# A stable $\infty$-category for equivariant $\mathrm{K\\!K}$-theory

Ulrich Bunke Fakultät für Mathematik, Universität Regensburg, 93040
Regensburg, GERMANY
<EMAIL_ADDRESS>Alexander Engel Institut für
Mathematik und Informatik der Universität Greifswald
Walther-Rathenau-Str. 47 in 17489 Greifswald, GERMANY
<EMAIL_ADDRESS>Markus Land Mathematisches Institut, Ludwig-
Maximilians-Universität München, 80333 München, GERMANY
<EMAIL_ADDRESS>
###### Abstract

For a countable group $G$ we construct a small, idempotent complete, symmetric
monoidal, stable $\infty$-category $\mathrm{KK}_{\mathrm{sep}}^{G}$ whose
homotopy category recovers the triangulated equivariant Kasparov category of
separable $G$-$C^{*}$-algebras, and exhibit its universal property. Likewise,
we consider an associated presentably symmetric monoidal, stable
$\infty$-category $\mathrm{KK}^{G}$ which receives a symmetric monoidal
functor $\mathrm{kk}^{G}$ from possibly non-separable $G$-$C^{*}$-algebras and
discuss its universal property. In addition to the symmetric monoidal
structures, we construct various change-of-group functors relating these KK-
categories for varying $G$. We use this to define and establish key properties
of a (spectrum valued) equivariant analytic $K$-homology theory on proper and
locally compact $G$-topological spaces, allowing for coefficients in arbitrary
$G$-$C^{*}$-algebras. Finally, we extend the functor $\mathrm{kk}^{G}$ from
$G$-$C^{*}$-algebras to $G$-$C^{*}$-categories. These constructions are key in
a companion paper about a form of equivariant Paschke duality and assembly
maps.

###### Contents

  1. 1 Introduction and statements
    1. 1.1 A small stable $\infty$-category of separable $G$-$C^{*}$-algebras
    2. 1.2 A presentable stable $\infty$-category for $G$-$C^{*}$-algebras
    3. 1.3 Equivariant analytic $K$-homology
    4. 1.4 The s-finitary extension
    5. 1.5 Change of groups functors
    6. 1.6 Extensions to $C^{*}$-categories
  2. 2 A stable $\bm{\infty}$-category of separable $\bm{G}$-$\bm{C^{*}}$-algebras
  3. 3 The s-finitary extension
  4. 4 Change of groups functors
  5. 5 Analytic $\bm{K}$-homology
  6. 6 Extension to $\bm{C^{*}}$-categories
  7. 7 Tensor products of $\bm{C^{*}}$-categories
  8. A Applications to assembly maps

#### Acknowledgements.

Ulrich Bunke was supported by the SFB 1085 (Higher Invariants) funded by the
Deutsche Forschungsgemeinschaft (DFG).

Alexander Engel acknowledges financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the Priority
Programme SPP 2026 “Geometry at Infinity” (EN 1163/5-1, project number
441426261, Macroscopic invariants of manifolds) and through Germany’s
Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics –
Geometry – Structure.

Markus Land was supported by the research fellowship DFG 424239956, and by the
Danish National Research Foundation through the Copenhagen Centre for Geometry
and Topology (DNRF151).

##  1 Introduction and statements

###  1.1 A small stable $\infty$-category of separable $G$-$C^{*}$-algebras

Let $G$ be a countable group and $A$, $B$ be $G$-$C^{*}$-algebras, i.e.
$C^{*}$-algebras with an action of $G$ by automorphisms. In this situation, we
have the abelian group $K\\!K^{G}(A,B)$ introduced by Kasparov in his work on
the Novikov conjecture [Kas88]. This group depends contravariantly on the
first algebra $A$ and covariantly on the second algebra $B$.

The construction of $K\\!K^{G}{(A,B)}$ can be generalized to graded
$G$-$C^{*}$-algebras, to families of $C^{*}$-algebras parametrized by a space
as in [Kas88], or to the case where $G$ is a locally compact groupoid [LG99].
Though we think that many of our constructions also work in more general
situations, in the present paper we will stick to the classical situation
since this is what immediately generalizes to $C^{*}$-categories, and what we
need for the applications to Paschke duality and assembly maps in [BEL].

If we restrict to separable $G$-$C^{*}$-algebras, then the Kasparov product

$K\\!K^{G}(A,B)\otimes K\\!K^{G}(B,C)\to K\\!K^{G}(A,C)\,,$

which has also been introduced in [Kas88], can be considered as the
composition law of an ${\mathbf{Ab}}$-enriched category $\mathrm{KK}_{0}^{G}$.
The objects of this category are the separable $G$-$C^{*}$-algebras and the
morphism groups are given by

${\mathtt{Hom}}_{\mathrm{KK}_{0}^{G}}(A,B)\coloneqq K\\!K^{G}(A,B)\,.$

The category $\mathrm{KK}_{0}^{G}$ is additive and the sum is represented by
the direct sum of $C^{*}$-algebras. As observed and exploited in [MN06],
$\mathrm{KK}_{0}^{G}$ has a canonical refinement to a triangulated category.

Often, triangulated categories arise as homotopy categories of stable
$\infty$-categories [Lur, Sec. 1.1.2]. The first objective of this paper is to
show that the triangulated categories $\mathrm{KK}_{0}^{G}$ are no exception
to this principle. Standard references for the language of $\infty$-categories
in general are [Lur09], [Cis19]. For the definition and properties of stable
$\infty$-categories we refer to [Lur, Ch. 1].

The first result of this paper is the construction of a stable
$\infty$-category $\mathrm{KK}_{\mathrm{sep}}^{G}$ whose homotopy category is
canonically equivalent, as a triangulated category, to $\mathrm{KK}_{0}^{G}$.
This generalizes a construction of [LN18] from the non-equivariant to the
equivariant case. In the following we provide the precise statement.

Let ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ denote
the category of separable, possibly non-unital $C^{*}$-algebras with
$G$-action. We then have a canonical functor

$\mathrm{kk}_{0}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{0}^{G}$
(1.1)

which is the identity on objects and sends $f\colon A\to B$ to the element
$[f]$ in $K\\!K^{G}(A,B)$ represented by the Kasparov $(A,B)$-module
$(B,f,0)$.

######  Definition 1.1.

A morphism $f$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ is called
a $\mathrm{kk}_{0}^{G}$-equivalence if $\mathrm{kk}_{0}^{G}(f)$ is an
isomorphism.

We let $W_{\mathrm{kk}_{0}^{G}}$ denote the collection of all
$\mathrm{kk}_{0}^{G}$-equivalences. The following definition is the direct
generalization of [LN18] to the equivariant case.

######  Definition 1.2.

We define the $\infty$-category

$\mathrm{KK}_{\mathrm{sep}}^{G}\coloneqq{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})[W_{\mathrm{kk}_{0}^{G}}^{-1}]$

and let

$\mathrm{kk}_{\mathrm{sep}}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{\mathrm{sep}}^{G}$

denote the canonical functor.

Here the $\infty$-category
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})[W_{\mathrm{kk}_{0}^{G}}^{-1}]$
denotes the Dwyer–Kan localization of
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ at the
collection $W_{\mathrm{kk}_{0}^{G}}$ of the
$\mathrm{kk}_{0}^{G}$-equivalences. Such a Dwyer–Kan localization exists and
is characterized by a universal property, see Remark 2.5 for the precise
statement. We have the following theorem:

######  Theorem 1.3.

  1. 1.

The $\infty$-category $\mathrm{KK}_{\mathrm{sep}}^{G}$ is stable.

  2. 2.

We have a canonical factorization

$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{kk}_{0}^{G}}$$\scriptstyle{\mathrm{kk}_{\mathrm{sep}}^{G}}$$\textstyle{\mathrm{KK}_{0}^{G}}$$\textstyle{\mathrm{KK}_{\mathrm{sep}}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ho}}$
(1.2)

and $\mathrm{ho}$ is an equivalence of triangulated categories.

The proof of Theorem 1.3 is a modification of the argument given for the non-
equivariant case in [LN18]. It will be given in Section 2.111Note that the
link points to the end of the proof.

The specific motivation for the present paper was the need, in the companion
paper [BEL], to refine the classical equivariant analytic $K$-homology functor
with coefficients in a $G$-$C^{*}$-algebra $A$

$X\mapsto K_{A,*}^{G,\mathrm{an}}(X)\coloneqq K\\!K_{*}^{G}(C_{0}(X),A)$

to a spectrum valued functor. Using Theorem 1.3 we get such a refinement by
setting

$X\mapsto
K_{A}^{G,\mathrm{an}}(X)\coloneqq\mathrm{KK}_{\mathrm{sep}}^{G}(C_{0}(X),A)\,,$
(1.3)

where for the moment $A$ must be separable, and is $X$ a locally compact and
second countable topological $G$-space so that $C_{0}(X)$ is separable, too.
Here $\mathrm{KK}_{\mathrm{sep}}^{G}(B,A)$ is a short-hand notation for the
spectrum
${\mathtt{map}}_{\mathrm{KK}_{\mathrm{sep}}^{G}}(\mathrm{kk}_{\mathrm{sep}}^{G}(B),\mathrm{kk}_{\mathrm{sep}}^{G}(A))$.
In Definition 1.14 below we will remove the restrictions on $A$ and $X$.

The next theorem lists properties of the functor
$\mathrm{kk}_{\mathrm{sep}}^{G}$ and the $\infty$-category
$\mathrm{KK}_{\mathrm{sep}}^{G}$ which reflect well-known properties of
Kasparov’s bifunctor $K\\!K^{G}$. We first explain some of the notions
appearing in the statement. We consider a functor from $G$-$C^{*}$-algebras to
a stable $\infty$-category. It is called reduced if it sends the zero algebra
to a zero object. It is semiexact if it sends every semisplit exact sequence
to a fibre sequence, where an exact sequence is semisplit if it admits an
equivariant cpc (completely positive and contractive) split (see also
Definition 2.13.1). It is $\mathbb{K}^{G}$-stable if it sends morphisms of the
form

$A\otimes K(H)\to A\otimes K(H^{\prime})$ (1.4)

to equivalences, where $H\to H^{\prime}$ is an equivariant isometric inclusion
of non-zero separable $G$-Hilbert spaces. It is homotopy invariant if it sends
the morphisms of the form

$A\to C([0,1])\otimes A$ (1.5)

given by the embedding $a\mapsto{1\otimes a}$ to equivalences. Finally, we
refer to [MN06, Def. 2.5] for the notion of admissibility of a diagram
$A\colon{\rm
Nature}\to{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$.

######  Theorem 1.4.

  1. 1.

$\mathrm{kk}_{\mathrm{sep}}^{G}$ is reduced.

  2. 2.

$\mathrm{kk}_{\mathrm{sep}}^{G}$ is semiexact.

  3. 3.

$\mathrm{kk}_{\mathrm{sep}}^{G}$ is $\mathbb{K}^{G}$-stable.

  4. 4.

$\mathrm{kk}_{\mathrm{sep}}^{G}$ is homotopy invariant.

  5. 5.

$\mathrm{KK}_{\mathrm{sep}}^{G}$ admits countable colimits and is therefore
idempotent complete.

  6. 6.

$\mathrm{kk}_{\mathrm{sep}}^{G}$ preserves countable sums.

  7. 7.

$\mathrm{kk}_{\mathrm{sep}}^{G}$ preserves colimits of admissible diagrams
$A\colon{\rm
Nature}\to{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$.

The proof of this theorem will be given in Section 2.19.

The functor (1.1) has a characterization by universal properties for functors
to additive categories [Hig87], [Tho98], [Mey00], see Proposition 2.2. Our
next theorem states that $\mathrm{kk}_{\mathrm{sep}}^{G}$ has a similar
universal property for functors from
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ to objects
of the large category ${\mathbf{Cat}}^{\mathrm{ex}}_{\infty}$ of small stable
$\infty$-categories and exact functors.

######  Theorem 1.5.

The functor
$\mathrm{kk}_{\mathrm{sep}}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{\mathrm{sep}}^{G}$
is initial among functors from
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ to objects
of ${\mathbf{Cat}}^{\mathrm{ex}}_{\infty}$ which are reduced, semiexact and
$\mathbb{K}^{G}$-stable.

The proof of this theorem will be given in Section 2. We also have a version
for additive targets, see Theorem 2.23, in which semiexactness is replaced by
split-exactness.

######  Remark 1.6.

Note that the universal property stated in Theorem 1.5 is different from the
obvious one stating that $\mathrm{kk}_{\mathrm{sep}}^{G}$ is the initial
functor to $\infty$-categories which inverts
${\mathrm{kk}^{G}_{0}}$-equivalences. The latter holds true by definition of
$\mathrm{KK}_{\mathrm{sep}}^{G}$ as a Dwyer–Kan localization.

Note also that in most references, the universal property is stated for
functors which are in addition homotopy invariant. In the unequivariant
situation it was first proven by Higson that homotopy invariance follows from
split-exactness and $\mathbb{K}$-stability, and in the equivariant situation
the same is true, see also Remark 2.3. ∎

In the following, we consider the minimal and maximal tensor products
$\otimes_{\min}$ and $\otimes_{\max}$ of $C^{*}$-algebras. Both of them equip
the category ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ of possibly
non-unital $C^{*}$-algebras with $G$-action with a symmetric monoidal
structure and preserve separable algebras.

