Proof. By the Yoneda lemma, the space of maps from $\Delta _{\operatorname {Top}}^n$ to $X_{\bullet }$ in $\operatorname {PreStk}$ is naturally equivalent to $X_{\bullet }(\Delta _{\operatorname {Top}}^n)$ (where we view a groupoid via the nerve as an object in $\mathcal {S}$). This, in turn, is represented by the simplicial set $\operatorname {Hom}_{\operatorname {Top}}(\Delta ^n_{\operatorname {Top}}, X_{\bullet }) = (\operatorname {Sing} X_{\bullet })_n$. Since every simplicial set is the homotopy colimit along $\Delta ^{\operatorname {op}}$ of its levels, we obtain $X_{\bullet }(\Delta _{\operatorname {Top}}^n) \simeq |(\operatorname {Sing} X_{\bullet })_n|$. Thus,

\[ |\operatorname{Map}_{\operatorname{PreStk}}(\Delta^{\ast}, X_{\bullet})| \simeq \|(\operatorname{Sing} X_{\bullet})_{\ast}\|, \]

where the inner realization is with respect to $\bullet$ and the outer with respect to $\ast$.Footnote ⁶

To obtain the first equivalence in the statement, we just have to use the natural equivalence $|(\operatorname {Sing} X_n)_{\ast }| \to X_n$ for every $n$ and that the order of geometric realization does not matter (as these are just colimits over $\Delta ^{\operatorname {op}}$).

The second equivalence in the statement of the lemma holds by definition.

Proof. We recall from [Reference Gepner and HenriquesGH07, § 2.5] that there is a class of fibrant topological groupoids. Moreover, there is a fibrant replacement functor $X_{\bullet } \to \operatorname {fib}(X_{\bullet })$, which induces by [Reference Gepner and HenriquesGH07, Proposition 3.7] an equivalence $X_{\bullet }^{{\dagger} } \to \operatorname {fib}(X_{\bullet })^{{\dagger} }$. By [Reference Gepner and HenriquesGH07, § 3.4], this implies that $|X_{\bullet }| \to |\operatorname {fib}(X_{\bullet })|$ is an equivalence. This reduces the statement of the proposition to the class of fibrant topological groupoids $X_{\bullet }$. By [Reference Gepner and HenriquesGH07, § 3.3], $\operatorname {fib}(X_{\bullet }) \to \operatorname {fib}(X_{\bullet })^{{\dagger} }$ induces an equivalence on paracompact spaces such as $\Delta ^n_{\operatorname {Top}}$.

Thus,

\begin{align*} \operatorname{\underline{Map}}_{\operatorname{Stk}_\infty} (\operatorname{pt}, X_{\bullet}^{{\dagger}})\simeq |\operatorname{Map}_{\operatorname{Stk}}(\Delta^n_{\operatorname{Top}},X_{\bullet}^{{\dagger}})|\simeq |\operatorname{Map}_{\operatorname{PreStk}}(\Delta^n_{\operatorname{Top}}, X_\bullet)|. \end{align*}

Lemma 2.10 gives $\operatorname {Map}_{\operatorname {PreStk}}(\Delta ^n_{\operatorname {Top}}, X_\bullet )\simeq |X_\bullet |$, finishing the proof.

Proof. By Lemma B.5, there is a natural equivalence

\[ \operatorname{Map}_{\operatorname{Orb}}(\mathbf{B} H,\mathbf{B} G) \simeq |\operatorname{\underline{Map}}_{\operatorname{TopGpd}}(H {\rightrightarrows} \operatorname{pt}, G {\rightrightarrows} \operatorname{pt})|. \]

One easily calculates that the mapping groupoid $\underline {\operatorname {Map}}_{\operatorname {TopGpd}}(H{\rightrightarrows} \operatorname {pt},G{\rightrightarrows} \operatorname {pt})$ is isomorphic to the action groupoid $\{\operatorname {Map}_{\operatorname {Lie}}(H,G)\times G {\rightrightarrows} \operatorname {Map}_{\operatorname {Lie}}(H,G)\}$, with the action given by conjugation. As the realization of an action groupoid is the homotopy quotient, the result follows.

Proof. Let $X$ and $Y$ be topological spaces equipped with an action of the topological group $G$. The space of topological groupoid morphisms between the associated topological groupoids is the subspace of the product

\[ \underline{\operatorname{Map}}_{\operatorname{Top}}(Y,X)\times\underline{\operatorname{Map}}_{\operatorname{Top}}(G\times Y,G\times X) \]

consisting of those maps which are compatible with the groupoid structure as well as the projection to $\{G{\rightrightarrows} \operatorname {pt}\}$. These conditions are exactly what is needed to ensure that for an allowed pair $(f,h)$, the map $h$ is determined by $f$ and $f$ is $G$-equivariant: writing $\underline {\operatorname {Map}}_{\operatorname {Top}}(G\times Y,G\times X)\cong \underline {\operatorname {Map}}_{\operatorname {Top}}(G\times Y,G)\times \underline {\operatorname {Map}}_{\operatorname {Top}}(G\times Y,X)$, we see that the map $G\times Y\to G$ must be the projection in order to be compatible with the map down to $\{G{\rightrightarrows} \operatorname {pt}\}$. Moreover, in order to be compatible with the groupoid structure, the map $G\times Y\to X$ must factor as the composite of the projection $G\times Y\to Y$ followed by $f$ and the map $f$ must be equivariant. It is straightforward to verify that the topologies agree.

