pi_i^* B $$

in ${\mathcal{R}}_Y$ . To see this we may, for example, compare universal properties:

\begin{align*} \operatorname{Map}_{{\mathcal{R}}_Y}(A,B) &\simeq \lim_i\operatorname{Map}_{{\mathcal{R}}_{Y_i}}(\pi_i^* A,\pi_i^*B) \\ &\simeq \lim_i\operatorname{Map}_{{\mathcal{R}}_{Y}}(A, (\pi_i)_*\pi_i^* B) \\ &\simeq \operatorname{Map}_{{\mathcal{R}}_{Y}}(A, \lim_i (\pi_i)_*\pi_i^* B).\end{align*}

Finally, note that the canonical faithful map $f\colon Y \rightarrow {\mathbf{B}} G$ is induced by the universal property of a colimit by the maps $f_i\colon Y_i \rightarrow {\mathbf{B}} G$ . We can now calculate

\begin{align*} \mathbb{H}_{{G}}(X,{\mathcal{R}}) = f_* \mathbb{1}_{{\mathcal{R}}_{X/{\kern1pt}\!\!/ G}} \simeq f_* \lim_i (\pi_i)_*\pi_i^* \mathbb{1}_{{\mathcal{R}}_{X/{\kern1pt}\!\!/ G}} &\simeq \lim_i f_* (\pi_i)_* \mathbb{1}_{{\mathcal{R}}_{{X_i/{\kern1pt}\!\!/ G}}} \\&\simeq \lim_i (f_i)_* \mathbb{1}_{{\mathcal{R}}_{{X_i/{\kern1pt}\!\!/ G}}} \simeq \lim_i\mathbb{H}_{{G}}(X_i,{\mathcal{R}}). \end{align*}

As the main theme of this work, we show that a naive global 2-ring, in fact, endows its decategorification with significantly more structure. Before we dive into this, let us first give some examples of naive global 2-rings.

Example 6.8. Given any presentably symmetric monoidal stable $\infty$ -category ${\mathcal{C}}$ , we may define a functor

$$ {\mathcal{C}}_{\mathrm{bor}}\colon {\operatorname{Glo}_{}}^{\operatorname{op}}\rightarrow \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}),\quad {\mathbf{B}}G\mapsto {\mathcal{C}}^{B G}. $$

The limit extension of this functor defines a naive global 2-ring, which we call a Borel 2-ring.

Example 6.9. Consider the functor ${\operatorname{Glo}_{}}^{\operatorname{op}} \to \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}})$ which sends ${\mathbf{B}} G$ to the $\infty$ -category of genuine G-spectra $\mathrm{Sp}_G$ , and a morphism ${\mathbf{B}}\alpha \colon {\mathbf{B}} H \to {\mathbf{B}} G$ to the restriction–inflation functor $\alpha^* \colon \mathrm{Sp}_G \to \mathrm{Sp}_H$ . Limit extending, we obtain a naive global 2-ring

$$ \mathrm{Sp}_\bullet \colon \mathcal{S}_{\mathrm{gl}}^{\operatorname{op}} \to \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}). $$

The resulting decategorified naive global ring is equivalent to the underlying naive global ring associated to the global sphere spectrum $\mathbb{S}_{\mathrm{gl}}$ .

Example 6.10. We may also consider the limit extension of the functor

$$\mathrm{Sp}_{\bullet\text{-}\mathrm{gl}}\colon \operatorname{Glo}_{\mathrm{fin}}^{\operatorname{op}}\to \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}), \quad {\mathbf{B}} G\mapsto \mathrm{Sp}_{G\text{-}\mathrm{gl}} $$

defined for the family of finite groups in [Reference Cnossen, Lenz and LinskensCLL25a]. This functor sends ${\mathbf{B}} G$ to the $\infty$ -category of G-global spectra in the sense of [Reference LenzLen25]. Once again, the associated decategorified global ring is equivalent to the global sphere spectrum.

Example 6.11. Generalizing Example 6.9, one can associate to any $R\in \operatorname{CAlg}(\mathrm{Sp}_{\mathcal{E}-\mathrm{gl}})$ the functor

$$\operatorname{Mod}_{R_\bullet}\colon \operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}\rightarrow \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}), \quad {\mathbf{B}} G \mapsto \operatorname{Mod}_{\mathrm{res}_G R}(\mathrm{Sp}_G). $$

We refer to [Reference Linskens, Nardin and PolLNP25, Theorem 5.10] for a construction of this functor. Taking the limit extension, we obtain a naive global 2-ring. Its decategorification agrees with the underlying naive global ring of R.

The following two examples form the motivation for this article. We introduce two multiplicative global cohomology theories, and then categorifications thereof. Here we are brief; we refer the reader to Part IV for a more extensive account of these examples.

Example 6.12. Let ${\mathbf{G}}$ be a preoriented $\mathbf{P}$ -divisible group over a commutative ring spectrum R, in the sense of [Reference LurieLur19, Definition 2.6.1]. Given these data, Lurie [Reference LurieLur19, Construction 4.0.3] constructs a multiplicative global cohomology theory

$$R_{{\mathbf{G}}}^\bullet\colon \mathcal{S}_{\mathrm{fab}\text{-}\mathrm{gl}}^{\operatorname{op}}\to \operatorname{CAlg}, $$

called ${\mathbf{G}}$ -tempered cohomology. Here, fab refers to the family of finite abelian groups. Given a finite abelian group H, there is a noncanonical equivalence

$$ \mathrm{Spec}(R_{{\mathbf{G}}}^{{\mathbf{B}} H})\simeq {\mathbf{G}}[\widehat{H}] $$

between the spectrum of $R_{{\mathbf{G}}}^{{\mathbf{B}} H}$ and the $\widehat{H}$ -torsion points in ${\mathbf{G}}$ , where $\widehat{H}$ denotes the Pontryagin dual of H.

Tempered cohomology is categorified by the notion of tempered local systems.