######  Proposition 1.7 (Proposition 2.20).

The tensor product $\otimes_{?}$ for $?$ in $\\{\min,\max\\}$ descends to a
bi-exact symmetric monoidal structure on $\mathrm{KK}_{\mathrm{sep}}^{G}$, and
$\mathrm{kk}_{\mathrm{sep}}^{G}$ refines to a symmetric monoidal functor

$\mathrm{kk}_{\mathrm{sep}}^{G,\otimes_{?}}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})^{\otimes_{?}}\to\mathrm{KK}_{\mathrm{sep}}^{G,\otimes_{?}}\,.$

Moreover, the tensor structure $\otimes_{?}$ on
$\mathrm{KK}_{\mathrm{sep}}^{G}$ preserves countable colimits in each
variable.

###  1.2 A presentable stable $\infty$-category for $G$-$C^{*}$-algebras

For the purpose of the application in [BEL], the restriction of the definition
of analytic $K$-homology in (1.3) to separable coefficient algebras $A$ is not
sufficient. Therefore we must extend the functor
$\mathrm{kk}_{\mathrm{sep}}^{G}$ from separable $C^{*}$-algebras to all
$C^{*}$-algebras with $G$-action. In order to fix size issues we choose an
increasing sequence of three Grothendieck universes whose elements will be
called small, large and very large sets. All $C^{*}$-algebras are assumed to
be small. The category
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ is
essentially small, and it follows from the details of the proof of Theorem 1.3
that $\mathrm{KK}_{\mathrm{sep}}^{G}$ is also essentially small. In contrast,
the category ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ is large,
but locally small.

######  Definition 1.8.

We define the $\infty$-category

$\mathrm{KK}^{G}\coloneqq\mathrm{Ind}(\mathrm{KK}_{\mathrm{sep}}^{G})$

as the $\mathrm{Ind}$-completion of $\mathrm{KK}_{\mathrm{sep}}^{G}$ and let

$y^{G}\colon\mathrm{KK}_{\mathrm{sep}}^{G}\to\mathrm{KK}^{G}$ (1.6)

denote the canonical functor.

######  Remark 1.9.

If ${\mathcal{C}}$ is a small, stable $\infty$-category, then by [BGT13, Prop.
3.2] an explicit model for the canonical functor
${\mathcal{C}}\to\mathrm{Ind}({\mathcal{C}})$ is given by the Yoneda embedding

${\mathcal{C}}\to{\mathbf{Fun}}^{\mathrm{ex}}({\mathcal{C}}^{\mathrm{op}},\mathbf{Sp}^{\mathrm{}})\,,\quad
C\mapsto{\mathtt{map}}_{{\mathcal{C}}}(-,C)\,.$

The $\infty$-category $\mathrm{Ind}({\mathcal{C}})$ is compactly generated,
presentable and stable. Moreover, the functor
${\mathcal{C}}\to\mathrm{Ind}({\mathcal{C}})^{\omega}$ exhibits the full
subcategory $\mathrm{Ind}({\mathcal{C}})^{\omega}$ of compact objects in
$\mathrm{Ind}({\mathcal{C}})$ as the idempotent completion of ${\mathcal{C}}$.
∎

######  Definition 1.10.

We define the functor

$\mathrm{kk}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to\mathrm{KK}^{G}$

as the left Kan extension of $y^{G}\circ\mathrm{kk}_{\mathrm{sep}}^{G}$ along
the inclusion $\mathrm{incl}$ as indicated in

$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{incl}}$$\scriptstyle{\mathrm{kk}_{\mathrm{sep}}^{G}}$$\textstyle{\mathrm{KK}_{\mathrm{sep}}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y^{G}}$$\textstyle{\mathrm{KK}^{G}\,.}$$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{kk}^{G}}$$\scriptstyle{\Downarrow}$
(1.7)

###  1.3 Equivariant analytic $K$-homology

Using the Definitions 1.8 and 1.10 we can now extend the definition of the
spectrum valued analytic $K$-homology functor (1.3) to all
$G$-$C^{*}$-algebras (or even objects of $\mathrm{KK}^{G}$) $A$ and locally
compact $G$-spaces $X$.

We let $G\mathbf{LCH}^{\mathrm{prop}}$ denote the category of locally compact
Hausdorff spaces with $G$-action and equivariant, continuous and proper maps.
We further consider the category $G\mathbf{LCH}^{\mathrm{prop}}_{+}$ with the
same objects, but with the larger set of maps

${\mathtt{Hom}}_{G\mathbf{LCH}_{+}^{\mathrm{prop}}}(X,Y)\coloneqq{\mathtt{Hom}}_{G\mathbf{LCH}^{\mathrm{prop}}}((X^{+},\infty_{X}),(Y^{+},\infty_{Y}))\,,$

where $X^{+}$ and $Y^{+}$ are the one-point compactifications of $X$ and $Y$,
respectively, and a map $f\colon(X^{+},\infty_{X})\to(Y^{+},\infty_{Y})$ is a
continuous equivariant map $X^{+}\to Y^{+}$ with $f(\infty_{X})=\infty_{Y}$.
Equivalently, a morphism $f\colon X\to Y$ in
$G\mathbf{LCH}_{+}^{\mathrm{prop}}$ is a partially defined map $X\supseteq
U\stackrel{{\scriptstyle f}}{{\to}}Y$ on an open subset $U$ of $X$ with $f$ in
$G\mathbf{LCH}^{\mathrm{prop}}$ which corresponds to the map $f^{+}\colon
X^{+}\to Y^{+}$ such that $f^{+}_{|U}=f$ and $f(X^{+}\setminus
U)=\\{\infty_{Y}\\}$.

Two morphisms $X\to Y$ in $G\mathbf{LCH}_{+}^{\mathrm{prop}}$ are called
properly homotopic if there exists a homotopy $[0,1]\times X\to Y$ in
$G\mathbf{LCH}_{+}^{\mathrm{prop}}$ between them.

The category $G\mathbf{LCH}_{+}^{\mathrm{prop}}$ is the natural domain of the
functor

$C_{0}(-)\colon(G\mathbf{LCH}_{+}^{\mathrm{prop}})^{\mathrm{op}}\to{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$

which sends a locally compact $G$-space to the $C^{*}$-algebra of continuous
functions vanishing at $\infty$ with the induced $G$-action. More precisely,
$C_{0}(X)$ is the kernel of the evaluation map

$C_{0}(X)\coloneqq{\mathtt{ker}}(C(X^{+})\to{\mathbb{C}})\,,$

where the evaluation map takes the value at the point $\infty$. Since we do
not assume that $X$ is second countable, the algebra $C_{0}(X)$ is in general
not separable.

If $Y$ is an invariant closed subset of $X$, then we have an exact sequence

$0\to C_{0}(X\setminus Y)\to C_{0}(X)\to C_{0}(Y)\to 0$ (1.8)

in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$.

######  Definition 1.11.

We say that $Y$ is split-closed in $X$, if the sequence (1.8) is semisplit,
i.e. if it admits an equivariant cpc (completely positive and contractive)
split.

The following proposition provides sufficient criteria for $Y$ being split-
closed.

######  Proposition 1.12 (Proposition 5.1).

The closed invariant subset $Y$ of $X$ is split-closed in the following cases:

  1. 1.

$G$ acts properly on an invariant neighbourhood of $Y$ in $X$ and $Y$ is
second countable.

  2. 2.

$Y$ admits a $G$-invariant tubular neighbourhood.

The notion of split-closedness is introduced since $\mathrm{kk}^{G}$ only
sends semisplit exact sequences to fibre sequences.

######  Example 1.13.

Let ${\rm Nature}\to\beta{\rm Nature}$ be the Stone–Čech compactification of
the discrete space ${\rm Nature}$ and $\partial{\rm Nature}\coloneqq\beta{\rm
Nature}\setminus{\rm Nature}$. Then $\partial{\rm Nature}$ is not split-closed
in $\beta{\rm Nature}$. In fact, it is known that $C_{0}({\rm Nature})$ does
not even have a closed linear complementary subspace in $C_{0}(\beta{\rm
Nature})=C_{b}({\rm Nature})$.∎

######  Definition 1.14.

We define the equivariant analytic $K$-homology functor

$K^{G,\mathrm{an}}\colon
G\mathbf{LCH}_{+}^{\mathrm{prop}}\times\mathrm{KK}^{G}\to\mathbf{Sp}^{\mathrm{}}$

by

$(X,A)\mapsto K^{G,\mathrm{an}}_{A}(X)\coloneqq\mathrm{KK}^{G}(C_{0}(X),A)\,.$

The following theorem states the basic properties of equivariant analytic
$K$-homology. All spaces in the statement belong to
$G\mathbf{LCH}_{+}^{\mathrm{prop}}$, and $A$ is in $\mathrm{KK}^{G}$ or
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$, where in the latter case
we drop the functor $\mathrm{kk}^{G}$ in order to simplify the notation. In
Remark 1.16 below we will explain these statements in more detail.

######  Theorem 1.15.

  1. 1.

If $X$ is second countable and $A$ is a $\sigma$-unital $G$-$C^{*}$-algebra,
then we have an isomorphism of ${\mathbb{Z}}$-graded abelian groups

$K_{A,*}^{G,\mathrm{an}}(X)\cong K\\!K_{*}^{G}(C_{0}(X),A)\,.$

  2. 2.

The functor $K^{G,\mathrm{an}}$ is homotopy invariant.

  3. 3.

If $Y$ is a split-closed $G$-invariant subspace of $X$, then we have a fibre
sequence

$K^{G,\mathrm{an}}_{A}(Y)\to K^{G,\mathrm{an}}_{A}(X)\to
K^{G,\mathrm{an}}_{A}(X\setminus Y)\,.$ (1.9)

  4. 4.

We consider an exact sequence $0\to A\to B\to C\to 0$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$. If the sequence is
semisplit, or if $X$ is properly homotopy equivalent to a finite $G$-CW
complex with finite stabilizers, then we have a fibre sequence

$K^{G,\mathrm{an}}_{A}(X)\to K^{G,\mathrm{an}}_{B}(X)\to
K^{G,\mathrm{an}}_{C}(X)\ .$ (1.10)

If $X$ is second countable, then

$\mathrm{KK}^{G}\ni A\mapsto K^{G,\mathrm{an}}_{A}(X)$ (1.11)

preserves filtered colimits.

  5. 5.

We have

$K^{G,\mathrm{an}}_{A}([0,\infty)\times X)\simeq 0\,.$

Furthermore, if $(X_{n})_{n\in{\rm Nature}}$ is a family of second countable
spaces and $A$ is separable, then we have a canonical equivalence.

$K^{G,\mathrm{an}}_{A}\big{(}\bigsqcup_{n\in{\rm
Nature}}X_{n}\big{)}\xrightarrow{\simeq}\prod_{n\in{\rm
Nature}}K^{G,\mathrm{an}}_{A}(X_{n})\,.$

  6. 6.

If $X_{0}\supseteq X_{1}\supseteq\ldots\supseteq X_{n}\supseteq\ldots$ is a
decreasing sequence of closed invariant subspaces of a second countable space
$X_{0}$ such that $\bigcap_{n}X_{n}\to{X_{0}}$ is split-closed, and $A$ is
separable, then we have an equivalence

$K_{A}^{G,\mathrm{an}}\big{(}\bigcap_{n\in{\rm
Nature}}X_{n}\big{)}\xrightarrow{\simeq}\operatorname*{\mathtt{lim}}_{n\in{\rm
Nature}}K_{A}^{G,\mathrm{an}}(X_{n})\,.$

  7. 7.

If $H$ is a finite subgroup of $G$, then we have an equivalence

$K^{G,\mathrm{an}}_{A}(G/H)\simeq
K^{C^{*}\mathbf{Alg}}(\mathrm{Res}^{G}_{H}(A)\rtimes H)\,.$

  8. 8.

The functor $K^{G,\mathrm{an}}$ has a lax symmetric monoidal refinement

${G\mathbf{LCH}_{+}^{\mathrm{prop}}}^{,\otimes}\times{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})^{\otimes_{?}}\to\mathbf{Sp}^{\mathrm{}\otimes}$

for $?$ in $\\{\min,\max\\}$.

######  Remark 1.16.

In this remark we explain the meaning of the assertions of Theorem 1.15 in
greater detail.

The Assertion 1.15.1 shows that, under the conditions on $X$ and $A$ as
stated, the functor $K^{G,\mathrm{an}}_{A}$ is a spectrum valued refinement of
the classical equivariant analytic $K$-homology functor.

The homotopy invariance in $X$ stated in Assertion 1.15.2 first of all means
that the functor $K^{G,\mathrm{an}}_{A}$ sends the projection $[0,1]\times
X\to X$ to an equivalence. Since we work in the category of proper equivariant
maps, this implies invariance of $K^{G,\mathrm{an}}_{A}$ under proper
equivariant homotopies. Furthermore, for fixed $X$ the functor
$K^{G,\mathrm{an}}_{-}(X)$ is also homotopy invariant in the algebra variable.

The Assertion 1.15.3 implies that $K^{G,\mathrm{an}}_{A}$ satisfies excision
for invariant split-closed decompositions $(Z,Y)$ of $X$, i.e., invariant
decompositions such that $Y$ is split-closed in $X$ and $Y\cap Z$ is split-
closed in $Z$. Furthermore, the fact that the fibre of
$K^{G,\mathrm{an}}_{A}(Y)\to K^{G,\mathrm{an}}_{A}(X)$ only depends on the
complement $X\setminus Y$ is often referred to as the strong excision axiom.
Note that the map $K^{G,\mathrm{an}}_{A}(X)\to
K^{G,\mathrm{an}}_{A}(X\setminus Y)$ in (1.9) is induced by the partially
defined map $X\supseteq X\setminus Y\xrightarrow{{\mathtt{id}}_{X\setminus
Y}}X\setminus Y$.