Proof. Let $S$ be the set of morphisms $\mathbf {B} G \to \mathbf {B} G/K$ for closed subgroups $K\subset G$. By [Reference LurieLur09a, Proposition 5.5.5.7], there exists a factorization system $(L,R)$ on $\mathcal {S}_{\operatorname {Orb}}$ such that $L$ is the saturation of $S$ and $R$ consists of those morphisms that are right-orthogonal to $L$ (or, equivalently, to $S$). By definition, these are exactly the representable morphisms.

In $\operatorname {Orb}$, we can factor a morphism $\mathbf {B} G \to \mathbf {B} H$ corresponding to a group homomorphism $\varphi \colon G \to H$ as $\mathbf {B} G \to \mathbf {B}(G/\ker (\varphi )) \simeq \mathbf {B}\operatorname {im}(\varphi )\to \mathbf {B} H$. The first arrow lies in $L$ and the second in $R$; thus, the factorization system restricts.

Proof. Consider for every $f\colon Y \to X$ with $Y\in \mathcal {D}$, the full subcategory $\mathcal {E}$ of $(\mathcal {D}_{/X})_{f/}$ on those factorizations

with $h\in R$. We have to show that $\mathcal {E}$ is contractible. Choose a factorization $Y \xrightarrow {l} T \xrightarrow {r} X$ with $T\in \mathcal {D}$, $l\in L$, and $r\in R$. We claim that this is terminal in $\mathcal {E}$. Equivalently, the space of dashed arrows in

is contractible. However, this is part of the statement that $L$ and $R$ are orthogonal.

Proof. The equivalence $\mathcal {P}(\operatorname {Orb}_{/\mathbf {B} G})\simeq \mathcal {P}(\operatorname {Orb})_{/\mathbf {B} G}$ is formal. Moreover, since $\operatorname {Orb}_G \to \operatorname {Orb}_{/\mathbf {B} G}$ is fully faithful, so is $\mathcal {P}(\operatorname {Orb}_G) \to \mathcal {P}(\operatorname {Orb}_{/\mathbf {B} G})$. Since mapping out of any $\mathbf {B} H$ in $\mathcal {S}_{\operatorname {Orb}} = \mathcal {P}(\operatorname {Orb})$ is colimit-preserving, the class of representable morphisms is closed under colimits in $\mathcal {P}(\operatorname {Orb})_{/\mathbf {B} G}$. Since the image of $\operatorname {Orb}_G$ in $\operatorname {Orb}$ consists of representable morphisms, the image of $\mathcal {P}(\operatorname {Orb}_G) \to \mathcal {P}(\operatorname {Orb})_{/\mathbf {B} G}$ consists of representable morphisms as well. It remains to show that every representable morphism lies in the image.

Let $X \to \mathbf {B} G$ be representable. Restricting $X \in \mathcal {P}(\operatorname {Orb}_{/\mathbf {B} G})$ to the subcategory of representables $\operatorname {Orb}^{\operatorname {rep}}_{/\mathbf {B} G}$ and left Kan extending again, results in the morphism

\[ \operatorname{colim}_{(\mathbf{B} H \to \mathbf{B} G)\in(\operatorname{Orb}_{/\mathbf{B} G}^{\operatorname{rep}})_{/X}}\mathbf{B} H\to \operatorname{colim}_{(\mathbf{B} H \to \mathbf{B} G)\in(\operatorname{Orb}_{/\mathbf{B} G})_{/X}}\mathbf{B} H \simeq X, \]

approximating $X$ as a colimit of representable orbits $\mathbf {B} H$ over $\mathbf {B} G$. Thus, it suffices to show that $(\operatorname {Orb}_{/\mathbf {B} G}^{\operatorname {rep}})_{/X}$ is final in $(\operatorname {Orb}_{/\mathbf {B} G})_{/X}$. This follows from Lemma 2.18 and the following observation: given any $\mathbf {B} H \to X$, factor $\mathbf {B} H \to X \to \mathbf {B} G$ as $\mathbf {B} H \to \mathbf {B} H/Q \to \mathbf {B} G$ with $\mathbf {B} H/Q \to \mathbf {B} G$ representable, resulting in the following diagram.

Since $X \to \mathbf {B} G$ is representable, we obtain a lift $\mathbf {B} H/Q \to X$, which is automatically representable as well. This gives the required factorization $\mathbf {B} H \to \mathbf {B} H/Q \to X$.

Proof. We need to show that $\operatorname {Map}_{\operatorname {Stk}_{\infty }}(\mathbf {B} H, [-/G])\colon \mathcal {S}^G \to \mathcal {S}$ preserves colimits for every compact Lie group $H$. Letting $H$ vary, this implies that the functor

\begin{align*} \mathcal{S}^G\to\operatorname{Stk}_{\infty} \to \mathcal{S}_{\operatorname{Orb}},\quad X \mapsto [X/G] \end{align*}

preserves colimits.