Example 6.13. Let ${\mathbf{G}}$ be a preoriented $\mathbf{P}$ -divisible group over R. As discussed in [Reference LurieLur19, Remark 5.2.11], there exists a limit-preserving functor

$$\mathrm{LocSys}_{\mathbf{G}} \colon \mathcal{S}_{\mathrm{fab}\text{-}\mathrm{gl}}^{\operatorname{op}}\rightarrow \widehat{\mathrm{Cat}}, \quad \mathcal{X} \mapsto \mathrm{LocSys}_{\mathbf{G}}(\mathcal{X}), $$

which sends a global space $\mathcal{X}$ to the $\infty$ -category of ${\mathbf{G}}$ -tempered local systems on $\mathcal{X}$ , and a morphism of global spaces $f\colon \mathcal{X} \to \mathcal{Y}$ to the pullback functor $f^* \colon \mathrm{LocSys}_{{\mathbf{G}}}(\mathcal{Y}) \to \mathrm{LocSys}_{{\mathbf{G}}}(\mathcal{X})$ (see § 15 for more details). If, in addition, ${\mathbf{G}}$ is oriented in the sense of [Reference LurieLur18b, Proposition 4.3.23], then Proposition 15.8 below implies that $\mathrm{LocSys}_{\mathbf{G}}$ is a naive global 2-ring.

So far, we have only given examples of naive global 2-rings coming from limit-preserving functors. The following example motivates our more general definition.

Example 6.14. Let $\mathsf{S}$ be a nonconnective spectral Deligne–Mumford stack and let ${\mathbf{E}}$ be an oriented elliptic curve in nonconnective spectral Deligne–Mumford stacks over $\mathsf{S}$ , in the sense of [Reference Gepner and MeierGM23, Definition 3.1]. Following [Reference Gepner and MeierGM23], we can associate to ${\mathbf{E}}$ a global cohomology theory

$$\text{H}_{{\mathrm{gl}}}(-,{\mathbf{E}}) \colon \mathcal{S}_{\mathrm{ab}\text{-}\mathrm{gl}}^{\operatorname{op}}\rightarrow \operatorname{CAlg}(\mathrm{Sp}),\quad {\mathbf{B}} A \mapsto \Gamma({\mathbf{E}}[\widehat{A}], {\mathcal{O}}_{{\mathbf{E}}[\widehat{A}]}). $$

Similarly to the case of tempered cohomology, if A is a compact abelian Lie group, we denote by ${\mathbf{E}}[\widehat{A}]$ the $\widehat{A}$ -torsion points of ${\mathbf{E}}$ .

Example 6.15. As we explain in § 14, there is also a way to associate to ${\mathbf{E}}\to \mathsf{S}$ a naive global 2-ring:

$${\mathcal{Q}}_\bullet^{\mathbf{E}} \colon \mathcal{S}_{\mathrm{ab}\text{-}\mathrm{gl}}^{\operatorname{op}}\rightarrow \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}). $$

Morally, this comes about by taking the categories of quasi-coherent sheaves on a suitable geometric extension of ${\mathbf{E}}$ to compact ab-global spaces. This assignment will not, however, be limit-preserving, so this is an example of a naive global 2-ring not coming from Lemma 6.7. Considering the natural equivalence $\mathrm{End}_{\operatorname{QCoh}(X)}(\mathbb{1})\simeq \Gamma(X,\mathcal{O}_X)$ , we see that ${\mathcal{Q}}_\bullet^{\mathbf{E}}$ categorifies $\text{H}_{{\mathrm{gl}}}(-,{\mathbf{E}})$ .

7. Relative global sections

In this section, we construct the functoriality which is required to prove Theorem 6.2. To motivate the steps of our construction, we first sketch a construction of the functor

$$\mathbb{H}_{{G}}(-,{\mathcal{R}})\colon \mathcal{S}_G^{\operatorname{op}}\rightarrow \operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G}), \quad [X/{\kern1pt}\!\!/ G \xrightarrow{p} {\mathbf{B}} G] \mapsto p_*\mathbb{1}_{X/{\kern1pt}\!\!/ G}.$$

1. (1) First we claim that there exists an oplax cone which sends a G-space X to the unit $\mathbb{1}_{X/{\kern1pt}\!\!/ G}$ of the symmetric monoidal $\infty$ -category ${\mathcal{R}}_{X/{\kern1pt}\!\!/ G}$ .

   
   The triangles are filled by the map $\mathbb{1}_{Y/{\kern1pt}\!\!/ G}\rightarrow f_*\mathbb{1}_{X/{\kern1pt}\!\!/ G}$ coming from the lax monoidal structure of $f_*$ .

2. (2) Next, we observe that because the $\infty$ -category of G-spaces admits a final object, there exists a unique cone $\operatorname{CAlg}({\mathcal{R}}_{\bullet /{\kern1pt}\!\!/ G})\Rightarrow \Delta(\operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G}))$ which sends a G-space $p\colon X/{\kern1pt}\!\!/ {\mathbf{B}} G \rightarrow {\mathbf{B}} G$ to the functor $p_*\colon \operatorname{CAlg}({\mathcal{R}}_{X/{\kern1pt}\!\!/ {\mathbf{B}} G})\rightarrow \operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G})$ .

3. (3) Given the previous two constructions, we define the functor $\mathbb{H}_{{G}}(-,{\mathcal{R}})$ by pasting the previous two oplax natural transformations together

   
   using the identifications of functors $\mathcal{S}_G^{\operatorname{op}}\rightarrow \operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G})$ with $\mathcal{S}_G$ -shaped oplax cones $\Delta(\ast)\Rightarrow \Delta(\operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G}))$ (see Example 2.8).

Clearly, the resulting functor will send $p\colon X/{\kern1pt}\!\!/ G\to {\mathbf{B}}G$ to $p_* \mathbb{1}_{X/{\kern1pt}\!\!/ G}$ and so agrees with (1) of Definition 6.3. Similarly, considering the definition, one sees that it will agree on morphisms with the desideratum (2) of Definition 6.3.