It immediately follows from Definition 1.14 that for fixed $X$, the functor

$\mathrm{KK}^{G}\ni A\mapsto K^{G,\mathrm{an}}_{A}(X)\in\mathbf{Sp}$

preserves limits. The Assertion 1.15.4 states additional exactness properties
of $K^{G,\mathrm{an}}_{A}(X)$ as a functor on
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ rather than
$\mathrm{KK}^{G}$.

The Assertion 1.15.5 is our expression of local finiteness of
$K^{G,\mathrm{an}}_{A}$. The second part is often referred to as the cluster
axiom.

The Assertion 1.15.6 is also called the continuity axiom.

Following [LN18], in Assertion 1.15.7 we use the spectrum valued $K$-theory
functor for $C^{*}$-algebras

$K^{C^{*}\mathbf{Alg}}(-)\coloneqq\mathrm{KK}({\mathbb{C}},-)\colon
C^{*}\mathbf{Alg}^{\mathrm{nu}}\to\mathbf{Sp}$ (1.12)

which is equivalent to the one constructed in e.g. [Joa03], [Joa04]; see
[LN18, Prop. 3.7.1]. Let $G\mathbf{Orb}$ be the orbit category of $G$, i.e.
the full subcategory of $G$-sets consisting of transitive $G$-sets, and let
$G_{\mathbf{Fin}}\mathbf{Orb}$ denote its full subcategory on orbits with
finite stabilizers. Considering $G$-sets as discrete topological spaces, we
have an embedding $G_{\mathbf{Fin}}\mathbf{Orb}\to
G\mathbf{LCH}_{+}^{\mathrm{prop}}$ and can define a functor

$K^{G,\mathrm{an}}_{A}\colon G_{\mathbf{Fin}}\mathbf{Orb}\to
G\mathbf{LCH}_{+}^{\mathrm{prop}}\xrightarrow{K^{G,\mathrm{an}}_{A}}\mathbf{Sp}^{\mathrm{}}\,.$
(1.13)

The Assertion 1.15.7 implies that this functor has the same values as the
Davis–Lück functor used in [DL98], [Joa03], [LNS17] (for $A={\mathbb{C}}$) and
[Kra] (in general). In [BEL], we will upgrade this to an equivalence of
functors, see (A.9) for a precise statement. In the Appendix A we explain how
the functor $K^{G,\mathrm{an}}_{A}$, which is involved in the definition of
the domain of the spectrum valued Baum–Connes assembly map, features in a
comparison of assembly maps, see Diagram LABEL:refoijoergwergrefw. Its
construction is one of the motivations for the companion paper [BEL].

In Assertion 8 the symmetric monoidal structure on
${G\mathbf{LCH}_{+}^{\mathrm{prop}}}^{,\otimes}$ is given by the cartesian
product of the underlying topological spaces.∎

The functor $K^{G,\mathrm{an}}$ will be derived in (5.6) from the more
fundamental functor

$\mathrm{kk}^{G}C_{0}\coloneqq\mathrm{kk}^{G}\circ C_{0}\colon
G\mathbf{LCH}_{+}^{\mathrm{prop}}\to\mathrm{KK}^{G}$

whose properties will be stated in Theorem 5.2.

The proof of Theorem 1.15 (and of Theorem 5.2) will employ almost all of the
general results about $\mathrm{kk}^{G}$ stated below. It will be completed in
Section 5.

###  1.4 The s-finitary extension

We now come back to the properties of the functor $\mathrm{kk}^{G}$ from
Definition 1.10.

######  Definition 1.17.

A functor $F$ defined on ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$
is called s-finitary if for every $A$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ the canonical map

$\operatorname*{\mathtt{colim}}_{A^{\prime}\subseteq_{{\mathrm{sep}}}A}F(A^{\prime})\to
F(A)$ (1.14)

is an equivalence, where $A^{\prime}$ runs through the separable $G$-invariant
subalgebras of $A$.

The prefix ‘s’ stands for separable. In contrast, in the literature a finitary
functor is usually required to preserve all filtered colimits. The following
theorem lists the basic properties of $\mathrm{kk}^{G}$.

######  Theorem 1.18.

  1. 1.

$\mathrm{kk}^{G}$ is s-finitary.

  2. 2.

$\mathrm{kk}^{G}$ is reduced.

  3. 3.

$\mathrm{kk}^{G}$ is semiexact.

  4. 4.

$\mathrm{kk}^{G}$ is homotopy invariant.

  5. 5.

$\mathrm{kk}^{G}$ is $\mathbb{K}^{G}$-stable.

The proof of this theorem will be finished in Section 3.

In order to formulate the universal property of $\mathrm{kk}^{G}$ we consider
the very large $\infty$-category category
$\mathbf{CAT}_{\infty}^{\mathrm{ccpl}\cap\mathrm{ex}}$ of cocomplete (with
respect to small colimits) stable $\infty$-categories and functors preserving
small colimits. As noted in Remark 1.9 the $\infty$-category $\mathrm{KK}^{G}$
is presentable stable and therefore belongs to
$\mathbf{CAT}_{\infty}^{\mathrm{ccpl}\cap\mathrm{ex}}$.

######  Theorem 1.19 (Theorem 3.3).

$\mathrm{kk}^{G}$ is initial among functors from
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ to objects of
$\mathbf{CAT}_{\infty}^{\mathrm{ccpl}\cap\mathrm{ex}}$ which are s-finitary,
reduced, $\mathbb{K}^{G}$-stable and semiexact.

Assume that $A,B$ are in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$.
Then on the one hand, we have the abelian group $K\\!K_{0}^{G}(A,B)$ defined
by Kasparov in [Kas88]. On the other hand, we have the abelian group
$\pi_{0}\mathrm{KK}^{G}(A,B)$ defined by the abstract categorical procedure in
Definition 1.8. If both $A$ and $B$ are separable, then these two groups are
identified by the morphism $\mathrm{ho}$ in (2.15). The next proposition
extends this isomorphism to all degrees and from separable to $\sigma$-unital
$B$.222Using [Ska85, Sec. 3] on can remove the assumption that $B$ is
$\sigma$-unital. For separable $A$ the functor $K\\!K_{0}(A,-)$ is
$s$-finitary. Thereby various definitions of the functor coincide. We let
$C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\sigma}$ be the full subcategory of
$C^{*}\mathbf{Alg}^{\mathrm{nu}}$ of $\sigma$-unital $C^{*}$-algebras.

######  Proposition 1.20 (Proposition 3.5).

For any objects $A$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ and $B$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\sigma})$, the functor
$\mathrm{ho}$ induces an isomorphism of ${\mathbb{Z}}$-graded abelian groups

$\pi_{*}\mathrm{KK}^{G}(A,B)\cong K\\!K^{G}_{*}(A,B)\,.$

Note that this proposition for separable $B$ is a direct consequence of the
compatibility of $\mathrm{ho}$ with the triangulated structure stated in
Theorem 1.3.2.

The following result extends Proposition 1.7 from the separable to the general
case.

######  Proposition 1.21 (Proposition 3.8).

The symmetric monoidal structure $\otimes_{?}$ on
$\mathrm{KK}_{\mathrm{sep}}^{G}$ for $?$ in $\\{\min,\max\\}$ canonically
induces a presentably symmetric monoidal structure on $\mathrm{KK}^{G}$ and
$\mathrm{kk}^{G}$ refines to a symmetric monoidal functor

$\mathrm{kk}^{G,{\otimes}_{?}}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})^{\otimes_{?}}\to\mathrm{KK}^{G,{\otimes}_{?}}\,.$

As an immediate consequence of Proposition 1.21 we can define for $?$ in
$\\{\min,\max\\}$ an internal morphism functor

$\mathrm{kk}^{G}_{?}(-,-)\colon(\mathrm{KK}^{G})^{\mathrm{op}}\times\mathrm{KK}^{G}\to\mathrm{KK}^{G}\,.$
(1.15)

It is characterized by a natural equivalence
$\mathrm{KK}^{G}(A,\mathrm{kk}^{G}_{?}(B,C))\simeq\mathrm{KK}^{G}(A\otimes_{?}B,C)$
for all $A,B,C$ in $\mathrm{KK}^{G}$ and preserves limits in both arguments.

###  1.5 Change of groups functors

In the following we consider various change of groups functors. If $H\to G$ is
a homomorphism of groups, then we have an obvious restriction functor

$\mathrm{Res}^{G}_{H}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to{\mathbf{Fun}}(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}})\,.$

If $H$ is a subgroup of $G$, then we have an induction functor

$\mathrm{Ind}_{H}^{G}\colon{\mathbf{Fun}}(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$

which will be explained in Construction 4.9. Finally, we have maximal and
reduced crossed product functors

$-\rtimes_{\max}G,-\rtimes_{r}G\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to
C^{*}\mathbf{Alg}^{\mathrm{nu}}$

whose details will be recalled in Construction 4.14. The following results say
that these functors descend to functors (denoted by the same symbols) between
the corresponding stable $\infty$-categories.

######  Theorem 1.22.

  1. 1.

There exists a factorization

$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}_{H}^{G}}$$\scriptstyle{\mathrm{kk}^{G}}$$\textstyle{{\mathbf{Fun}}(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{kk}^{H}}$$\textstyle{\mathrm{KK}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}_{H}^{G}}$$\textstyle{\mathrm{KK}^{H}}$

and $\mathrm{Res}_{H}^{G}\colon\mathrm{KK}^{G}\to\mathrm{KK}^{H}$ preserves
colimits and compact objects.

  2. 2.

There exists a factorization

$\textstyle{{\mathbf{Fun}}(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ind}_{H}^{G}}$$\scriptstyle{\mathrm{kk}^{H}}$$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{kk}^{G}}$$\textstyle{\mathrm{KK}^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ind}_{H}^{G}}$$\textstyle{\mathrm{KK}^{G}}$

and $\mathrm{Ind}_{H}^{G}\colon\mathrm{KK}^{H}\to\mathrm{KK}^{G}$ preserves
colimits and compact objects.

  3. 3.

There exists a factorization

$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rtimes_{?}G}$$\scriptstyle{\mathrm{kk}^{G}}$$\textstyle{C^{*}\mathbf{Alg}^{\mathrm{nu}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{kk}}$$\textstyle{\mathrm{KK}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\rtimes_{?}G}$$\textstyle{\mathrm{KK}}$

for $?\in\\{r,\max\\}$ and $-\rtimes_{?}G\colon\mathrm{KK}^{G}\to\mathrm{KK}$
preserves colimits and compact objects.

The functor in Assertion 3 induces on mapping spectra a spectrum level version
of Kasparov’s descent morphism [Kas88].

Since $\mathrm{KK}_{\mathrm{sep}}^{G}$ is idempotent complete by Theorem
1.4.5, the functor $y^{G}$ in (1.6) identifies
$\mathrm{KK}_{\mathrm{sep}}^{G}$ with the full subcategory of
$\mathrm{KK}^{G}$ of compact objects, see Remark 1.9. Hence the assertion that
the functors preserve compact objects means that they induce functors between
the separable versions of the respective $\mathrm{KK}$-categories. It then
follows from the assertion about preservation of colimits that the functors
appearing in Theorem 1.22 are equivalent to the canonical extensions of their
separable versions. The proof of Theorem 1.22 will be given in Section 4.

There exists a canonical natural transformation

$\iota\colon{\mathtt{id}}\to\mathrm{Res}^{G}_{H}\circ\mathrm{Ind}_{H}^{G}\,,$
(1.16)

of endofunctors of ${\mathbf{Fun}}(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ which
will be explained in detail after (4.5). It looks like the unit of an
adjunction, and it indeed becomes the unit of an adjunction after application
of $\mathrm{kk}^{G}$ (and it is not one before), see (1.20) below. The
transformation $\iota$ induces the first transformation of functors in

${-}\rtimes_{?}H\to\mathrm{Res}^{G}_{H}\circ\mathrm{Ind}^{G}_{H}(-)\rtimes_{?}H\to\mathrm{Ind}^{G}_{H}(-)\rtimes_{?}G\,,$
(1.17)

where the second is canonically induced by the inclusion of $H$ into $G$, see
(4.13).

We now assume that $H$ is a finite group. If $A$ is in
$C^{*}\mathbf{Alg}^{\mathrm{nu}}$, then we consider the homomorphism

$\epsilon_{A}\colon A\to\mathrm{Res}_{H}(A)\rtimes H\,,\quad
a\mapsto\frac{1}{|H|}\sum_{h\in H}(a,h)\,,$

where $\mathrm{Res}_{H}(A)$ denotes $A$ equipped with the trivial $H$-action
and we refer to Construction 4.14 for the notation $(a,h)$ for elements in
$\mathrm{Res}_{H}(A)\rtimes H$. The family $\epsilon=(\epsilon_{A})_{A\in
C^{*}\mathbf{Alg}^{\mathrm{nu}}}$ is a natural transformation

$\epsilon\colon{\mathtt{id}}\to\mathrm{Res}_{H}(-)\rtimes H$ (1.18)

of endofunctors of $C^{*}\mathbf{Alg}^{\mathrm{nu}}$.