Let $X$ be a $G$-space and $\Phi \colon \mathbf {B} H \to \mathbf {B} G$ be a map in $\operatorname {Stk}_{\infty }$, which after choice of basepoints corresponds to a homomorphism $\varphi \colon H \to G$. As in the proof of Lemma B.5, one shows using that all principal bundles on $\Delta ^n$ are trivial that the natural maps

\[ \operatorname{Map}_{\operatorname{TopGpd}/(G {\rightrightarrows} \operatorname{pt})}(H \times \Delta^n{\rightrightarrows} \Delta^n, G\times X{\rightrightarrows} X) \to \operatorname{Map}_{\operatorname{Stk}/\mathbf{B} G}(\mathbf{B} H \times \Delta^n, [X/G]) \]

are equivalences of (discrete) groupoids for all $n$. After geometrically realizing the resulting simplicial objects, the right-hand side becomes $\operatorname {Map}_{(\operatorname {Stk}_{\infty })_{/\mathbf {B} G}}(\mathbf {B} H, [X/G])$, while we can identify the left-hand side with the space of $\varphi$-equivariant maps $\operatorname {pt} \to X$ (using essentially the argument of Proposition 2.15). This, in turn, is the same as the $\operatorname {im}(\varphi )$-fixed points of $X$. As a colimit of $G$-spaces induces a colimit of $\operatorname {im}(\varphi )$-fixed points, the functor $\operatorname {Map}_{(\operatorname {Stk}_{\infty })_{/\mathbf {B} G}}(\mathbf {B} H, [-/G])\colon \mathcal {S}^G \to \mathcal {S}$ preserves colimits. As it is the fiber of $\operatorname {Map}_{\operatorname {Stk}_{\infty }}(\mathbf {B} H, [-/G]) \to \operatorname {Map}_{\operatorname {Stk}_{\infty }}(\mathbf {B} H, \mathbf {B} G)$ over $\Phi$, we see that $\operatorname {Map}_{\operatorname {Stk}_{\infty }}(\mathbf {B} H, [-/G])$ preserves colimits as well.

The last point follows since the functor $\operatorname {Orb}_G \to \operatorname {Orb}\to \mathcal {S}_{\operatorname {Orb}}$ agrees with $X \mapsto [X/G]$ and a colimit-preserving functor from $\mathcal {S}^G \simeq \mathcal {P}(\operatorname {Orb}_G)$ is determined by its restriction to $\operatorname {Orb}_G$.

Proof. Since the forgetful functor $\operatorname {PreAb}(\mathcal {S}) \to \mathcal {S}$ is conservative and preserves sifted colimits, [Reference Gepner, Groth and NikolausGGN15, Proposition B.7] implies $\operatorname {PreAb}(\mathcal {S}) \simeq \operatorname {Fun}^{\Pi }(\mathcal {T}_{\mathrm {preab}}, \mathcal {S})$. This implies the analogous equivalence for presheaf $\infty$-categories.

For a general $i$-category $\mathcal {X}$ with finite products, consider the composite

\[ \operatorname{PreAb}(\mathcal{X}) \to \operatorname{PreAb}(\mathcal{P}(X)) \xrightarrow{\simeq} \operatorname{Fun}^{\Pi}(\mathcal{T}_{\mathrm{preab}}, \mathcal{P}(\mathcal{X})), \]

where we used that the Yoneda embedding $\mathcal {X} \to \mathcal {P}(\mathcal {X})$ preserves products. (Note that we have potentially passed to a larger universe.)

Since the Yoneda embedding is fully faithful, it suffices to show that for any preoriented abelian group $A \in \operatorname {PreAb}(\mathcal {X})$, the corresponding functor $\mathcal {T}_{\mathrm {preab}} \to \mathcal {P}(\mathcal {X})$ takes values in representable presheaves. Thus, let $L \in \mathcal {T}_{\mathrm {preab}}$ be free on a set with $n$ elements. Then for any $A\in \operatorname {PreAb}(\mathcal {X})$, the functor

\[ \operatorname{Map}_{\operatorname{PreAb}(\mathcal{S})}(L, \operatorname{Map}_{\mathcal{X}}(-, A))\colon \mathcal{X}^{\operatorname{op}} \to \mathcal{S} \]

is indeed representable by the underlying object of $A^n$.

Construction 3.8 Morally, the elliptic cohomology groups for a compact abelian Lie group $G$ are related to the sheaf cohomology groups of the elliptic curve and other related abelian varieties. In order to make this precise, we invoke a stacky version of Pontryagin duality, which we refer to as shifted Pontryagin duality. More precisely, it is given by the functor

\[ \operatorname{Map}_{\operatorname{Orb}}(-,\mathbf{B}\mathbf{T})\colon(\operatorname{Orb}^{\operatorname{ab}})^{\operatorname{op}}\to\operatorname{Ab}(\mathcal{S})_{B\mathbf{T}/} = \operatorname{PreAb}(\mathcal{S}),\quad X \mapsto \widehat{X}, \]

which sends the abelian orbit stack $X$ to the preoriented abelian group

\[ B\mathbf{T}\simeq\operatorname{Map}(\operatorname{pt},\mathbf{B}\mathbf{T})\to\operatorname{Map}(X,\mathbf{B}\mathbf{T}). \]

Here the preorientation is induced from the projection $X\to \operatorname {pt}$, and the abelian group structure on the mapping space $\operatorname {Map}(X,\mathbf {B}\mathbf {T})$ is induced pointwise from that on $\mathbf {B}\mathbf {T}$.

By Proposition 2.12, we have $\widehat {\mathbf {B} G} \simeq \widehat {G} \times B \mathbf {T}$, where $\widehat {G}$ is the Pontryagin dual, i.e. the space of homomorphisms from $G$ to $\mathbf {T}$.