We construct the oplax transformations sketched in step (1) and step (2) above in §§ 7.1 and 7.2, respectively. In both cases, our task is made more difficult by the fact that we require a construction which is functorial enough to allow us to deduce Theorem 6.2. Finally, in § 7.3, we combine these two constructions together, as in step (3).

7.1 Constructing unit sections

In this section, we construct the oplax cone $\mathbb{1}_\bullet\colon\Delta(\ast)\Rightarrow {\mathcal{R}}_{-/{\kern1pt}\!\!/ G}$ , in the guise of a section of the cartesian unstraightening of the functor $\operatorname{CAlg}({\mathcal{R}}_{-/{\kern1pt}\!\!/ G})$ . We begin with a simplification of the task. Note that the unit $\mathbb{1}_{{\mathcal{R}}_X}$ is the initial object of $\operatorname{CAlg}({\mathcal{R}}_X)$ . In particular, given $f\colon X\rightarrow Y$ in $\mathcal{S}_{\mathcal{E}-\mathrm{gl}}$ , the map $\mathrm{lax}_f\colon \mathbb{1}_{{\mathcal{R}}_Y}\rightarrow f_* \mathbb{1}_{{\mathcal{R}}_X}$ can be characterized as the unique map of commutative algebras from $\mathbb{1}_{{\mathcal{R}}_Y}$ to $f_*\mathbb{1}_{{\mathcal{R}}_X}$ . Therefore, the construction of a unit section as above is subsumed by the construction of initial object sections.

Definition 7.1. We write $\operatorname{Cart}$ for the subcategory of $\operatorname{Ar}(\mathrm{Cat})$ spanned on objects by the cartesian fibrations and on morphisms by those squares such that F sends p-cartesian edges to q-cartesian edges.

We define $\operatorname{Cart}^{(\emptyset)}$ to be the full subcategory of $\operatorname{Cart}$ on those cartesian fibrations $p\colon X\rightarrow S$ such that, for each object $s\in S$ , the fiber $X_s$ of p over s admits an initial object.

Proposition 7.2. The inclusion $\operatorname{Cart}^{(\emptyset)}\subset \operatorname{Ar}(\mathrm{Cat})$ factors through the subcategory $\operatorname{Fun}_{\mathrm{R}}^{\mathrm{oplax}}([1],\mathbf{\mathrm{Cat}})$ . In particular, postcomposing with the first mate equivalence (Theorem 2.15), we obtain a functor

$$ s_\emptyset\colon \operatorname{Cart}^{(\emptyset)} \rightarrow \operatorname{Fun}_{\mathrm{L}}^{\mathrm{lax}}([1],\mathbf{\mathrm{Cat}}). $$

Proof. Consider a cartesian fibration $p\colon X\rightarrow S$ , which classifies a functor $F\colon S^{\operatorname{op}}\rightarrow \mathrm{Cat}$ . It suffices to show that $p\colon X \to S$ has a left adjoint. We claim that the object $(\emptyset,s)$ is a left-adjoint object of $s\in S$ , where $\emptyset$ is the initial object of $X_s$ . To see this, consider the map

$$ \phi \colon \operatorname{Map}_X((\emptyset,s),(x,t))\rightarrow \operatorname{Map}_S(s,t) $$

induced by p. By [Reference LurieLur09b, Proposition 2.4.4.2], the fiber over $f\colon s\to t$ is equivalent to

$$\operatorname{Map}_{X_s}(\emptyset, F(f)(x))\simeq \ast.$$

Thus, $\phi$ is an equivalence.

Remark 7.3. The argument given in the proof of Proposition 7.2 also shows that the left adjoint s to $p\colon X\rightarrow S$ is fully faithful. In particular, the composite ps is canonically equivalent to the identity on S via the unit of the adjunction $s\dashv p$ , and so s is canonically a section of p.

Example 7.4. Consider a cartesian fibration $p\in \operatorname{Cart}^{(\emptyset)}$ , which classifies a functor $F\colon S^{\operatorname{op}}\rightarrow \mathrm{Cat}$ , and let s be a section constructed by this procedure, that is, $s=s_{\emptyset}(p)$ . Then, s may be interpreted as oplax cones $\ast\Rightarrow F$ (see the dual of Remark 2.7). The component at $i\in {\mathcal{I}}$ is given by the functor $\{\emptyset_{F(i)}\}\colon \ast \rightarrow F(i)$ . If we begin with a functor $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}^{\otimes,\mathrm{lax}}$ , then applying the previous construction to the functor $\operatorname{CAlg}(F)\colon {\mathcal{I}} \rightarrow \mathrm{Cat}^{(\emptyset)}$ , we obtain a lax cocone, which we denote by $\mathbb{1}_\bullet\colon \ast\Rightarrow \operatorname{CAlg}(F)$ .

7.2 Cartesian transport to the initial object

Having constructed the unit section, we now build the natural transformation $\operatorname{CAlg}({\mathcal{R}}_{\bullet/{\kern1pt}\!\!/ G})\Rightarrow \Delta(\operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G}))$ . Note that such a natural transformation unstraightens to a functor $\mathrm{Un}^{\mathrm{ct}}(\operatorname{CAlg}({\mathcal{R}}_{\bullet/{\kern1pt}\!\!/ G}))\rightarrow \operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G})\times \mathcal{S}_G^{\operatorname{op}}$ , which by adjunction is, in turn, equivalent to a functor $\mathrm{Un}^{\mathrm{ct}}(\operatorname{CAlg}({\mathcal{R}}_{\bullet/{\kern1pt}\!\!/ G}))\rightarrow \operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G})$ . It is in this guise that we build the natural transformation above. Of course, we define it more generally for any functor $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ such that ${\mathcal{I}}$ admits a final object. In this case, it will be a functor with signature $\mathrm{Un}^{\mathrm{ct}}(F)\rightarrow F(\ast)$ .