Let $B$ be in $C^{*}\mathbf{Alg}^{\mathrm{nu}}$. Then we have a canonical
homomorphism

$\lambda_{B}\colon\mathrm{Res}_{G}(B)\rtimes_{\max}G\to B$

of $C^{*}$-algebras which corresponds to the covariant representation
$({\mathtt{id}}_{B},\mathrm{triv})$ consisting of the identity of $B$ and the
trivial representation of $G$. The family $\lambda=(\lambda_{B})_{B\in
C^{*}\mathbf{Alg}^{\mathrm{nu}}}$ is a natural transformation

$\lambda\colon\mathrm{Res}_{G}(-)\rtimes_{\max}G\to{\mathtt{id}}$ (1.19)

of endofunctors of $C^{*}\mathbf{Alg}^{\mathrm{nu}}$.

The Assertions 1 and 3 of the following theorem are $\infty$-categorical level
versions of Green’s Imprimitivity Theorem [Gre78] (for $?=\max$), [Kas88] and
the Green–Julg theorem [Jul81]. Both statements generalize the versions for
the triangulated categories stated in [MN06]. Assertion 4 is known as the dual
Green–Julg theorem.

######  Theorem 1.23.

  1. 1.

The natural transformation (1.16) induces the unit of an adjunction

$\mathrm{Ind}_{H}^{G}:\mathrm{KK}^{H}\rightleftarrows\mathrm{KK}^{G}:\mathrm{Res}^{G}_{H}\,.$
(1.20)

  2. 2.

The transformation (1.17) naturally induces an equivalence of functors

${-}\rtimes_{?}H\to\mathrm{Ind}_{H}^{G}(-)\rtimes_{?}G\colon\mathrm{KK}^{H}\to\mathrm{KK}$

for $?$ in $\\{{r},\max\\}$.

  3. 3.

If $H$ is finite, then the natural transformation (1.18) induces the unit of
an adjunction

${\mathrm{Res}_{H}}:\mathrm{KK}\rightleftarrows\mathrm{KK}^{H}:-\rtimes H\,.$

  4. 4.

The natural transformation (1.19) induces the counit of an adjunction

$-\rtimes_{\max}G:\mathrm{KK}^{G}\rightleftarrows\mathrm{KK}:{\mathrm{Res}_{G}}\,.$

Since $H$ is finite, the crossed product in Assertion 3 need not be decorated.
It is simply the algebraic crossed product which happens to be equal to the
reduced and maximal one in this case. Note that Assertion 2 in the case
$?=\max$ is also a consequence of Assertions 1 and 4. The proof of this
theorem will be completed in Section 4.

Note that $\mathrm{Ind}_{H}^{G}({\mathbb{C}})\cong C_{0}(G/H)$. Let

$r^{G}_{H}\colon\mathrm{KK}^{G}(C_{0}(G/H),-)\simeq\mathrm{KK}^{H}({\mathbb{C}},\mathrm{Res}^{G}_{H}(-))$
(1.21)

denote the equivalence of the mapping spectrum functors given by the
adjunction in the Theorem 1.23.1. Furthermore, let

$\mathrm{GJ}^{H}\colon\mathrm{KK}^{H}({\mathbb{C}},\mathrm{Res}^{G}_{H}(-))\simeq\mathrm{KK}({\mathbb{C}},\mathrm{Res}^{G}_{H}(-)\rtimes
H)$

denote the equivalence of the mapping spectrum functors given by the
adjunction in the Theorem 1.23.3.

######  Corollary 1.24.

If $H$ is a finite subgroup of $G$, then we have an equivalence

$\mathrm{GJ}^{H}\circ
r^{G}_{H}\colon\mathrm{KK}^{G}(C_{0}(G/H),-)\to{K^{C^{*}\mathbf{Alg}}(\mathrm{Res}_{H}^{G}(-)\rtimes
H)}$ (1.22)

of functors from ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ to
$\mathbf{Sp}^{\mathrm{}}$.

By Theorem 1.18.3 the functor $\mathrm{kk}^{G}$ sends exact sequences in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ admitting equivariant cpc
splits to fibre sequences. As an immediate consequence, the functor

$\mathrm{KK}^{G}(A,-)\colon{\mathbf{Fun}}(BG,{C^{*}\mathbf{Alg}^{\mathrm{nu}}})\to\mathbf{Sp}^{\mathrm{}}$

has the same property for every $A$ in $\mathrm{KK}^{G}$. The Theorem 1.27
shows that under certain restrictions on $A$ this functor in fact sends all
exact sequences to fibre sequences. The conditions are formulated so that we
can deduce this exactness using Corollary 1.24 from the fact that the usual
$K$-theory functor for $C^{*}$-algebras in (1.12) sends all exact sequences to
fibre sequences.

Recall that a thick subcategory of a stable $\infty$-category is a full stable
subcategory which is closed under taking retracts. A localizing subcategory of
a presentable stable $\infty$-category is a full stable subcategory which is
closed under all colimits.

######  Definition 1.25.

  1. 1.

The objects of the thick subcategory of $\mathrm{KK}^{G}$ generated by the
objects $\mathrm{kk}^{G}(C_{0}(G/H))$ for all finite subgroups $H$ of $G$ are
called $G$-proper.

  2. 2.

The objects of the localizing subcategory of $\mathrm{KK}^{G}$ generated by
the $G$-proper objects are called ind-$G$-proper.

Note that $G$-proper objects are ind-$G$-proper. The following result provides
examples of $G$-proper objects in $\mathrm{KK}^{G}$.

######  Proposition 1.26 (Proposition 5.5).

If $X$ is in $G\mathbf{LCH}^{\mathrm{prop}}$ and homotopy equivalent (in
$G\mathbf{LCH}^{\mathrm{prop}}$) to a finite $G$-CW complex with (necessarily)
finite stabilizers, then $\mathrm{kk}^{G}(C_{0}(X))$ is $G$-proper.

Let $P$ be an object of $\mathrm{kk}^{G}$.

######  Theorem 1.27 (Theorem 5.6).

  1. 1.

If $P$ is ind-$G$-proper, then the functor

$\mathrm{KK}^{G}(P,-)\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to\mathbf{Sp}^{\mathrm{}}$

sends all exact sequences to fibre sequences.

  2. 2.

If $P$ is $G$-proper, then the functor

$\mathrm{KK}^{G}(P,-)\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to\mathbf{Sp}^{\mathrm{}}$

preserves filtered colimits.

######  Remark 1.28.

We define the thick subcategory of $G$-nuclear objects of $\mathrm{KK}^{G}$
generated by objects of the form $\mathrm{kk}^{G}(\mathrm{Ind}_{H}^{G}(A))$
for finite subgroups $H$ and nuclear $A$ in
$C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}$ equipped with the trivial
$H$-action. We further define the localizing subcategory of $\mathrm{KK}^{G}$
of ind-$G$ nuclear objects generated by the $G$-nuclear objects. Since
${\mathbb{C}}$ is nuclear, (ind)-$G$-proper objects are (ind)-$G$-nuclear. It
follows from [Ska88] that the functor
$\mathrm{KK}_{\mathrm{sep}}(A,-):C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}\to\mathbf{Sp}$
sends all exact sequences to fibre sequences provided $A$ is nuclear. Using
this fact one can show that the assertions of Theorem 1.27 remain true (with
essentially the same proof) with (ind)-$G$-proper replaced by
(ind)-$G$-nuclear.

For instance, if $X$ in $\mathbf{LCH}^{\mathrm{prop}}$ is separable and $H$ is
a finite subgroup of $G$, then $\mathrm{kk}^{G}(C_{0}(G/H\times X))$ is
$G$-nuclear, but not necessarily (ind)-$G$-proper. ∎

###  1.6 Extensions to $C^{*}$-categories

Again motivated by the applications in [BEL], we extend the functor
$\mathrm{kk}^{G}$ from $C^{*}$-algebras to $C^{*}$-categories with $G$-action.
We refer to the beginning of Section 6 for a more detailed introduction to
$C^{*}$-categories. We have a fully faithful inclusion of the category
$C^{*}\mathbf{Alg}^{\mathrm{nu}}$ of (possibly non-unital) $C^{*}$-algebras
into the category $C^{*}\mathbf{Cat}^{\mathrm{nu}}$ of small (possibly non-
unital) $C^{*}$-categories which considers a $C^{*}$-algebra as a
$C^{*}$-category with a single object. This inclusion fits into an adjunction

$A^{f}:C^{*}\mathbf{Cat}^{\mathrm{nu}}\rightleftarrows
C^{*}\mathbf{Alg}^{\mathrm{nu}}:\mathrm{incl}$ (1.23)

first considered by [Joa03] (see [Bun] for the non-unital case).

######  Definition 1.29.

We define the functor

$\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\to\mathrm{KK}^{G}$

as the composition

${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\xrightarrow{A^{f}}{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\xrightarrow{\mathrm{kk}^{G}}\mathrm{KK}^{G}\,.$

Let ${\mathbf{C}}$ be in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$.

######  Definition 1.30.

We call ${\mathbf{C}}$ separable if ${\mathbf{C}}$ has countably many objects
and all morphism spaces of ${\mathbf{C}}$ are separable.

The following extends Definition 1.17 from $C^{*}$-algebras to
$C^{*}$-categories.

######  Definition 1.31.

A functor $F$ defined on ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$
is called s-finitary, if for every ${\mathbf{C}}$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ the canonical map

$\operatorname*{\mathtt{colim}}_{{\mathbf{C}}^{\prime}}F({\mathbf{C}}^{\prime})\to
F({\mathbf{C}})$ (1.24)

is an equivalence, where ${\mathbf{C}}^{\prime}$ runs through the poset of
separable $G$-invariant subcategories of ${\mathbf{C}}$.

As in the case of $C^{*}$-algebras, we added the letter ‘s’ (for separable),
since this notion differs from the definition of a finitary functor in [BE,
Def. 13.7]. The latter requires the preservation of all filtered colimits. As
an example, using that $G$ is countable one can check that the identity
functor on ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ is
$s$-finitary.

In the following theorem we list the properties of the functor
$\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$ for $C^{*}$-categories. For notions
appearing in its statement we refer to the following sources: unitary
equivalence: [BE, Def. 3.19], weak Morita equivalence: [BE, Def. 18.3] and
Definition 6.12, exact sequence: [Bun, Def. 8.5], flasque: Definition 6.18 and
[BE, Def. 11.3], maximal crossed product $-\rtimes G$: [Bun, Def. 5.9],
reduced crossed product $-\rtimes_{r}G$: [BE, Thm. 12.1] and relative Morita
equivalence: [BE, Def. 17.1].

######  Theorem 1.32.

  1. 1.

The functor $\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$ is s-finitary.

  2. 2.

The functor $\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$ sends unitary equivalences
to equivalences.

  3. 3.

The functor $\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$ sends weak Morita
equivalences to equivalences.

  4. 4.

We have an equivalence

$\mathrm{kk}_{C^{*}\mathbf{Cat}}(-\rtimes_{{?}}G)\simeq(-\rtimes_{{?}}G)\circ\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$

of functors ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\to\mathrm{KK}$
for $?\in\\{r,\max\\}$.

  5. 5.

If $P$ in $\mathrm{KK}^{G}$ is ind-$G$-proper, then the functor
$\mathrm{KK}^{G}(P,\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(-))$ sends all exact
sequences in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ to fibre
sequences.

  6. 6.

If $P$ in $\mathrm{KK}^{G}$ is $G$-proper, then
$\mathrm{KK}^{G}(P,\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(-))$ preserves filtered
colimits.

  7. 7.

If $P$ in $\mathrm{KK}^{G}$ is ind-$G$-proper, then
$\mathrm{KK}^{G}(P,\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(-))$ annihilates
flasques in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})$.

  8. 8.

If $P$ in $\mathrm{KK}^{G}$ is ind-$G$-proper, then
$\mathrm{KK}^{G}(P,\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(-))$ sends relative
Morita equivalences in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ to
equivalences.

The proof of Theorem 1.32 will be given in Section 6.

######  Remark 1.33.

Using Remark 1.28 in Assertions 5 we could replace the conditions
(ind)-$G$-proper by (ind)-$G$-nuclear.

######  Remark 1.34.

Let us consider here the case of the trivial group. In [BE, Def. 13.4] we
introduced the notion of a homological functor $\mathrm{Hg}\colon
C^{*}\mathbf{Cat}^{\mathrm{nu}}\to\mathbf{M}$. By definition, it is a functor
to a stable $\infty$-category $\mathbf{M}$ which sends unitary equivalences of
$C^{*}$-categories to equivalences and exact sequences of $C^{*}$-categories
to fibre sequences. If $\mathrm{Hg}$ in addition preserves filtered colimits,
then it is called finitary.

If $P$ in $\mathrm{KK}$ is ind-proper (or more generally ind-nuclear), then it
follows from Theorem 1.32 (and Remark 1.33) that
$\mathrm{KK}(P,\mathrm{kk}_{C^{*}\mathbf{Cat}}(-))\colon
C^{*}\mathbf{Cat}^{\mathrm{nu}}\to\mathbf{Sp}$ is a homological functor. If
$P$ is proper (or more generally nuclear), then this functor is also finitary.

As an example, if $X$ is locally compact and homotopy equivalent to a finite
CW-complex, then $P\coloneqq\mathrm{kk}(C_{0}(X))$ is proper by Proposition
1.26, and therefore $\mathrm{KK}(C_{0}(X),\mathrm{kk}_{C^{*}\mathbf{Cat}}(-))$
is a homological functor. More generally, if $A$ in
$C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}$ is such that $\mathrm{kk}(A)$
is nuclear (e.g. if $A$ is $\mathrm{kk}^{G}_{0}$-equivalent to a separable
nuclear $C^{*}$-algebra), then
$\mathrm{KK}(A,\mathrm{kk}_{C^{*}\mathbf{Cat}}(-))$ is a homological functor.