Proof. Let $\operatorname {Orb}^{\operatorname {ab}}_{n\ast }\simeq \operatorname {Orb}^{\operatorname {ab}}_*\times _{\operatorname {Orb}^{\operatorname {ab}}}\cdots \times _{\operatorname {Orb}^{\operatorname {ab}}}\operatorname {Orb}^{\operatorname {ab}}_*$ denote the iterated fibered product, which we might regard as the $\infty$-category of $n$-pointed orbits. We have a morphism of augmented simplicial $\infty$-categories

(1)

in which the vertical maps are given by shifted Pontryagin duality, sending an $n$-pointed $X$ to the abelian group object $\operatorname {Map}_{\operatorname {Orb}^{\operatorname {ab}}_{n*}}(X, \mathbf {B} \mathbf {T})$. The vertical maps factor as claimed, since the undercategories in question are determined by the shifted Pontryagin dual

\[ \operatorname{Map}_{\operatorname{Orb}^{\operatorname{ab}}_{n\ast}}(\operatorname{pt},\mathbf{B}\mathbf{T})\simeq\mathbf{T}^{n-1} \]

of the initial object $\operatorname {pt}$ in $(\operatorname {Orb}^{\operatorname {ab}})^{\operatorname {op}}$.

For every $n>0$, we claim a natural equivalence

(2)\begin{equation} \operatorname{Map}_{\operatorname{Orb}^{\operatorname{ab}}_{n\ast}}(\mathbf{B} H, \mathbf{B} G) \simeq \operatorname{Map}_{\operatorname{Lie}}(H,G) \times G^{n-1} \simeq \operatorname{Map}_{\operatorname{Lie}_{/\mathbb{Z}^{n-1}}}(H\times\mathbb{Z}^{n-1},G\times\mathbb{Z}^{n-1}). \end{equation}

Indeed, the $\infty$-category $\operatorname {Orb}_*^{\operatorname {ab}}$ is equivalent to $\operatorname {CptLie}^{\operatorname {ab}}$ by Remark 2.14, settling the case $n=1$. Every two pointings of orbits are equivalent and we can thus restrict for $n>1$ to the case that all pointings of $\mathbf {B} G$ are the same canonical map $\operatorname {pt} \xrightarrow {p} \mathbf {B} G$. By definition of the slice category, $\operatorname {Map}_{\operatorname {Orb}^{\operatorname {ab}}_{n\ast }}(\mathbf {B} H, \mathbf {B} G)$ is the product of $\operatorname {Map}_{\operatorname {CptLie}}(H, G) \simeq \operatorname {Map}_{\operatorname {Orb}^{\operatorname {ab}}_{\ast }}(\mathbf {B} H, \mathbf {B} G)$ and the $(n-1)$-fold product of the space of self-homotopies of $p$ in $\operatorname {Map}_{\operatorname {Orb}^{\operatorname {ab}}}(\operatorname {pt}, \mathbf {B} G)$, i.e. $G^{n-1}$, giving the claim. One checks that the equivalence in (2) is compatible with composition, at least up to homotopy.

On mapping spaces, the $n$th vertical arrow in (1) is

\[ \operatorname{Map}_{\operatorname{Orb}^{\operatorname{ab}}_{n*}}(\mathbf{B} H, \mathbf{B} G) \xrightarrow{P} \operatorname{Map}_{\operatorname{Ab}(\mathcal{S})_{/T^{n-1}}} (\operatorname{Map}_{\operatorname{Orb}_{n*}^{\operatorname{ab}}}(\mathbf{B} G, \mathbf{B} T), \operatorname{Map}_{\operatorname{Orb}_{n*}^{\operatorname{ab}}}(\mathbf{B} H, \mathbf{B} T)). \]

Using (2), for $n>0$, the target is equivalent to

\[ \operatorname{Map}_{\operatorname{Ab}(\mathcal{S})_{/T^{n-1}}}(\operatorname{Map}_{\operatorname{Lie}}(G,T)\times T^{n-1}, \operatorname{Map}_{\operatorname{Lie}}(H,T)\times T^{n-1}). \]

Note that $\operatorname {Map}_{\operatorname {Lie}}(G,T)\times T^{n-1}$ is the Pontryagin dual of $G\times \mathbb {Z}^{n-1}$. By the fully faithfulness of Pontryagin duality and of the embedding $\operatorname {FGLie} \to \operatorname {Ab}(\mathcal {S})$ from Example 3.3, this is equivalent to

\begin{align*} \operatorname{Map}_{\operatorname{Lie}_{/\mathbb{Z}^{n-1}}}(H\times\mathbb{Z}^{n-1},T\times\mathbb{Z}^{n-1}) \simeq \operatorname{Map}_{\operatorname{Orb}^{\operatorname{ab}}_{n*}}(\mathbf{B} H, \mathbf{B} G). \end{align*}

Tracing through the equivalences shows that this provides the required inverse up to homotopy of $P$. Thus, the vertical maps in (1) (except, possibly, for that on the far left) are fully faithful.

Observe now that the leftmost horizontal maps in (1) are quotient functors (see Appendix A for details). It follows that the left-hand vertical map is also fully faithful (cf. the description of mapping spaces in the proof of Proposition A.3).

Proof. By definition, $\mathcal {T}(\mathcal {S}) = \operatorname {Fun}^{\Pi }(\mathcal {T}, \mathcal {S})$ agrees with $P_{\Sigma }(\mathcal {T}^{\operatorname {op}})$ in the sense of [Reference LurieLur09a, § 5.5.8]. Introducing opposite categories in Proposition 5.5.8.15 in [Reference LurieLur09a] implies that the restriction functor

\begin{align*} \operatorname{res}\colon \operatorname{Fun}^R(\mathcal{T}(\mathcal{S})^{\operatorname{op}}, \mathcal{X}) \to \operatorname{Fun}^{\Pi}(\mathcal{T}, \mathcal{X})\simeq \mathcal{T}(\mathcal{X}) \end{align*}

is an equivalence. It remains to identify its inverse explicitly.