To be able to deduce Theorem 6.2, we require the functor above to itself be functorial in both natural transformations $F\Rightarrow G$ and precomposition by an arbitrary functor ${\mathcal{J}}\rightarrow {\mathcal{I}}$ , not just one which preserves the final object. This is the case because the induction functors generally do not preserve the final object. To make this precise, we introduce the following definition.

Definition 7.5. Let $\operatorname{Cart}_{(\emptyset)}$ be the full subcategory of $\operatorname{Cart}$ spanned by those cartesian fibrations $p\colon X\rightarrow {\mathcal{I}}$ such that ${\mathcal{I}}$ has an initial object.

Note that the base of a cartesian fibration $p\colon X\rightarrow {\mathcal{I}}$ is given by the opposite of the source of its straightening ${\mathcal{I}}^{\operatorname{op}}\to\mathrm{Cat}$ , explaining the switch from final to initial objects. With the previous definition, we may state the goal of this section.

Proposition 7.6. There exists a functor $\operatorname{Cart}_{(\emptyset)}\rightarrow \operatorname{Fun}([1],\mathrm{Cat})$ which sends a cartesian fibration $p\colon X\rightarrow {\mathcal{I}}$ to the functor

$$\mathrm{Tr}_\emptyset\colon X\rightarrow X_\emptyset, \quad (x,i)\mapsto F(\emptyset \rightarrow i)(x). $$

To prove this statement, we appeal to the functoriality of cartesian lifts in a cartesian fibration. There is a convenient formulation of this, which goes via the notion of free fibrations, in the sense of [Reference Gepner, Haugseng and NikolausGHN17], which we recall now.

Proposition 7.7. The inclusion $\operatorname{Cart} \subseteq \operatorname{Ar}(\mathrm{Cat})$ admits a left adjoint $\mathrm{F}^{\mathrm{ct}}\colon \operatorname{Ar}(\mathrm{Cat}) \to \operatorname{Cart}$ .

Proof. Both Cart and Ar(Cat) are cartesian fibrations over Cat, and the inclusion is clearly a map of cartesian fibrations. Therefore, by [Reference LurieLur17, Proposition 7.3.2.6], it suffices to prove that the restriction of the inclusion to each fiber

$$ \mathrm{incl} \colon \operatorname{Cart}({\mathcal{I}})\rightarrow \mathrm{Cat}_{/{\mathcal{I}}} $$

has a left adjoint. This is precisely [Reference Gepner, Haugseng and NikolausGHN17, Theorem 4.5].

Definition 7.8. Concretely, the functor $\mathrm{F}^{\mathrm{ct}}$ sends a functor $p\colon X \to {\mathcal{I}}$ to the cartesian fibration whose total space is given by the following pullback square.

We view $\mathrm{F}_{{\mathcal{I}}}^{\mathrm{ct}}(X)$ as living over ${\mathcal{I}}$ via the composite $\mathrm{F}^{\mathrm{ct}}_{{\mathcal{I}}}(X) \to \mathrm{Ar}({\mathcal{I}}) \xrightarrow{\mathrm{ev}_0} {\mathcal{I}}$ . When the base $\infty$ -category ${\mathcal{I}}$ is clear from the context, we drop it from the notation and simply write $\mathrm{F}^{\mathrm{ct}}(X)$ . We refer to $\mathrm{F}^{\mathrm{ct}}(X)\rightarrow {\mathcal{I}}$ as the free fibration of p.

Definition 7.9. We write </unnumdisplay> for the functor which sends a cartesian fibration $p\colon X \rightarrow S$ to the counit of the adjunction $\mathrm{F}^{\mathrm{ct}}(X)\dashv \mathrm{incl}$ .

We write $\mathrm{Tr}\colon \operatorname{Cart} \rightarrow \operatorname{Ar}(\mathrm{Cat})$ for the projection to the top of the square. Write $G\colon S^{\operatorname{op}}\rightarrow \mathrm{Cat}$ for the straightening of $p\colon X\rightarrow S$ . Unwinding the definitions, we find that Tr sends the pair $(x\in X_i, f\colon i\rightarrow j)$ to $G(f)(x)\in X_j\subset X$ . The functor Tr encodes the functoriality of cartesian lifts. Therefore, we call it the cartesian transport functor.

We now apply the cartesian transport functor to construct the functor $\mathrm{Tr}_\emptyset\colon \mathrm{Un}^{\mathrm{ct}}(F)\rightarrow F(\emptyset)$ . Recall that we require this construction to be functorial in maps in the base which do not necessarily preserve the initial object. Nevertheless, we begin by assuming that it does, and then later reduce to this case.

Construction 7.10. Let $\mathrm{Cat}_{\emptyset}$ be the subcategory of Cat spanned by those $\infty$ -categories which have initial objects and those functors which preserve the initial object. We build a functor $\mathrm{Cat}_{\emptyset} \rightarrow \operatorname{Fun}([1],\mathrm{Cat})$ as follows. First, consider the functor $\mathrm{Cat} \rightarrow \operatorname{Fun}([1],\mathrm{Cat})$ which sends an $\infty$ -category ${\mathcal{C}}$ to the functor $\mathrm{ev}_1 \colon {\mathcal{C}}\times [1] \rightarrow {\mathcal{C}}$ . Now note that when ${\mathcal{C}}$ has an initial object, this functor admits a right adjoint, given by the unique functor

$$ c_\emptyset \colon {\mathcal{I}}\rightarrow \mathrm{Ar}({\mathcal{I}}),\quad x\mapsto [\emptyset \rightarrow x]. $$

Therefore, we may pass through the first mate equivalence (2.15) to obtain a functor

$$ c_\emptyset \colon \mathrm{Cat}^\ast \rightarrow \operatorname{Fun}^{\mathrm{lax}}([1],\mathrm{Cat}). $$

A quick check shows that the Beck–Chevalley transformations are in fact equivalences, and so this functor factors through $\operatorname{Fun}([1],\mathrm{Cat})$ .