The functor $\mathrm{kk}_{C^{*}\mathbf{Cat}}$ itself is not a
$\mathrm{KK}$-valued homological functor since it does not send all exact
sequences to fibre sequences. ∎

The minimal and maximal tensor products for $C^{*}$-algebras can naturally be
extended to $C^{*}$-categories, see [Del12] for $\otimes_{\max}$, and [AV20]
for both. In Section 7 we will give a comprehensive treatment of both cases.
The definition of the maximal tensor product in terms of its universal
property is stated in Definition 7.2, while the analoguous definition of the
minimal tensor product is given in Definition 7.6.

######  Theorem 1.35.

The functor $\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$ refines to a symmetric
monoidal functor

$\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G,{\otimes}_{?}}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\otimes_{?}}\to\mathrm{KK}^{G,{\otimes}_{?}}$

for $?$ in $\\{\min,\max\\}$.

The proof of this theorem will be given in Section 7.21.

##  2 A stable $\bm{\infty}$-category of separable
$\bm{G}$-$\bm{C^{*}}$-algebras

Let $C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}$ and
$C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\sigma}$ denote the full subcategories of
$C^{*}\mathbf{Alg}^{\mathrm{nu}}$ of separable and of $\sigma$-unital
$C^{*}$-algebras, respectively. In the present paper we work with the
bivariant $G$-equivariant Kasparov $K\\!K$-theory functor [Kas88]

$K\\!K_{*}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\sigma})^{\mathrm{op}}\times{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\sigma})\to{\mathbf{Ab}}^{{\mathbb{Z}}}\,,$

where ${\mathbf{Ab}}^{{\mathbb{Z}}}$ denotes the category of
${\mathbb{Z}}$-graded abelian groups. Further good references for this functor
and its properties are [Mey00] and [Bla98].

We will use the symbol $K\\!K^{G}\coloneqq K\\!K_{0}^{G}$ for the
${\mathbf{Ab}}$-valued functor obtained from $K\\!K^{G}_{*}$ by extracting the
degree-zero component. Using the Kasparov intersection product in order to
define the composition we construct the ${\mathbf{Ab}}$-enriched category
$\mathrm{KK}_{0}^{G}$ and the functor

$\mathrm{kk}_{0}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{0}^{G}$

(appearing in (1.1)) as explained in the introduction. The functor
$\mathrm{kk}_{0}^{G}$ is reduced, homotopy invariant and
$\mathbb{K}^{G}$-stable since it inherits these properties from $K\\!K^{G}$.

Recall the Definition 1.1 of a $\mathrm{kk}_{0}^{G}$-equivalence. The functor
$\mathrm{kk}_{0}^{G}$ can be characterized by the following universal
property:

######  Proposition 2.1.

The functor
$\mathrm{kk}_{0}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{0}^{G}$
exhibits the target as localisation of
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ at the set
of ${\mathrm{kk}_{0}^{G}}$-equivalences in the sense of ordinary categories.

###### Proof.

Let

$F\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to{\mathcal{D}}$

be any functor to an ordinary category ${\mathcal{D}}$ which sends
$\mathrm{KK}_{0}^{G}$-equivalences to isomorphisms in ${\mathcal{D}}$. We must
show that there exists a unique factorisation through $\mathrm{KK}_{0}^{G}$,
indicated by the dashed arrow in the following diagram

$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\mathrm{kk}_{0}^{G}}$$\textstyle{{\mathcal{D}}\,.}$$\textstyle{\mathrm{KK}_{0}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{F}}$
(2.1)

Since
$\mathrm{kk}_{0}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{0}^{G}$
is the identity on objects we are forced to define $\bar{F}(A)\coloneqq F(A)$
for all objects $A$ of
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$.

For the discussion of morphisms we use the Cuntz picture of $K\\!K^{G}$ due to
[Mey00, Sec. 6]. Let

$K\coloneqq K(\ell^{2}\otimes L^{2}(G))$ (2.2)

denote the algebra of compact operators on the $G$-Hilbert space
$\ell^{2}\otimes L^{2}(G)$, where $G$ acts on $L^{2}(G)$ by the left-regular
representation, and by conjugation on the operators. We furthermore let

$q_{s}(A):={\mathtt{ker}}(A\otimes K\sqcup A\otimes K\to A\otimes K)$

be the kernel of the fold map. Note that the coproduct in $C^{*}$-algebras is
realized by the free product. By [Mey00, Thm. 6.5] we have a bijection

$K\\!K^{G}(A,B)\cong[q_{s}(A)\otimes K,q_{s}(B)\otimes K]\,,$ (2.3)

where the notation $[-,-]$ stands for norm-continuous homotopy classes of
morphisms in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. Moreover,
the composition in $\mathrm{KK}_{0}^{G}$ corresponds to the composition of
homotopy classes.

We consider the $G$-Hilbert space
$H^{\prime}\coloneqq{\mathbb{C}}\oplus\ell^{2}\otimes L^{2}(G)$ and the
associated $G$-$C^{*}$-algebra $K^{\prime}\coloneqq K(H^{\prime})$ of compact
operators. The equivariant inclusion ${\mathbb{C}}\to H^{\prime}$ induces an
inclusion $d:{\mathbb{C}}\to K^{\prime}$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$, where
${\mathbb{C}}$ has the trivial $G$-action. We also have a canonical inclusion
$e\colon K\to K^{\prime}$.

Let $\pi_{A}\colon q_{s}(A)\to A\otimes K$ be the restriction of
${\mathtt{id}}_{A\otimes K}\sqcup 0$ to $q_{s}(A)$. It is shown in [Mey00,
Sec. 6] that $\pi_{A}$ is a ${\mathrm{kk}_{0}^{G}}$-equivalence. For every $A$
in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ we have
a zig-zag

$q_{s}(A)\otimes
K\stackrel{{\scriptstyle\pi_{A}\otimes{\mathtt{id}}_{K}}}{{\to}}{A}\otimes
K\otimes K\stackrel{{\scriptstyle{\mathtt{id}}_{A}\otimes
e\otimes{\mathtt{id}}_{K}}}{{\to}}A\otimes K^{\prime}\otimes
K\stackrel{{\scriptstyle{\mathtt{id}}_{A}\otimes
d\otimes{\mathtt{id}}_{K}}}{{\leftarrow}}A\otimes
K\stackrel{{\scriptstyle{\mathtt{id}}_{A}\otimes e}}{{\to}}A\otimes
K^{\prime}\stackrel{{\scriptstyle{\mathtt{id}}_{A}\otimes
d\otimes{\mathtt{id}}_{K}}}{{\leftarrow}}A$ (2.4)

which is natural in $A$. By the $\mathbb{K}^{G}$-stability of
$\mathrm{kk}_{0}^{G}$ all these maps are ${\mathrm{kk}_{0}^{G}}$-equivalences.
We now consider the diagram

$\displaystyle\mathclap{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\raise-21.21776pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
164.5152pt\raise 49.59023pt\hbox{{}\hbox{\kern 0.0pt\raise
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0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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K,q_{s}(B)\otimes
K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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190.4614pt\raise-56.9333pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
256.74225pt\raise-69.1944pt\hbox{{}\hbox{\kern 0.0pt\raise
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The left (or right) vertical isomorphism is induced by applying $F$ (or
$\mathrm{kk}_{0}^{G}$) to the zig-zag (2.4) using the fact that $F$ (or
$\mathrm{kk}_{0}^{G}$) sends $\mathrm{kk}_{0}^{G}$-equivalences to
isomorphisms. If $\bar{F}$ exists, then the outer square and the two triangles
involving dashed and bold arrows commute.

Using the lower right triangle we see that the arrow marked by $!$ is
surjective. This shows that the arrow marked by $!!$ is uniquely determined if
it exists.

Since $F$ inverts $\mathrm{KK}_{0}^{G}$-equivalences, it also sends homotopic
maps to equal maps. Consequently, the arrow $F$ marked by $!!!$ factorizes
uniquely through the dotted arrow $F^{\prime}$ as indicated. We are therefore
forced to define

$\bar{F}_{A,B}:K\\!K^{G}(A,B)\to{\mathtt{Hom}}_{{\mathcal{D}}}(A,B)$

as the composition going clockwise from the upper right-corner to the upper
left corner. The whole diagram is bi-natural in $A$ (contravariant) and $B$
(covariant). This implies that the family
$(\bar{F}_{A,B})_{A,B\in{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})}$
provides the action of the desired functor $\bar{F}$ on morphisms. The
triangle (2.1) then commutes by construction. ∎

The following universal property of $\mathrm{kk}_{0}^{G}$ (which differes from
the one in Proposition 2.1) was shown in [Tho98, Thm. 2.2], see also [Mey00,
Thm. 6.6]. A functor from
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ to an
additive category is called split exact if it preserves split-exact sequences,
see also Definition 2.13.2.

######  Proposition 2.2.

The functor

$\mathrm{kk}_{0}^{G}:{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\longrightarrow\mathrm{KK}^{G}_{0}$

is initial among all functors to an additive $1$-category which are reduced,
$\mathbb{K}^{G}$-stable and split exact.

######  Remark 2.3.

The statement of Meyer requires additivity and homotopy invariance of the
functors. But note that the condition of being split exact and additive are
equivalent to the condition of being split exact and reduced. Furthermore, we
use that split-exactness and $\mathbb{K}^{G}$-stability together imply
homotopy invariance (see [Hig88] for the non-equivariant case). We refer to
[CMR07, Thm. 3.35] for an argument which applies verbatim to the equivariant
case. ∎

######  Corollary 2.4.

A functor from
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ to an
additive $\infty$-category which is reduced, $\mathbb{K}^{G}$-stable and split
exact sends ${\mathrm{kk}_{0}^{G}}$-equivalences to equivalences.

###### Proof.

Let
$F:{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to{\mathcal{D}}$
be a functor as in the statement of the corollary. Since
$\mathrm{ho}({\mathcal{D}})$ is an additive $1$-category we can apply
Proposition 2.2 to the composition
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\stackrel{{\scriptstyle
F}}{{\to}}{\mathcal{D}}\to\mathrm{ho}({\mathcal{D}})$ in order to conclude
that it factorizes over $\mathrm{kk}_{0}^{G}$. In particular, it sends
${\mathrm{kk}_{0}^{G}}$-equivalences to isomorphisms. Since the canonical
functor ${\mathcal{D}}\to\mathrm{ho}({\mathcal{D}})$ detects equivalences we
conclude that $F$ sends ${\mathrm{kk}_{0}^{G}}$-equivalences to equivalences.
∎

######  Remark 2.5.

Let ${\mathcal{C}}$ be a small $\infty$-category and $W$ be a set of morphisms
in ${\mathcal{C}}$. Then we can form the Dwyer–Kan localization [Lur, Def.
1.3.4.1, Rem. 1.3.4.2]

$\ell\colon{\mathcal{C}}\to{\mathcal{C}}[W^{-1}]\,.$ (2.13)

It is characterized by the universal property that for any $\infty$-category
${\mathcal{D}}$ the restriction along $\ell$ induces an equivalence

${\mathbf{Fun}}({\mathcal{C}}[W^{-1}],{\mathcal{D}})\xrightarrow{\simeq}{\mathbf{Fun}}^{W}({\mathcal{C}},{\mathcal{D}})\,,$
(2.14)

where ${\mathbf{Fun}}^{W}({\mathcal{C}},{\mathcal{D}})$ is the full
subcategory of ${\mathbf{Fun}}({\mathcal{C}},{\mathcal{D}})$ of functors which
send morphisms in $W$ to equivalences. ∎

Note that we consider ordinary categories as $\infty$-categories using the
nerve, but we will not write the nerve explicitly. With these conventions we
can consider the functor

$\mathrm{kk}_{\mathrm{sep}}^{G}:{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{\mathrm{sep}}^{G}$

introduced in Definition 1.2.

We now start with the proof of Theorem 1.3 following the lines of [LN18] using
[Mey00] and [Uuy13]. First of all it follows from the universal property of
the Dwyer–Kan localization that there exists a factorization of
$\mathrm{kk}_{0}^{G}$ over a functor $\mathrm{ho}$ as indicated by the
following commuting triangle:

$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{kk}_{0}^{G}}$$\scriptstyle{\mathrm{kk}_{\mathrm{sep}}^{G}}$$\textstyle{\mathrm{KK}_{0}^{G}\,.}$$\textstyle{\mathrm{KK}_{\mathrm{sep}}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ho}}$
(2.15)

This is the commutative triangle in (1.2).

In order to show that ${\mathrm{KK}_{\mathrm{sep}}^{G}}$ is stable we need the
explicit model for the suspension functor

$S\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\,,\quad
A\mapsto C_{0}((0,1))\otimes A\,.$ (2.16)

######  Lemma 2.6.

  1. 1.

The functor $S$ uniquely descends to an equivalence $S_{0}$ of additive
categories such that

$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\scriptstyle{\mathrm{kk}_{0}^{G}}$$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{kk}_{0}^{G}}$$\textstyle{\mathrm{KK}_{0}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{0}}$$\textstyle{\mathrm{KK}_{0}^{G}}$
(2.17)

commutes.

  2. 2.