Fixing $A\in \mathcal {T}(\mathcal {X})$, consider the class of $B\in \mathcal {T}(\mathcal {S})$ such that the functor

\[ X \mapsto \operatorname{Map}_{\mathcal{T}(\mathcal{S})}(B, \operatorname{Map}_{\mathcal{X}}(X, A)) \]

is representable. Since $\mathcal {X}$ is complete, this class is closed under colimits. Moreover, for $B$ being free on one generator, the representing object is the underlying object in $\mathcal {X}$ of $A$ itself, yielding representability for all $B$. Thus, the functor

\[ \mathcal{T}(\mathcal{S})^{\operatorname{op}}\times \mathcal{T}(\mathcal{X}) \to \mathcal{P}(\mathcal{X}),\quad (A,B)\mapsto (X \mapsto \operatorname{Map}_{\mathcal{T}(\mathcal{S})}(B, \operatorname{Map}_{\mathcal{X}}(X, A))) \]

factors through a functor

\[ \mathcal{T}(\mathcal{S})^{\operatorname{op}}\times \mathcal{T}(\mathcal{X}) \to \mathcal{P}(\mathcal{X}) \]

we denote by $\operatorname {\underline {Map}}_{\mathcal {T}}$. This yields, in particular, the functor $\mathcal {T}(\mathcal {X}) \to \operatorname {Fun}^R(\mathcal {T}(\mathcal {S})^{\operatorname {op}}, \mathcal {X})$ from the claim of our proposition. This is inverse to $\operatorname {res}$ if $\operatorname {\underline {Map}}_{\mathcal {T}}(L, A)$ is naturally equivalent to $A(L)$ for all $L\in \mathcal {T}$. But the functor $\operatorname {Map}_{\mathcal {T}(\mathcal {S})}(L, \operatorname {Map}_{\mathcal {X}}(-, A)))$ is equivalent to $\operatorname {Map}_{\mathcal {X}}(-, A)(L) = \operatorname {Map}_{\mathcal {X}}(-, A(L))$ by Yoneda, which is indeed represented by $A(L)$.

Construction 3.13 Let $\mathcal {X}$ be a finitely complete $\infty$-category. Note that the shifted Pontryagin duality functor $\operatorname {Orb}^{\operatorname {ab}} \to \operatorname {PreAb}(\mathcal {S})$ actually takes values in $\operatorname {PreAb}(\mathcal {S})^{\operatorname {fin}}$ in the sense of Remark 3.12: $\widehat {\mathbf {B} G}$ is equivalent to the product of $\widehat {G}$ (with trivial preorientation) and $B\mathbf {T}$ (with tautological preorientation) and the Pontryagin dual of any compact abelian Lie group is a finitely generated discrete group. Precomposition with shifted Pontryagin duality turns thus the equivalence $\operatorname {PreAb}(\mathcal {X}) \to \operatorname {Fun}^{\mathrm {lex}}(\operatorname {PreAb}(\mathcal {S})^{\operatorname {op}, \operatorname {fin}}, \mathcal {X})$ from Remark 3.12 into a functor

\[ \operatorname{PreAb}(\mathcal{X}) \to \operatorname{Fun}(\operatorname{Orb}^{\operatorname{ab}}, \mathcal{X}),\quad A \mapsto (X \mapsto \operatorname{\underline{Map}}_{\operatorname{PreAb}}(\widehat{X}, A)). \]

This functor is also functorial in $\mathcal {X}$ in the following sense: given a functor $F\colon \mathcal {X} \to \mathcal {Y}$ preserving finite limits, we obtain the following commutative square.

Now assume that $\mathcal {X}$ is additionally cocomplete. Given $A \in \operatorname {PreAb}(\mathcal {X})$, left Kan extension along $\operatorname {Orb}^{\operatorname {ab}} \subset \operatorname {Orb}$ defines a functor $\operatorname {Orb} \to \mathcal {X}$. As $\mathcal {S}_{\operatorname {Orb}}$ is the presheaf $\infty$-category on $\operatorname {Orb}$, i.e. its universal cocompletion, this extends essentially uniquely to a colimit-preserving functor $\mathcal {S}_{\operatorname {Orb}}\to \mathcal {X}$. More precisely, we obtain a functor

\begin{align*} \operatorname{PreAb}(\mathcal{X}) \to \operatorname{Fun}^{\mathrm{L}}(\mathcal{S}_{\operatorname{Orb}}, \mathcal{X}) \end{align*}

into colimit-preserving functors. This construction is natural in $\mathcal {X}$ with respect to functors that preserve colimits and finite limits.

Proof. The forgetful functor $\operatorname {PreAb}(\mathcal {S}) \to \operatorname {Ab}(\mathcal {S})$ has a left adjoint sending $H$ to $H\times \mathbf {B}\mathbf {T}$, with preorientation given by inclusion into the second factor. As computed in Construction 3.8, the shifted Pontryagin dual of $\mathbf {B} G$ is equivalent to $\widehat {G} \times \mathbf {B} \mathbf {T}$, i.e. lies in the image of the left adjoint. Thus,

\begin{align*} F(\mathbf{B} G) = \operatorname{\underline{Map}}_{\operatorname{PreAb}}(\widehat{\mathbf{B} G}, A) \simeq \operatorname{\underline{Map}}_{\operatorname{Ab}}(\widehat{G}, A). \end{align*}

The last equivalence uses the defining property of $\operatorname {\underline {Map}}$ as in Proposition 3.10.

Note that Pontryagin duality sends limits to colimits. Thus, $F$ preserves limits and the last claim follows.