Construction 7.11. Consider the subcategory $\operatorname{Cart}_{\emptyset} = \operatorname{Cart}\times_{\mathrm{Cat}} \mathrm{Cat}_{\emptyset}$ of $\operatorname{Cart}$ spanned by those cartesian fibrations whose base admits an initial object and those maps of cartesian fibrations such that the functor in the base preserves the initial object. We build a functor $\operatorname{Cart}_{\emptyset} \rightarrow \operatorname{Fun}([1]\times\Lambda^2_2,\mathrm{Cat})$ by sending an object $p\colon X\rightarrow {\mathcal{I}}$ to the following diagram.

Taking the limit along $\Lambda^2_2$ results in a functor $\Psi\colon \operatorname{Cart}_{\emptyset} \rightarrow \operatorname{Fun}([1],\mathrm{Cat})$ . Note that the composite $\mathrm{ev}_1\circ \Psi$ is equivalent to the restriction of the functor $\mathrm{F}^{\mathrm{ct}}\colon \operatorname{Cart}\rightarrow \mathrm{Cat}$ to $\operatorname{Cart}_{\emptyset}$ and $\mathrm{ev}_0\circ \Psi$ is equivalent to the identity functor on $\operatorname{Cart}_{\emptyset}.$ In particular, $\Psi$ defines a natural transformation from the identity of $\operatorname{Cart}_{\emptyset}$ to $\mathrm{F}^{\mathrm{ct}}$ .

Construction 7.12. As observed before, the source of Tr agrees with the target of $\Psi$ , and so we may paste them to obtain a new functor

$$ \mathrm{Tr}_{\emptyset} \colon \operatorname{Cart}_{\emptyset} \rightarrow \operatorname{Fun}([1],\mathrm{Cat}). $$

By definition, $\mathrm{Tr}_{\emptyset}$ sends a cartesian fibration $p\colon X \to {\mathcal{I}}$ to the functor

$$ \mathrm{Tr}_{\emptyset}(X)\colon X \xrightarrow{(\mathrm{id},c_\emptyset)} \mathrm{F}^{\mathrm{ct}}(X) \xrightarrow{\mathrm{Tr}} X. $$

Thus, we find that $\mathrm{Tr}_{\emptyset}(X)(x,i) \simeq (\emptyset \to i)_*(x)$ . In particular, the composite factors through the fiber $X_\emptyset$ . We implicitly restrict the target of the functor to the fiber over the final object from hereon. Similarly, applying this functor to a map in $\operatorname{Cart}_{\emptyset}$ as follows, returns a commutative square.

We have built the functor $\mathrm{Tr}_\emptyset$ of Proposition 7.6. However, we have so far only exhibited its functoriality in natural transformations and maps in the base which preserve the initial object. Next, we explain how to proceed in the case that the map in the base does not necessarily preserve the final object.

Construction 7.13. Consider the functor

$$ (-)^{\triangleleft}\colon \mathrm{Cat}\rightarrow \mathrm{Cat}, \quad {\mathcal{I}}\mapsto {\mathcal{I}}^{\triangleleft} := \{\mathrm{pt}\}\ast {\mathcal{I}}, $$

where $\ast$ refers to the join of two $\infty$ -categories. Note that including the second component of the join induces a natural transformation $b\colon \mathrm{id}\Rightarrow (-)^{\triangleleft}$ . Consider the induced natural transformation

$$ b^*\colon \operatorname{Fun}((-)^{\triangleleft},\mathrm{Cat}) \Rightarrow \operatorname{Fun}(-,\mathrm{Cat}) $$

and note that each restriction functor admits a right adjoint, given by right Kan extension. Passing through the first mate equivalence (2.15), we obtain an oplax natural transformation

$$ \operatorname{Fun}((-),\mathrm{Cat}) \Rightarrow \operatorname{Fun}((-)^{\triangleleft},\mathrm{Cat}) $$

which, evaluated at an $\infty$ -category ${\mathcal{I}}$ , is given by right Kan extension along $b\colon {\mathcal{I}}\rightarrow {\mathcal{I}}^{\triangleleft}$ . Observe that the right Kan extension of a diagram $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ along b is given by the universal limit cone ${\mathcal{I}}^{\triangleright}\rightarrow \mathrm{Cat}$ associated to F. Such an oplax natural transformation induces an endofunctor

$$ \Phi\colon \operatorname{Cart} \rightarrow \operatorname{Cart} $$

of the cartesian unstraightening of the functor $\operatorname{Fun}((-)^{\operatorname{op}},\mathrm{Cat})$ .

Construction 7.14. Suppose we now restrict $\Phi$ to the full subcategory $\operatorname{Cart}_{(\emptyset)}\subseteq \operatorname{Cart}$ spanned by those fibrations whose base has an initial object. We emphasize that the functors do not necessarily preserve the initial object. The crucial observation, however, is that the induced functor $h^{\triangleleft}\colon {\mathcal{I}}^\triangleleft\rightarrow {\mathcal{J}}^\triangleleft$ does preserve the initial object. Therefore, $\Phi$ restricts to a functor

$$ \Phi\colon \operatorname{Cart}_{(\emptyset)}\rightarrow \operatorname{Cart}_{\emptyset}. $$

It is worth unraveling the effect of the functor $\Phi$ . Note that the limit of a functor $F \colon {\mathcal{I}}\rightarrow \mathrm{Cat}$ such that ${\mathcal{I}}$ has a initial object is simply given by $F(\emptyset)$ , and so $\Phi(F)\colon {\mathcal{I}}^{\triangleleft}\rightarrow \mathrm{Cat}$ agrees with F on ${\mathcal{I}}$ and sends the cone point to $F(\emptyset)$ again. Therefore, the effect of $\Phi$ on objects is not so interesting. Nevertheless, the effect of $\Phi$ on morphisms is useful. Suppose that the following is a morphism in $\operatorname{Cart}_{(\emptyset)}$ which represents the natural transformation $\eta\colon F\Rightarrow G\circ h$ .