The functor $S$ essentially uniquely descends to a functor (also denoted by)
$S$ completing the square

$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{kk}_{\mathrm{sep}}^{G}}$$\scriptstyle{S}$$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{kk}_{\mathrm{sep}}^{G}}$$\textstyle{\mathrm{KK}_{\mathrm{sep}}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\textstyle{\mathrm{KK}_{\mathrm{sep}}^{G}}$
(2.18)

###### Proof.

In order to obtain $S_{0}$, we apply Proposition 2.2 to the composition

$\mathrm{kk}_{0}^{G}\circ
S\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{0}^{G}\,.$

It straightforward to check that the functor $\mathrm{kk}_{0}^{G}\circ S$ is
reduced, $\mathbb{K}^{G}$-stable and split exact, using that
$\mathrm{kk}_{0}^{G}$ has these properties. The functor
$\mathrm{kk}_{0}^{G}\circ S$ therefore factorizes over an additive endofunctor
$S_{0}$ as required. It then follows from Bott periodicity that $S_{0}$ is an
equivalence of categories.

Since by definition $\mathrm{kk}_{0}^{G}$ detects
${\mathrm{kk}_{0}^{G}}$-equivalences, we conclude from Assertion 1 that $S$ in
(2.16) preserves $\mathrm{kk}_{0}^{G}$-equivalences. Applying the universal
property of $\mathrm{kk}_{\mathrm{sep}}^{G}$ being a Dwyer-Kan localization
(see Definition 1.2) we obtain the essentially unique factoriaztion of
$\mathrm{kk}_{\mathrm{sep}}^{G}\circ S$ through
$\mathrm{kk}_{\mathrm{sep}}^{G}$ as asserted in Assertion 2. ∎

As shown in [Kas88], for every semisplit exact sequence

$0\to I\to A\to B\to 0$ (2.19)

we have a boundary operator $\partial$ in $K\\!K^{G}(S(B),I)$ such that the
sequence

$K\\!K^{G}(D,S(A))\to
K\\!K^{G}(D,S(B))\stackrel{{\scriptstyle\partial}}{{\to}}K\\!K^{G}(D,I)\to
K\\!K^{G}(D,A)\to K\\!K^{G}(D,B)$ (2.20)

is exact for every $D$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. The
boundary operator is natural in the sequence in the following sense. If

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is a morphism to a second sequence of this kind, then the square

$\textstyle{K\\!K^{G}(D,S(B))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{K\\!K^{G}(D,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\\!K^{G}(D,S(B^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial^{\prime}}$$\textstyle{K\\!K^{G}(D,I^{\prime})}$

commutes.

The main problem in the proof of Theorem 1.3 is to control finite limits in
$\mathrm{KK}_{\mathrm{sep}}^{G}$. To this end we calculate the Dwyer-Kan
localization in Definition 1.2 using a category of fibrant objects structure
on ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. We now
recall the definition of a category of fibrant objects in the form used in
[Uuy13, Def. 1.1]. Let ${\mathcal{C}}$ be a category with a terminal object
and $F$ and $W$ be a collection of morphisms in ${\mathcal{C}}$.

######  Definition 2.7.

The triple $({\mathcal{C}},F,W)$ is a category of fibrant objects if the
following conditions hold.

  1. 1.

F0: $F$ is closed under compositions.

  2. 2.

F1: $F$ contains all isomorphisms.

  3. 3.

F2: Pull-backs of morphisms in $F$ exist and belong again to $F$.

  4. 4.

F3: For every $C$ in ${\mathcal{C}}$ the morphism $C\to*$ belongs to $F$.

  5. 5.

W1: $W$ contains all isomorphisms.

  6. 6.

W2: $W$ has the $2$-out-of-$3$-property.

  7. 7.

FW1: $W\cap F$ is stable under forming pull-backs.

  8. 8.

FW2: For every $C$ in ${\mathcal{C}}$ the morphism $C\to C\times C$ has a
factorization $C\stackrel{{\scriptstyle w}}{{\to}}C^{I}\stackrel{{\scriptstyle
f}}{{\to}}C\times C$, where $w$ is in $W$ and $f$ is in $F$.

Note that the product in the formulation of Condition FW2 exists by a
combination of Conditions F3 and F2.

######  Remark 2.8.

The object $C^{I}$ appearing in FW2 is called a path object. From the
existence of a path object $C^{I}$ one can construct a factorisation of an
arbitrary morphism $\phi\colon A\to C$ in ${\mathcal{C}}$ as a weak
equivalence followed by a fibration as follows. We choose a factorization
$C\stackrel{{\scriptstyle w}}{{\to}}C^{I}\stackrel{{\scriptstyle
f}}{{\to}}C\times C$ as in $FW2$. Then we consider the commutative diagram

${A}$${A\times_{C}C^{I}}$${C^{I}}$${A\times C}$${C\times
C}$${C}$$\scriptstyle{u}$$\scriptstyle{\phi}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{{\mathtt{pr}}}$$\scriptstyle{\phi\times{\mathtt{id}}}$
(2.21)

where the square is a pull-back. It exists by $F2$ which also implies that
$f^{\prime}$ is a fibration. Using $F3$ and $F2$ one checks that the
projection ${\mathtt{pr}}$ is also a fibration. We conclude by $F0$ that
${\mathtt{pr}}\circ f^{\prime}$ is a fibration. Furthermore, the map $u$ is
given by $A\cong A\times_{C}C\stackrel{{\scriptstyle A\times
w}}{{\to}}A\times_{C}C^{I}$ and therefore a weak equivalence by $FW1$ and $F3$
(applied to $A\to*$). Consequently

$A\stackrel{{\scriptstyle
u}}{{\longrightarrow}}A\times_{C}C^{I}\stackrel{{\scriptstyle{\mathtt{pr}}\circ
f^{\prime}}}{{\longrightarrow}}C$ (2.22)

is the desired factorization of $\phi$ as a composition of a weak equivalence
and a fibration. ∎

######  Remark 2.9.

The category
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ admits a
canonical path object $C\mapsto C([0,1],C)$. The factorization

$C\longrightarrow C([0,1],C)\longrightarrow C\times C$

required in $FW2$ is given by the inclusion of constant maps and the
evaluation at $0$ and $1$. The first map is a homotopy equivalence.
Furthermore, the map $C^{I}\to C\times C$ is surjective and admits an
equivariant contractive completely positive split. For example, one can take
the split

$(c_{0},c_{1})\mapsto(t\mapsto(1-t)c_{0}+tc_{1})\,.$ (2.23)

Applying the general factorization (2.22) to a morphism $A\to C$, one obtains
the usual cone sequence

$0\longrightarrow C(\phi)\longrightarrow\mathrm{Cyl}(\phi)\longrightarrow C\to
0\,.$

Explicitly,

$\mathrm{Cyl}(\phi)\coloneqq\\{(a,c)\in A\oplus
C([0,1],C)\>|\>c(0)=\phi(a)\\}\,,$ (2.24)

the homomorphism $\mathrm{Cyl}(\phi)\to C$ is given by $(a,c)\mapsto c(1)$,
and

$C(\phi):=\\{(a,c)\in\mathrm{Cyl}(\phi)\>|\>c(1)=0\\}$

with the induced $G$-actions. ∎

A morphism $\phi\colon A\to C$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ is called
a semisplit surjection if it is a surjection and admits an equivariant cpc
split. We let $\SS$ denote the set of semisplit surjections. Furthermore
recall that $W_{\mathrm{kk}_{0}^{G}}$ is the set of
${\mathrm{kk}_{0}^{G}}$-equivalences.

######  Proposition 2.10.

The triple
$({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}),\SS,W_{\mathrm{kk}_{0}^{G}})$
is a category of fibrant objects.

###### Proof.

The axioms $F0$, $F1$, $F3$, $W1$, and $W2$ are obvious.

We show $F2$. We consider a pullback

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D}$
(2.25)

in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ such
that $\phi$ is in $\SS$ with equivariant cpc split $s:D\to C$. The square of
the underlying Banach spaces is a pullback square of Banach spaces. Thus the
equivariant contractive split $s$ of $\phi$ induces an equivariant contractive
split $t:B\to A$. We now use the fact that an element in $A$ is positive if
its images in $C$ and $B$ are positive. The same is true for the extension of
the square to finite matrices. Using that $s$ is completely positive we can
now conclude that $t$ is completely positive as well. Hence $\psi$ is in
$\SS$, too.

To show $FW2$, we make use of the cylinder discussed in Remark 2.9. The map
$C\to C^{I}$ is a homotopy equivalence, hence in particular a
${\mathrm{kk}_{0}^{G}}$-equivalence. Furthermore, as explained in Remark 2.9
the projection $C^{I}\to C\times C$ has an equivariant contractive completely
positive split and therefore belongs to $\SS$.

In order to prepare the proof of $FW1$ we consider an exact sequence admitting
an equivariant cpc split.

$0\longrightarrow I\longrightarrow
A\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}B\longrightarrow 0\,.$ (2.26)

As explained above, then we have the boundary operator $\partial$ in
$K\\!K^{G}(S(B),I)$ such that the sequence

$K\\!K^{G}(D,S(A))\to
K\\!K^{G}(D,S(B))\stackrel{{\scriptstyle\partial}}{{\to}}K\\!K^{G}(D,I)\to
K\\!K^{G}(D,A)\stackrel{{\scriptstyle\psi_{*}}}{{\to}}K\\!K^{G}(D,B)$ (2.27)

is exact for every $D$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. By the
Yoneda Lemma, the morphism $\psi$ is a ${\mathrm{kk}_{0}^{G}}$-equivalence if
and only if $\psi_{*}$ is an isomorphism for every $D$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. We
conclude from the exactness of (2.27) that this is the case if and only if
$K\\!K^{G}(D,I)=0$ for every $D$.

We now consider a pull-back square (2.25) in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ such that
$\phi$ belongs to $\SS\cap W_{\mathrm{kk}_{0}^{G}}$. By $F2$ we know that
$\psi$ is in $\SS$ and it remains to show that $\psi$ belongs to
$W_{\mathrm{kk}_{0}^{G}}$. This follows from the fact that
${\mathtt{ker}}(\phi)\cong{\mathtt{ker}}(\psi)$. Since $\phi$ is in
$W_{\mathrm{kk}_{0}^{G}}$ we have
$K\\!K^{G}(D,{\mathtt{ker}}(\psi))=K\\!K^{G}(D,{\mathtt{ker}}(\phi))=0$ for
all $D$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$.
Consequently, $\psi$ belongs to $W_{\mathrm{kk}_{0}^{G}}$. ∎

For the formulation of universal properties it turned out to be useful to
reformulate various exactness properties of functors defined on
$G$-$C^{*}$-algebras using squares so that they make sense not only for stable
targets, but also for pointed ones. But we will show in Lemma 2.14 below that
these new definitions are equivalent to the old same-named conditions in the
situations where they have been used above. We do these considerations at this
point since we will use squares already in the proof of Proposition 2.15
below.

We consider a cartesian square

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
C}$$\textstyle{D}$ (2.28)

in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$. We will consider the
following two conditions on the square in (2.28).

######  Definition 2.11.

  1. 1.

The square (2.28) is called semisplit if there exist equivariant cpc’s
$s\colon C\to A$ and $t\colon D\to B$ such that $p\circ s={\mathtt{id}}_{C}$
and $q\circ t={\mathtt{id}}_{D}$.

  2. 2.

The square (2.28) is called split if there exist equivariant morphisms
$s\colon C\to A$ and $t\colon D\to B$ such that $p\circ s={\mathtt{id}}_{C}$
and $q\circ t={\mathtt{id}}_{D}$.

######  Remark 2.12.

The conditions on $q$ appearing in Definition 2.11 imply the corresponding
conditions on $p$, see the verification of the axiom F2 in the proof of
Proposition 2.10 for a proof of this fact. ∎

Note that an exact sequence

$0\to I\to A\xrightarrow{q}B\to 0$ (2.29)

in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ is semisplit or split,
if and only if the cartesian square

$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{B}}$
(2.30)

has the corresponding property as defined in Definition 2.11.

Let
$F\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to{\mathcal{D}}$
(or
$F\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to{\mathcal{D}}$)
be a functor to a pointed $\infty$-category ${\mathcal{D}}$. Corresponding to
the list of conditions in Definition 2.11 we introduce the following two
exactness conditions on the functor $F$.

######  Definition 2.13.

  1. 1.

$F$ is semiexact if it is reduced and sends any semisplit cartesian square to
a cartesian square in ${\mathcal{D}}$.

  2. 2.

F is split exact if it is reduced and sends any split cartesian square to a
cartesian square in ${\mathcal{D}}$.

Let $F$ be a functor as in Definition 2.13.

######  Lemma 2.14.

  1. 1.

If ${\mathcal{D}}$ is stable, then $F$ is semiexact in the sense of Definition
2.13.1 if and only if $F$ sends all semisplit exact sequences to fibre
sequences.

  2. 2.

If ${\mathcal{D}}$ is additive, then $F$ is split exact in the sense of
Definition 2.13.2 if and only if $F$ sends all split exact sequences to fibre
sequences.

###### Proof.

In each case, the ”only if“ implication is obvious. To see the ”if“
implication, we extend the square (2.28) to the left by adding a pullback of
$p$ along $0\to C$:

${I}$${A}$${B}$${0}$${C}$${D}$$\scriptstyle{p}$$\scriptstyle{q}$

The left and the outer squares are exact sequences which are send by $F$ to
fibre sequences. Thus applying $F$, in both cases we obtain the diagram

${F(I)}$${F(A)}$${F(B)}$${0}$${F(C)}$${F(D)}$$\scriptstyle{F(p)}$$\scriptstyle{F(q)}$
(2.31)

in which the left and the outer squares are cartesian and the lower left
corner is a zero object. In particular we can conclude that $F$ is reduced.
The right square is cartesian since $F(I)$ identifies with the fibre of both
$F(p)$ and $F(q)$. The stable case follows at once, and in the additive case
we use the existence of the splits of $F(p)$ and $F(q)$ to conclude.