Proof. Recall that there is a factorization system on $\operatorname {Orb}$: for a homomorphism $\varphi \colon H \to G$, the morphism $\mathbf {B} H \to \mathbf {B} G$ is in the left class if and only if $\varphi$ is surjective and in the right class if and only if $\varphi$ is injective. (See [Reference Linskens, Nardin and PolLNP22, Proposition 6.14] for further details.) Concretely, the factorization systems correspond to the factorization of some $\varphi \colon H\to G$ as $H \twoheadrightarrow \operatorname {im}(\varphi ) \hookrightarrow G$. This restricts clearly to $\operatorname {Orb}^{\mathcal {F}}$ for any global family $\mathcal {F}$ of groups. The conclusion follows from Lemma 2.18.

Proof. By the pointwise formula for Kan extensions, we need to show that the comparison map $KU^{\mathbf {B} G} \to \lim _{\operatorname {Orb}^{\mathcal {F}}_{/\mathbf {B} G}} KU^{(-)}$ is an equivalence. Consider the full subcategory $\mathcal {R} \subset \operatorname {Orb}^{\mathcal {F}}_{/\mathbf {B} G}$ of representable morphisms $\mathbf {B} H \to \mathbf {B} G$. By Lemma 4.2, the inclusion $\mathcal {R} \subset \operatorname {Orb}^{\mathcal {F}}_{/\mathbf {B} G}$ is final and, hence, $\mathcal {R}^{\operatorname {op}} \subset (\operatorname {Orb}^{\mathcal {F}}_{/\mathbf {B} G})^{\operatorname {op}}$ is initial. Thus, it suffices to show that the map $KU^{\mathbf {B} G} \to \lim _{\mathcal {R}}KU^{(-)}$ is an equivalence. By Proposition 2.16, $\mathcal {R} \simeq \operatorname {Orb}_G^{\mathcal {F}}$ for $\operatorname {Orb}_G^{\mathcal {F}} \subset \operatorname {Orb}_G$ the full subcategory on $G/H$ such that $H\in \mathcal {F}$. Moreover, $\operatorname {colim}_{G/H\in \operatorname {Orb}_G^{\mathcal {F}}} G/H$ is a universal space $E\mathcal {F}$ for the family of subgroups of $G$ that are in $\mathcal {F}$. Thus, $\pi _*\lim _{\mathcal {R}}KU^{(-)} \cong KU_G^{(-)*}(E\mathcal {F})$. By Corollary 1.3 from [Reference Adams, Haeberly, Jackowski and MayAHJM88], $K_G^*(\operatorname {pt}) \to K_G^*(E\mathcal {F})$ is an isomorphism and the result follows.

Proof. Let $X \in \operatorname {Orb}$. By adjunction, maps

\[ \Phi\colon ku \otimes_{\Sigma^{\infty}_+BU(1)} \Sigma^{\infty}_+\operatorname{Map}_{\operatorname{Orb}}(X, \mathbf{B} U(1)) \to ku^{X} \]

in $\operatorname {CAlg}_{ku}$ correspond to $E_{\infty }$-maps $\phi \colon \operatorname {Map}_{\operatorname {Orb}}(X, \mathbf {B} U(1)) \to \Omega ^{\infty } ku^{X}$ under $BU(1)$, with the multiplicative structure $E_{\infty }$-structure on the target.

The $E_{\infty }$-monoid $\operatorname {Map}_{\operatorname {Orb}}(X, \mathbf {B} U(1))$ includes the one-dimensional part into $\operatorname {Vect}(X)^{\simeq }$, whose group completion is $\Omega ^{\infty } ku^{X}$. Composing inclusion and group completion yields thus a map $\phi$ and a corresponding map $\Phi$, which is natural in $X$.

We will show that $\Phi$ is an equivalence if $X = \mathbf {B} G$ for $G$ abelian. Since

\[ \operatorname{Map}_{\operatorname{Orb}}(\mathbf{B} G, \mathbf{B} U(1)) \simeq \widehat{G} \times BU(1), \]

the source of $\Phi$ simplifies to $ku \otimes \widehat {G}_+$. More precisely, $\Phi$ arises as the group completion of

\begin{align*} \varphi\colon \operatorname{Vect}_{\mathbb{C}}^{\simeq}\otimes\, \widehat{G} \to \operatorname{Rep}_G,\quad (V_L)_{L\in \widehat{G}} \mapsto \bigoplus_{L\in \widehat{G}} V_L\otimes L; \end{align*}

here we interpret the tensor product $\operatorname {Vect}_{\mathbb {C}}^{\simeq }\otimes \, \widehat {G}$ of $E_{\infty }$-spaces as the finitely supported part in $\prod _{\widehat {G}} \operatorname {Vect}_{\mathbb {C}}^{\simeq }$, using that finite direct sums and finite products agree in $E_{\infty }$-spaces and the tensor product commutes with filtered colimits. By Schur's lemma, $\varphi$ is fully faithful, and it is essentially surjective since all $G$-representations decompose into one-dimensional representations.