Then, the induced morphism

represents the natural transformation

$$ \Phi(\eta)\colon \Phi(F) \rightarrow \Phi(G) $$

which agrees with $\eta$ on every object except at the cone point, where it is given by the composite

$$ F(\emptyset)\xrightarrow{\eta_\emptyset} G(h(\emptyset))\xrightarrow{G(h(\emptyset)\rightarrow \emptyset)} G(\emptyset). $$

Proof of Proposition 7.6. We may consider the composite

$$ \operatorname{Cart}_{(\emptyset)}\xrightarrow{\Phi} \operatorname{Cart}_{\emptyset}\xrightarrow{\mathrm{Tr}_{\emptyset}} \operatorname{Fun}([1],\mathrm{Cat}). $$

On objects, $\Phi \circ \mathrm{Tr}_{\emptyset}$ sends the cartesian fibration $p\colon \mathrm{Un}^{\mathrm{ct}}(F) \rightarrow {\mathcal{I}}$ to the functor $\mathrm{Tr}_{\emptyset}\colon \mathrm{Un}^{\mathrm{ct}}(F^\triangleleft)\rightarrow F(\emptyset).$ From now on, we restrict this functor in the source to the subcategory $\mathrm{Un}^{\mathrm{ct}}(F)$ . This is natural in ${\mathcal{I}}$ and so constitutes a new functor

$$ \mathrm{Tr}_{(\emptyset)}\colon \operatorname{Cart}_{(\emptyset)} \rightarrow \operatorname{Fun}([1],\mathrm{Cat}). $$

Note that, on objects, this functor agrees with $\mathrm{Tr}_{\emptyset}$ . However, on morphisms, $\mathrm{Tr}_{(\emptyset)}$ sends a square, where now the map h does not necessarily preserve the initial object, to a commutative square.

7.3 Combining the previous two constructions

We now combine the previous two sections to obtain the required result. To state it, we define ${\mathbf{Cat}}_{(\ast)}$ to be the full subcategory of Cat spanned by those $\infty$ -categories which admit a final object. Given an $\infty$ -category ${\mathcal{I}}$ with a terminal object and an object $i\in {\mathcal{I}}$ , we write $p_i\colon i\rightarrow \ast$ for the unique map from i to $\ast$ . Combining the constructions of this section, we obtain the following functor.

Theorem 7.15. There exists a functor

$$ \mathbb{P}(-)\colon {{\mathbf{Cat}}_{(\ast)}}\mathbin{\downarrow^{\operatorname{opl}}}{\mathrm{Cat}^{\otimes,\mathrm{lax}}}\rightarrow \operatorname{Fun}^{\mathrm{lax}}([1],{\mathbf{Cat}}) $$

with the following properties.

On objects, $\mathbb{P}(-)$ sends a functor $F\colon {\mathcal{I}}\rightarrow \mathrm{Cat}^{\otimes,\mathrm{lax}}$ to the functor $ \mathbb{P}(F)\colon {\mathcal{I}}^{\operatorname{op}}\rightarrow \operatorname{CAlg}(F(\ast)) $ such that:

1. (1) $\mathbb{P}(F)(i) \simeq F(p_i) \mathbb{1}_{F(i)}$ ; and

2. (2) given a map $f\colon X\rightarrow Y$ in ${\mathcal{I}}$ , the map $\mathbb{P}(F)(f)\colon F(p_j) \mathbb{1}_{F(j)} \rightarrow F(p_i)\mathbb{1}_{F(i)}$ is given by the composite

   $$ F(p_j) \mathbb{1}_{F(j)} \xrightarrow{F(p_j) \mathrm{lax}_f} F(p_j) F(f) \mathbb{1}_{F(i)} \simeq F(p_i) \mathbb{1}_{F(i)}, $$
   where $\mathrm{lax}_f$ is the lax unit map of the lax monoidal functor F(f).

   On morphisms, $\mathbb{P}(-)$ sends the following oplaxly commuting triangle to the square below, where:

3. (3) $\mathbb{P}(\eta)$ is given at an object $i \in {\mathcal{I}}$ by the composite

   \begin{align*}G(p_{F(i)})\mathbb{1}_{GF(i)} \xrightarrow{G(p_{F(i)})(\mathrm{lax}_{\eta_i})}G(p_{F(i)}) \eta_i \mathbb{1}_{H(i)} &\simeq G(p_{F(\ast)}) GF(p_i)\eta_{i}\mathbb{1}_{H(i)}\\& \simeq G(p_{F(\ast)})\eta_* H(p)(\mathbb{1}_{H(i)}),\end{align*}
   where $\mathrm{lax}_{\eta_i}$ is the lax unit map of $\eta_i$ .

Proof. Consider the composite

$$ {{\mathbf{Cat}}_{(\ast)}}\mathbin{\downarrow^{\operatorname{opl}}} {\mathrm{Cat}^{\otimes,\mathrm{lax}}} \xrightarrow{{\operatorname{CAlg}(-)}\circ{-}} {\mathbf{Cat}}\mathbin{\downarrow^{\operatorname{opl}}}{\mathrm{Cat}} {\operatorname{Cart}} $$

where the last equivalence is Corollary 2.12. Because the $\infty$ -category $\operatorname{CAlg}({\mathcal{D}})$ always has an initial object given by the unit $\mathbb{1}_{\mathcal{D}}$ and ${\mathcal{I}}^{\operatorname{op}}$ has an initial object by assumption, this composite lands in the intersection $\operatorname{Cart}^{(\emptyset)}\cap \operatorname{Cart}_{(\emptyset)}$ . We may therefore apply the functor

$$ (s_\emptyset,\mathrm{Tr}_{(\emptyset)})\colon \operatorname{Cart}^{(\emptyset)}\cap \operatorname{Cart}_{(\emptyset)}\rightarrow \operatorname{Fun}^{\mathrm{lax}}([1],\mathrm{Cat})\times_{\mathrm{Cat}} \operatorname{Fun}^{\mathrm{lax}}([1],\mathrm{Cat}). $$