∎

The following proposition shows all assertions of Theorem 1.3 except the one
about triangulated structures which will be obtained later in Proposition
2.18.

######  Proposition 2.15.

  1. 1.

The functor $\mathrm{kk}_{\mathrm{sep}}^{G}$ is semiexact.

  2. 2.

The functor
$\mathrm{ho}_{0}\colon\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})\to\mathrm{KK}_{0}^{G}$
is an equivalence.

  3. 3.

The $\infty$-category $\mathrm{KK}_{\mathrm{sep}}^{G}$ is stable.

###### Proof.

Assertion 1 is an immediate consequence of the fact that
$\mathrm{KK}_{\mathrm{sep}}^{G}$ is the $\infty$-category associated to a
category of fibrant objects whose fibrations are the semisplit exact
surjections. Indeed, consider a semisplit cartesian square (2.28). Since $p$
and $q$ are fibrations this square represents a cartesian square in the
$\infty$-category $\mathrm{KK}_{\mathrm{sep}}^{G}$, see [Cis19, Prop. 7.5.6].

We now prove Assertion 2. We will use that forming localizations is compatible
with going over to homotopy categories. Let ${\mathcal{C}}$ be an
$\infty$-category with a set of morphisms $W$. Then have a commutative square

$\textstyle{{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell}$$\textstyle{{\mathcal{C}}[W^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{ho}({\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ho}(\ell)}$$\textstyle{\mathrm{ho}({\mathcal{C}}[W^{-1}])}$
(2.32)

where
$\mathrm{ho}(\ell)\colon\mathrm{ho}({\mathcal{C}})\to\mathrm{ho}({\mathcal{C}}[W^{-1}])$
presents its target as the localization of $\mathrm{ho}({\mathcal{C}})$ at the
set $\mathrm{ho}(W)$ in the sense of ordinary categories. We apply this to
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ in place
of ${\mathcal{C}}$. Since
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ is an
ordinary category the left vertical functor in (2.32) is an equivalence. Hence
the functor

${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\xrightarrow{\mathrm{kk}_{\mathrm{sep}}^{G}}\mathrm{KK}_{\mathrm{sep}}^{G}\to\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})$

presents its target as the localization of
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ at the set
of ${\mathrm{kk}_{0}^{G}}$-equivalences in the sense of ordinary categories.
By Proposition 2.1 the functor

$\mathrm{kk}_{0}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{0}^{G}$

has the same universal property. This implies that
$\mathrm{ho}_{0}\colon\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})\to\mathrm{KK}_{0}^{G}$
is an equivalence.

The proof Assertion 3 that $\mathrm{KK}_{\mathrm{sep}}^{G}$ is stable can now
be copied from [LN18, Prop. 3.3]. First we note that
$\mathrm{KK}_{\mathrm{sep}}^{G}$, being the $\infty$-category associated to a
category of fibrant objects with a zero object, is pointed and has finite
limits. It remains to show that the loop functor $\Omega$ in
$\mathrm{KK}_{\mathrm{sep}}^{G}$ is an equivalence. It is well-known that it
suffices to show that $\Omega$ induces an equivalence on the homotopy
category, see for instance [LN18, Lem. 3.4].

Using the explicit description of the fibrant replacement of $0\to A$ given in
Remark 2.9 we see that

$\textstyle{\mathrm{kk}_{\mathrm{sep}}^{G}(S(A))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{kk}_{\mathrm{sep}}^{G}(A)}$
(2.33)

is a pull-back in $\mathrm{KK}_{\mathrm{sep}}^{G}$. Therefore $\Omega$ is
equivalent to the suspension functor given by the dotted arrow $S$ in (2.18).
In view of Assertion 2 shown above the induced action of this functor $S$ on
the homotopy category can be identified with the action of the functor $S_{0}$
in Lemma 2.6.2 on $\mathrm{KK}_{0}^{G}$. As asserted in the same statement
$S_{0}$ is an equivalence. ∎

Our next task is the verification of the properties of
$\mathrm{KK}_{\mathrm{sep}}^{G}$ and $\mathrm{kk}_{\mathrm{sep}}^{G}$ stated
in Theorem 1.4. Assertion 1.4.2 is already shown as a part of Proposition
2.15.

The next Lemma settles Assertions 1.4.1, 1.4.4 and 1.4.5 together.

######  Lemma 2.16.

The functor $\mathrm{kk}_{\mathrm{sep}}^{G}$ is reduced,
$\mathbb{K}^{G}$-stable, and homotopy invariant.

###### Proof.

We know that $\mathrm{kk}_{0}^{G}$ is reduced, $\mathbb{K}^{G}$-stable and
homotopy invariant. Therefore Proposition 2.15.2 together with the fact that
the canonical functor from an $\infty$-category to its homotopy category
detects equivalences and zero elements implies the assertion. ∎

The next lemma collects, for later reference, some simple consequences of the
details proof of Proposition 2.15.

######  Lemma 2.17.

  1. 1.

Every morphism in $\mathrm{KK}_{\mathrm{sep}}^{G}$ is equivalent to a morphism
$\mathrm{kk}_{\mathrm{sep}}^{G}(f)$ for a fibration $f$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$.

  2. 2.

Every product in $\mathrm{KK}_{\mathrm{sep}}^{G}$ is equivalent to the image
under $\mathrm{kk}_{\mathrm{sep}}^{G}$ of a product in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$.

  3. 3.

Every fibre sequence in $\mathrm{KK}_{\mathrm{sep}}^{G}$ is equivalent to the
image under $\mathrm{kk}_{\mathrm{sep}}^{G}$ of a cone sequence.

###### Proof.

These assertions follow from general facts about the associated
$\infty$-category of a category of fibrant objects, where for Assertion 3 we
use in addition that $\mathrm{KK}_{\mathrm{sep}}^{G}$ is stable. For
completeness we give alternative arguments which are specific to the present
situation as we will use some of the details later.

Assertion 1 follows from the equivalence
$\mathrm{ho}_{0}\colon\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})\stackrel{{\scriptstyle\simeq}}{{\to}}\mathrm{KK}_{0}^{G}$
and the surjectivity of the arrow marked by $!!$ in (LABEL:ewgregergggwggerg).

For Assertion 2 we note that any object in $\mathrm{KK}_{\mathrm{sep}}^{G}$ is
equivalent to an object of the form $\mathrm{kk}_{\mathrm{sep}}^{G}(A)$ for
some $A$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. Let
$\mathrm{kk}_{\mathrm{sep}}^{G}(B)$ be a second object. Then the square

$\textstyle{A\oplus
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is a split cartesian square in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$
(Definition 2.11.2). Since $\mathrm{kk}_{\mathrm{sep}}^{G}$ is semiexact by
Prop. 2.151 (and hence split exact) we conclude that the square

$\textstyle{\mathrm{kk}_{\mathrm{sep}}^{G}(A\oplus
B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{kk}_{\mathrm{sep}}^{G}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{kk}_{\mathrm{sep}}^{G}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(2.34)

is cartesian. This implies that

$\mathrm{kk}_{\mathrm{sep}}^{G}(A)\times\mathrm{kk}_{\mathrm{sep}}^{G}(B)\simeq\mathrm{kk}_{\mathrm{sep}}^{G}(A\oplus
B)\,.$ (2.35)

The assertion now follows from

$\mathrm{kk}_{\mathrm{sep}}^{G}(A\times
B)\simeq\mathrm{kk}_{\mathrm{sep}}^{G}(A\oplus
B)\stackrel{{\scriptstyle\eqref{ergverwfwerferf}}}{{\simeq}}\mathrm{kk}_{\mathrm{sep}}^{G}(A)\times\mathrm{kk}_{\mathrm{sep}}^{G}(B)\,,$

where we use that the product of $C^{*}$-algebras is given by the sum.

We finally show Assertion 3. Since $\mathrm{KK}_{\mathrm{sep}}^{G}$ is stable,
a morphism in $\mathrm{KK}_{\mathrm{sep}}^{G}$ can be extended to a fibre
sequence, and every fibre sequence is obtained in this way up to equivalence.
By Assertion 1 it suffices to consider fibre sequences obtained by extending
the morphism $\mathrm{kk}_{\mathrm{sep}}^{G}(p)$ for a morphism $p:A\to B$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. We next
argue that one can further assume that $p$ is a fibration. To this end we use
the cylinder defined in (2.24) in order to construct the commutative square

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{e}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Cyl}(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{B}$

in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$, where
$q(a,b)=b(1)$ and $e(a)=(a,{\mathtt{const}}_{p(a)})$. The morphism $e$ is a
${\mathrm{kk}_{0}^{G}}$-equivalence since it is an instance of the morphism
$u$ in (2.21). Therefore the two morphisms $\mathrm{kk}_{\mathrm{sep}}^{G}(p)$
and $\mathrm{kk}_{\mathrm{sep}}^{G}(q)$ are equivalent.

Note that $q$ is surjective and that the exact cone sequence

$0\to C(p)\to\mathrm{Cyl}(p)\stackrel{{\scriptstyle q}}{{\to}}B\to 0$ (2.36)

is semisplit. Since $\mathrm{kk}_{\mathrm{sep}}^{G}$ is semiexact by
Proposition 2.15.1, it sends this sequence to a fibre sequence. This finishes
the proof of Assertion 3. ∎

As a homotopy category of a stable $\infty$-category, the category
$\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})$ acquires a triangulated
structure [Lur, Thm. 1.1.2.14]. On the other hand $\mathrm{KK}_{0}^{G}$ has a
triangulated structure described in [MN06, Sec. 2.1].

######  Proposition 2.18.

The functor
$\mathrm{ho}_{0}\colon\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})\to\mathrm{KK}_{0}^{G}$
is an equivalence of triangulated categories.

###### Proof.

We know from Proposition 2.15.2 that $\mathrm{ho}_{0}$ is an equivalence of
categories. Since sums in $\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})$ and
$\mathrm{KK}_{0}^{G}$ are represented by sums in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ we
conclude further that $\mathrm{ho}_{0}$ preserves sums and is therefore
compatible with the ${\mathbf{Ab}}$-enrichment.

The inverse shift functor for $\mathrm{KK}_{\mathrm{sep}}^{G}$ and therefore
on $\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})$ is implemented by the
suspension functor (2.16) on the level of $C^{*}$-algebras via Lemma 2.6.2.
Similarly, by the description given in [MN06, Sec. 2.1] the inverse shift
functor on $\mathrm{KK}_{0}^{G}$ is also implemented by (2.16) via Lemma
2.6.2. This shows that $\mathrm{ho}_{0}$ commutes with the shift functor.

On the one hand, by [MN06, Sec. 2.1] the exact triangles in
$\mathrm{KK}_{0}^{G}$ are generated by the mapping cone sequences for
morphisms in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. On the
other hand, by Lemma 2.17.3 the fibre sequences in
$\mathrm{KK}_{\mathrm{sep}}^{G}$, and hence the exact triangles in
$\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})$ are also generated by the
mapping cone sequences for morphisms in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. We
conclude that the functor $\mathrm{ho}_{0}$ is compatible with the collections
of distinguished triangles. ∎

This finishes the proof of Theorem 1.3.

$\mathrm{KK}_{\mathrm{sep}}^{G}$ is stable by Proposition 2.15.2 and hence
admits all finite colimits. The following proposition strengthens this from
finite to countable colimits and settles the remaining Assertions 1.4.5 and
1.4.6. For the notion of an admissible diagram ${\rm
Nature}\to{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$
we refer to [MN06, Def. 2.5]. We will not repeat the definition here since the
exact details are not relevant. We will only use its consequence [MN06, Prop.
2.6].

######  Lemma 2.19.

  1. 1.

The category $\mathrm{KK}_{\mathrm{sep}}^{G}$ admits all countable colimits
and is therefore idempotent complete.

  2. 2.

The functor $\mathrm{kk}_{\mathrm{sep}}^{G}$ sends countable sums in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ to
coproducts.

  3. 3.

The functor $\mathrm{kk}_{\mathrm{sep}}^{G}$ preserves colimits of all
diagrams $A\colon{\rm
Nature}\to{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$
which are admissible.

###### Proof.

Since ${\mathrm{KK}_{\mathrm{sep}}^{G}}$ is stable, in order to show that
$\mathrm{KK}_{\mathrm{sep}}^{G}$ admits all countable colimits it suffices to
show that $\mathrm{KK}_{\mathrm{sep}}^{G}$ admits countable coproducts. The
functor $\mathrm{kk}_{\mathrm{sep}}^{G}$ is essentially surjective by
construction. Hence it suffices to show that for every countable family
$(A_{i})_{i\in I}$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ the family
$(\mathrm{kk}_{\mathrm{sep}}^{G}(A_{i}))_{i\in I}$ in
$\mathrm{KK}_{\mathrm{sep}}^{G}$ admits a coproduct. We consider the sum
$A\coloneqq\bigoplus_{i\in I}A_{i}$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ and let
$e_{i}\colon A_{i}\to A$ be the canonical inclusion for every $i$ in $I$. Then
we claim that
$(\mathrm{kk}_{\mathrm{sep}}^{G}(A),(\mathrm{kk}_{\mathrm{sep}}^{G}(e_{i}))_{i\in
I})$ represents the coproduct of the family
$(\mathrm{kk}_{\mathrm{sep}}^{G}(A_{i})_{i\in I})$. In general, coproducts in
a stable $\infty$-category can be detected on the homotopy category. In view
of the equivalence
$\mathrm{kk}_{0}^{G}\simeq\mathrm{ho}\circ\mathrm{kk}_{\mathrm{sep}}^{G}$
given by (1.2) it thus suffices to show that
$(\mathrm{kk}_{0}^{G}(A),(\mathrm{kk}_{0}^{G}(e_{i}))_{i\in I})$ represents
the coproduct in $\mathrm{KK}_{0}^{G}$. But this follows from [Kas88, Thm.
2.9] stating that the family of maps $(e_{i})_{i\in I}$ induces an isomorphism

$\mathrm{KK}_{0}^{G}(A,B)\xrightarrow{\cong}\prod_{i\in
I}\mathrm{KK}_{0}^{G}(A_{i},B)$

for every $B$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. The proof
of Assertion 1 is finished with the general observation that a stable and
countably cocomplete $\infty$-category is idempotent complete.