Proof of Proposition 4.4 For every $\mathbf {B} G\in \operatorname {Orb}^{\operatorname {ab}}$ and $R\in \operatorname {CAlg}_{ku}$, Lemma 4.5 implies a natural chain of equivalences

\begin{align*} \!\!\!\operatorname{Map}_{\operatorname{PreAb}(\mathcal{S})}(\operatorname{Map}(X, \mathbf{B}\mathbf{T}), \mathbb{G}_{m, ku}(R)) &\simeq \operatorname{Map}_{\operatorname{Ab}(\mathcal{S})} (\operatorname{Map}_{\operatorname{Orb}}(\mathbf{B} G, \mathbf{B} U(1)), \mathbb{G}_m(R))^{hBU(1)}\\ &\simeq \operatorname{Map}_{\operatorname{CAlg}}( \Sigma^{\infty}_+\operatorname{Map}_{\operatorname{Orb}}(\mathbf{B} G, \mathbf{B} U(1)), R)^{hBU(1)}\\ &\simeq \operatorname{Map}_{\operatorname{CAlg}_{ku}}\biggl(ku\!\!\!\!\!\!\underset{\Sigma^{\infty}_+BU(1)}{\otimes} \!\!\!\!\!\!\Sigma^{\infty}_+\operatorname{Map}_{\operatorname{Orb}}(\mathbf{B} G, \mathbf{B} U(1)), R\biggr)\\ &\simeq \operatorname{Map}_{\operatorname{CAlg}_{ku}}(ku^{\mathbf{B} G}, R). \end{align*}

Notation 5.5 We will often use the notation $\operatorname {pt}$ for the terminal object in the $\infty$-topos $\operatorname {Shv}(\mathsf {M})$; this is represented by $\operatorname {id}_{\tau _{\geq 0}\mathsf {M}}\colon \tau _{\geq 0}\mathsf {M} \to \tau _{\geq 0}\mathsf {M}$ in the big étale site of $\tau _{\geq 0}\mathsf {M}$.

Warning 5.10 We use the same notation for completion and for Pontryagin (or Picard) duality. The latter applies to a compact abelian topological group or an object in $\operatorname {Orb}^{\operatorname {ab}}$, while the former will typically be applied to a spectral elliptic curve. We trust that it should be clear in every case which of the two is meant.

Proof. The claim is equivalent to $\operatorname {cSpec}_{\mathsf {M}}$ being fully faithful on smooth coalgebras. One easily reduces to the case $\mathsf {M} = \operatorname {Spec} A$ for a connective $E_{\infty }$-ring $A$. As $\mathsf {M}$ is locally noetherian by assumption, we can further assume that $\pi _0A$ is noetherian and $\pi _iA$ is finitely generated over $\pi _0A$.

Lurie [Reference LurieLur18b, Proposition 1.5.9] proved that $\operatorname {cSpec}$ is fully faithful on smooth $A$-coalgebras as a functor into $\operatorname {Fun}(\operatorname {CAlg}^{\operatorname {cn}}, \mathcal {S})$. In contrast, we need to show fully faithfulness as a functor into $\operatorname {Fun}(\operatorname {CAlg}^{\operatorname {cn},\operatorname {afp}}_A, \mathcal {S})$, where the $\operatorname {afp}$ stands for almost of finite presentation. Tracing through the proof in [Reference LurieLur18b], we need to show that $\operatorname {Spf}$ is fully faithful as a functor into $\operatorname {Fun}(\operatorname {CAlg}^{\operatorname {cn},\operatorname {afp}}_A, \mathcal {S})$ on adic $E_{\infty }$-rings that arise as duals of smooth coalgebras.

Let $R$ and $S$ be adic $E_{\infty }$-rings. By [Reference LurieLur16, Lemma 8.1.2.2] we can find a tower

\[ \cdots \to R_3 \to R_2 \to R_1 \]

in $\operatorname {CAlg}_R^{\operatorname {cn}}$ such that $\operatorname {colim}_i \operatorname {Spec} R_i \to \operatorname {Spec} R$ factors over an equivalence $\operatorname {colim}_i \operatorname {Spec} R_i \simeq \operatorname {Spf} R$ in $\operatorname {Fun}(\operatorname {CAlg}^{\operatorname {cn}}, \mathcal {S})$ and every $R_i$ is almost perfect as a $R$-module. We claim that $R_i$ is almost of finite presentation over $A$ if $R$ is the dual of a smooth coalgebra $C$ over $A$. By [Reference LurieLur12, Proposition 7.2.4.31], this is equivalent to $\pi _0R_i$ being a finitely generated $\pi _0A$-algebra and $\pi _kR_i$ being a finitely generated $\pi _0R_i$-module for all $k$. By the definition of smooth coalgebras, $\pi _0R$ is of the form $(\pi _0A)[\kern-1pt[ t_1, \ldots, t_n]\kern-1pt]$ for some $n$ and $\pi _kR \cong \pi _kA \otimes _{\pi _0A}\pi _0R$. By [Reference LurieLur12, Proposition 7.2.4.17], $\pi _kR_i$ is finitely generated over $\pi _0R$. The claim follows as $\operatorname {Spec} R_i \to \operatorname {Spec} R$ factors over $\operatorname {Spf} R$ by construction and, thus, a power of the ideal $(t_1, \ldots, t_n)$ is zero in $R_i$.

Now let $R$ and $S$ be duals of smooth coalgebras. In particular, we see that $R$ is complete so that $R = \lim _i R_i$ in adic $E_{\infty }$-rings (cf. [Reference LurieLur16, Lemma 8.1.2.3]):

\begin{align*} \operatorname{Map}_{\operatorname{CAlg}^{\operatorname{cn}, ad}_A}(S, R) &\simeq \lim_i \operatorname{Map}_{\operatorname{CAlg}^{\operatorname{cn}, ad}_A}(S, R_i) \\ &\simeq \lim_i (\operatorname{Spf} S)(R_i) \\ &\simeq \lim_i \operatorname{Map}_{\operatorname{Fun}(\operatorname{CAlg}^{\operatorname{cn}, \operatorname{afp}}_A, \mathcal{S})}(\operatorname{Spec} R_i, \operatorname{Spf} S) \\ &\simeq \operatorname{Map}_{\operatorname{Fun}(\operatorname{CAlg}^{\operatorname{cn}, \operatorname{afp}}_A, \mathcal{S})}(\operatorname{Spf} R, \operatorname{Spf} S). \end{align*}

In the third step we have used the Yoneda lemma.