Here, the pullback is taken along evaluation at source and target, respectively. The fact that $(s_\emptyset,\mathrm{Tr}_{(\emptyset)})$ factors through the pullback is justified by the fact that the target of $s_\emptyset$ agrees with the source of $\mathrm{Tr}_{(\emptyset)}$ : both are the identity on $\operatorname{Cart}^{(\emptyset)}\cap \operatorname{Cart}_{(\emptyset)}$ . Now, we may paste these two lax natural transformations to obtain the functor $\mathbb{P}$ . More formally, this means we postcompose by the functor

$$ \mathrm{paste}\colon \operatorname{Fun}^{\mathrm{lax}}([1],\mathrm{Cat})\times_{\mathrm{Cat}} \operatorname{Fun}^{\mathrm{lax}}([1],\mathrm{Cat})\rightarrow \operatorname{Fun}^{\mathrm{lax}}([1],\mathrm{Cat}). $$

We define the result to be $\mathbb{P}(-)$ . Unwinding the construction, one finds it has the properties required.

Remark 7.16. We note the similarity between the properties of $\mathbb{P}(-)$ and the desiderata of Definition 6.3.

8. Families of equivariant cohomology theories

After the technical work of the last section, we can cash in and prove Theorem 6.2.

Proposition 8.1. Consider a functor ${\mathcal{R}} \colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}}\to\mathrm{Cat}^{\otimes,\mathrm{lax}}$ . Then there exists a functor

$$ \mathbb{H}_{{\bullet}}(-,{\mathcal{R}})\colon \operatorname{Glo}_{{\mathcal{E}}} \to \operatorname{Fun}^{\mathrm{lax}}([1], {\mathbf{Cat}}^{\otimes,\mathrm{lax}}) $$

such that the following hold.

1. (1) The functor $\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})$ sends ${\mathbf{B}} G$ to a functor

   $$ \mathbb{H}_{{G}}(-,{\mathcal{R}})\colon \mathcal{S}_G^{\operatorname{op}}\rightarrow \operatorname{CAlg}({\mathcal{R}}_{{\mathbf{B}} G}) $$
   which agrees on objects and morphisms with (1) and (2) of Definition 6.3, respectively.

2. (2) The functor $\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})$ sends a morphism ${\mathbf{B}} \alpha \colon {\mathbf{B}} H \to {\mathbf{B}} G$ to the following lax square, filled by a natural transformation

   $$ T_\alpha\colon \mathbb{H}_{{G}}(\alpha_!(-),{\mathcal{R}}) \Rightarrow \alpha_*\mathbb{H}_{{H}}(-,{\mathcal{R}}) $$
   which agrees pointwise with (3) of Definition 6.3.
   (8.1)

   

Proof. We first pass ${\mathcal{R}}$ through the composite

$$ \operatorname{Fun}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}},\mathrm{Cat}^{\otimes,\mathrm{lax}})\xrightarrow{ \small{\ref{prop:glquotient-oplaxlim}}} \operatorname{oplaxlim^\dagger} \operatorname{Fun}(\mathcal{S}_G,\mathrm{Cat}^{\otimes,\mathrm{lax}}) \xrightarrow{\mathrm{forget}} \operatorname{Fun}(\operatorname{Glo}_{{\mathcal{E}}},{\mathbf{Cat}}\mathbin{\downarrow^{\operatorname{opl}}} \mathrm{Cat}^{\otimes,\mathrm{lax}}), $$

where the second functor is justified by the identification of Construction 2.13. Next, we observe that the resulting object factors through the subcategory ${\mathbf{Cat}}_{(\ast)}\mathbin{\downarrow^{\operatorname{opl}}} \mathrm{Cat}^{\otimes,\mathrm{lax}}$ . Therefore, we may postcompose by the functor

$$ \mathbb{P}\colon {\mathbf{Cat}}_{(\ast)}\mathbin{\downarrow^{\operatorname{opl}}} \mathrm{Cat}^{\otimes,\mathrm{lax}}\rightarrow \operatorname{Fun}^{\mathrm{lax}}([1],{\mathbf{Cat}}) $$

from Theorem 7.15 to obtain

$$ \mathbb{H}_{{\bullet}}(-,{\mathcal{R}})\colon \operatorname{Glo}_{{\mathcal{E}}} \to \operatorname{Fun}^{\mathrm{lax}}([1], {\mathbf{Cat}}). $$

Finally, we observe that this functor factors through the subcategory $\operatorname{Fun}^{\mathrm{lax}}([1], {\mathbf{Cat}}^{\amalg})$ , and so, by Remark 8.3, we can equivalently view $\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})$ as a functor

$$ \mathbb{H}_{{\bullet}}(-,{\mathcal{R}}) \colon \operatorname{Glo}_{{\mathcal{E}}} \to \operatorname{Fun}^{\mathrm{lax}}([1], {\mathbf{Cat}}^{\otimes,\mathrm{lax}}). $$

This functor has the properties advertised by Theorem 7.15.

If we now assume that the diagram ${\mathcal{R}}\colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}}\to \mathrm{Cat}^{\otimes,\mathrm{lax}}$ is obtained from a naive global 2-ring by passing to right adjoints, then one can pass the functor $\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})$ constructed above through the mate equivalence, a fact which we record here.

Corollary 8.4. Let ${\mathcal{R}} \colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}}^{\operatorname{op}} \to \operatorname{CAlg}(\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}})$ be a functor. Then, there is a functor

$$ \mathbb{H}_{{\bullet}}(-,{\mathcal{R}})\colon \operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}\rightarrow \operatorname{Fun}^{\mathrm{oplax}}([1],{\mathbf{Cat}}^{\otimes,\mathrm{lax}}) $$

which agrees with the functor from Proposition 8.1 on objects, but sends a map ${\mathbf{B}} \alpha\colon {\mathbf{B}} H\rightarrow {\mathbf{B}} G$ in $\operatorname{Glo}_{{\mathcal{E}}}$ to the following oplax square, which is filled by the Beck–Chevalley transformation of $T_\alpha$ .

(8.2)

Proof. Consider the functor ${\mathcal{R}} \colon \mathcal{S}_{\mathcal{E}-\mathrm{gl}} \to \mathrm{Cat}^{\otimes,\mathrm{lax}}$ given by passing to adjoints. Applying the previous proposition to this, we obtain a functor

$$ \mathbb{H}_{{\bullet}}(-,{\mathcal{R}})\colon \operatorname{Glo}_{{\mathcal{E}}}\rightarrow \operatorname{Fun}([1],{\mathbf{Cat}}). $$

Observe that, because $\alpha^*$ is strong monoidal, the adjunction $\alpha^*\dashv \alpha_*$ induces an adjunction on commutative algebra objects. Therefore, $\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})$ factors through the subcategory $\operatorname{Fun}^{\mathrm{R},\mathrm{lax}}([1],\mathrm{Cat})$ from Definition 2.17. We may therefore pass $\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})$ through the second mate equivalence (2.18) to obtain the functor

$$ \mathbb{H}_{{\bullet}}(-,{\mathcal{R}})\colon \operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}\rightarrow \operatorname{Fun}^{\mathrm{oplax}}([1],{\mathbf{Cat}}). $$

Once again, we observe that the diagram factors through ${\mathbf{Cat}}^\amalg$ , and so lifts uniquely to a functor into $\operatorname{Fun}^{\mathrm{oplax}}([1],{\mathbf{Cat}}^{\otimes,\mathrm{lax}})$ .

The unfurling construction is functorial in the following sense.

Proposition 8.5. The previous construction constitutes a functor

$$\mathbb{H}\colon \mathrm{2CAlg}^{\mathrm{naive}}_{\mathcal{E}-\mathrm{gl}} \rightarrow \operatorname{Fun}(\operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}},\operatorname{Fun}^{\mathrm{oplax}}([1],{\mathbf{Cat}}^{\otimes,\mathrm{lax}})). $$

Proof. The construction is given by the following composite of functors:

and so functorial.

Remark 8.6. Unwinding the effect of the construction above, we find that it sends a map $F\colon {\mathcal{R}}\rightarrow {\mathcal{R}}'$ of naive global rings to a $\operatorname{Glo}_{{\mathcal{E}}}^{\operatorname{op}}$ -indexed collection of oplaxly commuting squares as follows.

The transformation $u_{F,G}$ is given at a G-space X by the map

$$F\mathbb{H}_{{G}}(X,{\mathcal{R}}) := F_{{\mathbf{B}} G} p_*\mathbb{1}_{{\mathcal{R}}(X/{\kern1pt}\!\!/ G)} \xrightarrow{BC} p_* F_{X/{\kern1pt}\!\!/ G} \mathbb{1}_{{\mathcal{R}}(X/{\kern1pt}\!\!/ G)} \simeq p_* \mathbb{1}_{{\mathcal{R}}'(X/{\kern1pt}\!\!/ G)} \simeq =: \mathbb{H}_{{G}}(X,{\mathcal{R}}), $$

where BC denotes the Beck–Chevalley transformation of the naturality equivalence $ F_{{\mathbf{B}} G} p^* \simeq p^* F_{X/{\kern1pt}\!\!/ G}$ .

All in all, we have now proven Theorem 6.2.

Proof of Theorem 6.2. The natural transformation constructed in Corollary 8.4 has all of the required properties by Proposition 8.1.

9. Global sections and equivariant cohomology theories

In this section, we show that the family of equivariant cohomology theories constructed before recovers the naive global ring $\text{H}_{{\mathrm{gl}}}(-,{\mathcal{R}})$ . More precisely, we exhibit an equivalence

(9.1) \begin{equation} \text{H}_{{\mathrm{gl}}}(-/{\kern1pt}\!\!/ \bullet,{\mathcal{R}}) \simeq \Gamma\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})\end{equation}

as objects of $\operatorname{laxlim}^{\dagger} \operatorname{Fun}(\mathcal{S}_\bullet^{\operatorname{op}},\operatorname{CAlg})$ . Of course, we should begin by explaining the notation used and how precisely we interpret both sides of this expression as objects of the partially lax limit. We begin with the left-hand side, which we have previously seen.

Construction 9.1. We may apply the equivalence of Proposition 5.18(2),

$$ \Phi \colon \operatorname{Fun}^{\mathrm{R}}(\mathcal{S}_{\mathcal{E}-\mathrm{gl}}^{\operatorname{op}},\operatorname{CAlg})\rightarrow \operatorname{laxlim}^{\dagger} \operatorname{Fun}^{\mathrm{R}}(\mathcal{S}_\bullet^{\operatorname{op}}, \operatorname{CAlg}), $$

to the functor $\text{H}_{{\mathrm{gl}}}(-,{\mathcal{R}})$ to obtain the object $\text{H}_{{\mathrm{gl}}}(-/{\kern1pt}\!\!/ \bullet\,,{\mathcal{R}})$ of $\operatorname{laxlim}^{\dagger} \operatorname{Fun}^{\mathrm{R}}(\mathcal{S}_\bullet^{\operatorname{op}}, \operatorname{CAlg})$ .

Before we can explain how to view $\Gamma\mathbb{H}_{{\bullet}}(-,{\mathcal{R}})$ as an object of the partially lax limit, we must reinterpret the global section functor $\Gamma$ .