Construction 6.1 To the preoriented elliptic curve $p\colon \mathsf {E} \to \mathsf {M}$ we can associate the $\infty$-topos $\mathcal {X}=\operatorname {Shv}(\mathsf {M})$ of space-valued sheaves on the big étale site of $\mathsf {M}$. As $\mathsf {E}$ defines a preoriented abelian group object in $\mathcal {X}$, Construction 3.13 yields a colimit-preserving functor

\[ \operatorname{Ell}\colon \mathcal{S}_{\operatorname{Orb}} \to \mathcal{X}, \]

where we leave the dependency on $\mathsf {E}$ implicit.

Proof. We claim first that for a surjection $h\colon A \to B$ of finitely generated abelian groups and a classical elliptic curve $E$ over a base scheme $S$, the map $\operatorname {Hom}(B, E) \to \operatorname {Hom}(A, E)$ is a closed immersion and, hence, affine. As the kernel of $h$ is a sum of cyclic groups and a composition of closed immersions is a closed immersion, we can assume that $\ker (h)$ is cyclic. The pushout square

induces the following pullback square.

Furthermore, $S \to \operatorname {Hom}(\ker (h), E)$ is a closed immersion as it is a right inverse of the structure morphism $\operatorname {Hom}(\ker (h), E) \to S$ and the latter is a separated morphism. (See, e.g., [Reference Meier and OzornovaMO14, Lemma 2.4] for this criterion for closed immersions.) Thus, its pullback $\operatorname {Hom}(B,E) \to \operatorname {Hom}(A,E)$ is also a closed immersion and thus affine.

Recall that $\operatorname {Ell}(\mathbf {B} G)\simeq \operatorname {Hom}(\widehat {G}, E)$, with $\widehat {G}$ denoting the Pontryagin dual, and that an inclusion $H\subset G$ induces a surjection $\widehat {G} \to \widehat {H}$. Thus, we know from the preceding paragraph that the underlying map of $\operatorname {Ell}(\mathbf {B} H) \to \operatorname {Ell}(\mathbf {B} G)$ is affine. We can, moreover, assume that the base of the elliptic curve $\mathsf {E}$ is an affine derived scheme $\operatorname {Spec} R$. By [Reference LurieLur18a, Remark 1.5.3], the elliptic curve $\mathsf {E}$ is based changed from $\operatorname {Spec} \tau _{\geq 0}R$. Thus, we can assume additionally that $R$ is connective and, hence, that $\operatorname {Ell}(\mathbf {B} G)$ and $\operatorname {Ell}(\mathbf {B} H)$ are spectral schemes (and not more general non-connective ones).

Let $\operatorname {Spec} A \to \operatorname {Ell}(\mathbf {B} G)$ be a map from an affine and let $(\mathcal {X}, \mathcal {O}_{\mathcal {X}})$ denote the pullback $\operatorname {Spec} A \times _{\operatorname {Ell}(\mathbf {B} G)} \operatorname {Ell}(\mathbf {B} H)$. As a pullback of spectral schemes, $(\mathcal {X}, \mathcal {O}_{\mathcal {X}})$ is a spectral scheme again, and the underlying scheme of $(\mathcal {X}, \mathcal {O}_{\mathcal {X}})$ is the pullback of the underlying scheme $(\mathcal {X},\pi _0\mathcal {O}_{\mathcal {X}})$, hence affine. Using [Reference LurieLur16, Corollary 1.1.6.3], we conclude that $(\mathcal {X},\mathcal {O}_{\mathcal {X}})$ is affine.

Construction 6.4 Let $G$ be a compact abelian Lie group and $\operatorname {Orb}_G$ its orbit $\infty$-category, which we identify using Proposition 2.16 with the full subcategory of $\operatorname {Orb}_{/\mathbf {B} G}$ on morphisms $\mathbf {B} H \to \mathbf {B} G$ inducing an injection $H\to G$. We consider the functor

\[ \operatorname{Orb}_G \to \operatorname{QCoh}(\operatorname{Ell}(\mathbf{B} G))^{\operatorname{op}},\quad G/H \mapsto f_*\mathcal{O}_{\operatorname{Ell}(\mathbf{B} H)}, \]

where $f\colon \operatorname {Ell}(\mathbf {B} H) \to \operatorname {Ell}(\mathbf {B} G)$ is the map induced by the inclusion $H\subset G$. As $f$ is affine by the preceding proposition, $f_*$ sends quasi-coherent sheaves to quasi-coherent sheaves by [Reference LurieLur16, Proposition 2.5.11]. The functor from $\operatorname {Orb}_G$ extends to a colimit-preserving functor

\[ \mathcal{E}ll_G\colon \mathcal{S}^G \to \operatorname{QCoh}(\operatorname{Ell}(\mathbf{B} G))^{\operatorname{op}}, \]

where $\mathcal {S}^G \simeq \operatorname {Fun}(\operatorname {Orb}_G^{\operatorname {op}}, \mathcal {S})$ denotes the $\infty$-category of $G$-spaces. As the target is pointed, this functor factors canonically to define a reduced version

\[ \widetilde{\mathcal{E}ll}_G\colon \mathcal{S}^G_* \to \operatorname{QCoh}(\operatorname{Ell}(\mathbf{B} G))^{\operatorname{op}}. \]