The claim also implies Assertion 2.

We finally show the Assertion 3. Let $A\colon{\rm
Nature}\to{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$
be an admissible diagram. We must show that the canonical map
$\operatorname*{\mathtt{colim}}_{{\rm
Nature}}\mathrm{kk}_{\mathrm{sep}}^{G}(A)\to\mathrm{kk}_{\mathrm{sep}}^{G}(\operatorname*{\mathtt{colim}}_{{\rm
Nature}}A)$ is an equivalence. To this end it suffices to show that the map
$\mathrm{ho}(\operatorname*{\mathtt{colim}}_{{\rm
Nature}}\mathrm{kk}_{\mathrm{sep}}^{G}(A){)}\to\mathrm{ho}(\mathrm{kk}_{\mathrm{sep}}^{G}(\operatorname*{\mathtt{colim}}_{{\rm
Nature}}A))$ obtained by composing with $\mathrm{ho}$ from (1.2) is an
isomorphism. Using the isomorphism

$\mathrm{ho}(\operatorname*{\mathtt{colim}}_{{\rm
Nature}}\mathrm{kk}_{\mathrm{sep}}^{G}(A))\cong{\mathtt{hocolim}}_{{\rm
Nature}}\mathrm{ho}(\mathrm{kk}_{\mathrm{sep}}^{G}(A))\cong{\mathtt{hocolim}}_{{\rm
Nature}}\mathrm{kk}_{0}^{G}(A)$

and Proposition 2.18 (to translate homotopy colimits formed in the homotopy
category of the stable $\infty$-category $\mathrm{KK}_{\mathrm{sep}}^{G}$ to
homotopy colimits formed in the triangulated category $\mathrm{KK}_{0}^{G}$
used in [MN06]), this is exactly the assertion of [MN06, Prop. 2.6]. ∎

The existence of coproducts in $\mathrm{KK}_{0}^{G}$ and the analogue of
Assertion 2.19.2 for $\mathrm{kk}_{0}^{G}$ has previously been shown in [MN06,
Prop. 2.1].

The proof of Theorem 1.4 is now complete.

In the following we consider the minimal and the maximal tensor products
$\otimes_{\min}$ and $\otimes_{\max}$ of $C^{*}$-algebras. Both equip
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ with a
symmetric monoidal structure. Recall that a symmetric monoidal structure on a
stable $\infty$-category is called bi-exact if the tensor product preserves
cofibre sequences, and hence finite colimits, in each variable. In this case
the $\infty$-category together with its symmetric monoidal structure is called
stably symmetric monoidal.

######  Proposition 2.20.

The tensor product $\otimes_{?}$ for $?$ in $\\{\min,\max\\}$ descends to a
bi-exact symmetric monoidal structure on $\mathrm{KK}_{\mathrm{sep}}^{G}$ and
$\mathrm{kk}_{\mathrm{sep}}^{G}$ refines to a symmetric monoidal functor

$\mathrm{kk}_{\mathrm{sep}}^{G,\otimes_{?}}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})^{\otimes_{?}}\to\mathrm{KK}_{\mathrm{sep}}^{G,\otimes_{?}}\,.$

Moreover, the tensor structure $\otimes_{?}$ on
$\mathrm{KK}_{\mathrm{sep}}^{G}$ preserves countable colimits in each
variable.

###### Proof.

In order to show that the tensor product descends along the localization

$\mathrm{kk}_{\mathrm{sep}}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{\mathrm{sep}}^{G}$

such that $\mathrm{kk}_{\mathrm{sep}}^{G}$ refines to a symmetric monoidal
functor we use [Hin16, Prop. 3.2.2]. By this result, it suffices to show that
for every $A$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ the
functor $A\otimes_{?}-$ preserves ${\mathrm{kk}_{0}^{G}}$-equivalences, where
$?$ is in $\\{\min,\max\\}$. It is easy to see that the functor
$\mathrm{kk}_{0}^{G}\circ(A\otimes_{?}-):{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{0}^{G}$
preserves zero objects and is $\mathbb{K}^{G}$-stable and split exact since
$\mathrm{kk}_{0}^{G}$ has these properties. We now apply Corollary 2.4 in
order to conclude that this composition sends
$\mathrm{kk}_{0}^{G}$-equivalences to isomorphisms. Hence $A\otimes_{?}-$
preserves $\mathrm{kk}_{0}^{G}$-equivalences.

In order to show that the resulting symmetric monoidal structure on
$\mathrm{KK}_{\mathrm{sep}}^{G}$ is exact we must show that
$\mathrm{kk}_{\mathrm{sep}}^{G}(A)\otimes_{?}-$ in
$\mathrm{KK}_{\mathrm{sep}}^{G}$ preserves fibre sequences for every $A$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. We will
use the observation made in the proof of Proposition 2.18 that every fibre
sequence in $\mathrm{KK}_{\mathrm{sep}}^{G}$ is equivalent to a cone sequence.

It suffices to show that $\mathrm{kk}_{\mathrm{sep}}^{G}(A\otimes_{?}-)$ sends
cone sequences to fibre sequences.

For the maximal tensor product, using the exactness of $A\otimes_{\max}-$ (see
e.g. [BO08, Prop. 3.7.1]) and the explicit description of the cone sequences
in Remark 2.9 one checks that $A\otimes_{\max}-$ preserves cone sequences in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. We then
use that the functor $\mathrm{kk}_{\mathrm{sep}}^{G}$ sends cone sequences to
fibre sequences.

For the minimal tensor product we use the fact that cone sequences admit
completely positive contractive splits and that the minimal tensor product is
functorial for completely positive contractive maps. We conclude that
$A\otimes_{\min}-$ sends cone sequences to semisplit exact sequences. We now
use that $\mathrm{kk}_{\mathrm{sep}}^{G}$ is semiexact.

Since we already know that $\otimes_{?}$ on $\mathrm{KK}_{\mathrm{sep}}^{G}$
is bi-exact, in order to show that the corresponding tensor structure on
$\mathrm{KK}_{\mathrm{sep}}^{G}$ preserves countable colimits in each argument
it suffices to show that it preserves countable sums. As observed in the proof
of Lemma 2.19, countable sums in $\mathrm{KK}_{\mathrm{sep}}^{G}$ are
presented by countable sums in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. It
follows from Lemma 7.13 and the fact that the tensor products of
$C^{*}$-algebras preserves finite sums, that $\otimes_{?}$ on
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ preserves
countable sums in each argument. Since $\mathrm{kk}_{\mathrm{sep}}^{G}$
preserves countable sums by Lemma 2.19.2 we can conclude that $\otimes_{?}$ on
$\mathrm{KK}_{\mathrm{sep}}^{G}$ preserves countable sums, too. ∎

######  Remark 2.21.

Alternatively, the argument in the proof of Proposition 1.7 given for the
minimal tensor product also applies to the maximal tensor product since the
latter is also functorial for completely positive contractive maps [Pis, Cor.
4.18]. ∎

######  Remark 2.22.

As a consequence of Proposition 2.20 the homotopy category
$\mathrm{ho}(\mathrm{KK}_{\mathrm{sep}}^{G})$ and therefore
$\mathrm{KK}_{0}^{G}$ has two tensor triangulated structures induced by
$\otimes_{\min}$ and $\otimes_{\max}$, respectively. For $\otimes_{\min}$ this
fact is well known, see e.g. [MN06, Sec. 2.5]. It seems that the symmetric
monoidal structure coming from the maximal tensor product has not been studied
so much, but for the non-equivariant case see e.g. [LN18, Lem. 3.13]. ∎

Note that $\mathrm{kk}_{\mathrm{sep}}^{G}$ is, by Definition 1.2, the initial
functor from
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ to
$\infty$-categories which sends ${\mathrm{kk}_{0}^{G}}$-equivalences to
equivalences. In Theorem 1.5 we stated a different universal property which
better reflects the standard properties of $K\\!K$-theory. We will now first
state an intermediate Theorem 2.23 involving additive $\infty$-categories in
order to formulate a direct $\infty$-categorial analog of [Mey00, Thm. 6.6]
which appeared in the present paper as Proposition 2.2.

Recall from [Lur, 6.1.6.13] that an $\infty$-category is called semi-additive
if it is pointed, admits finite products and coproducts, and if the canonical
morphism from a coproduct to the product of any two objects is an equivalence.
The homotopy category of a semi-additive category is canonically enriched in
abelian monoids. If all these morphism monoids are abelian groups then the
$\infty$-category is said to be additive. Stable $\infty$-categories are
additive. In particular, since $\mathrm{KK}_{\mathrm{sep}}^{G}$ is stable by
Proposition 2.15, it is additive.

######  Theorem 2.23.

The functor
$\mathrm{kk}_{\mathrm{sep}}^{G}\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to\mathrm{KK}_{\mathrm{sep}}^{G}$
is initial among functors from
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ to objects
of ${\mathbf{Cat}}^{\mathrm{add}}_{\infty}$ which are reduced,
$\mathbb{K}^{G}$-stable, and split exact.

###### Proof.

For an additive $\infty$-category ${\mathcal{D}}$ we consider the full
subcategory

${\mathbf{Fun}}^{{rse}}({\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}}_{\mathrm{sep}}),{\mathcal{D}})$
(2.37)

of
${\mathbf{Fun}}({\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}}_{\mathrm{sep}}),{\mathcal{D}})$
on functors which are reduced, $\mathbb{K}^{G}$-stable, and split exact. By
Corollary 2.4 the functor category (2.37) is a full subcategory of the
category
${\mathbf{Fun}}^{{\mathrm{kk}_{0}^{G}}}({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}),{\mathcal{D}})$
of functors sending ${\mathrm{kk}_{0}^{G}}$-equivalences to equivalences.

We now build the following commutative diagram

${{\mathbf{Fun}}^{\coprod}(\mathrm{KK}_{\mathrm{sep}}^{G},{\mathcal{D}})}$${{\mathbf{Fun}}^{{rse}}({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}),{\mathcal{D}})}$${{\mathbf{Fun}}(\mathrm{KK}_{\mathrm{sep}}^{G},{\mathcal{D}})}$${{\mathbf{Fun}}^{{\mathrm{kk}_{0}^{G}}}({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}),{\mathcal{D}})}$$\scriptstyle{\simeq}$
(2.38)

The vertical morphisms are inclusions of full subcategories and the horizontal
functors are induced by precomposition with $\mathrm{kk}_{\mathrm{sep}}^{G}$.
The superscript $\coprod$ stands for coproduct preserving functors which are
the morphisms in ${\mathbf{Cat}}^{\mathrm{add}}_{\infty}$. By the universal
property of $\mathrm{kk}_{\mathrm{sep}}^{G}$ as a Dwyer-Kan localization the
lower horizontal functor is an equivalence. We now justify that the dashed
arrow exists, making the diagram commute. For this, we note that the
localisation functor $\mathrm{kk}_{\mathrm{sep}}^{G}$ is reduced,
$\mathbb{K}^{G}$-stable and semiexact by Theorem 1.4. In particular it is
split-exact. Thus if $H\colon\mathrm{KK}_{\mathrm{sep}}^{G}\to{\mathcal{D}}$
preserves coproducts, then the composition
$H\circ\mathrm{kk}_{\mathrm{sep}}^{G}$ is reduced, $\mathbb{K}^{G}$-stable and
split exact.

In order to prove Theorem 2.23 we must show that the dashed arrow in (2.38) is
an equivalence. As all other functors in the diagram 2.38 are fully faithful,
so is the dashed one. It remains to show that it is essentially surjective.

To this end we consider $F$ in
${\mathbf{Fun}}^{{rse}}({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}),{\mathcal{D}})$.
In view of the lower horizontal equivalence in (2.38) there exists a functor
$\bar{F}$ in ${\mathbf{Fun}}(\mathrm{KK}_{\mathrm{sep}}^{G},{\mathcal{D}})$
and an equivalence $\bar{F}\circ\mathrm{kk}_{\mathrm{sep}}^{G}\simeq F$. We
must check that
$\bar{F}\in{\mathbf{Fun}}^{\coprod}(\mathrm{KK}_{\mathrm{sep}}^{G},{\mathcal{D}})$,
i.e., that $\bar{F}$ preserves coproducts.

First of all $F$ preserves the empty coproduct since
$\mathrm{kk}_{\mathrm{sep}}^{G}$ and $F$ are reduced.

We now show that $\bar{F}$ preserves binary coproducts. Since
$\mathrm{kk}_{\mathrm{sep}}^{G}$ and ${\mathcal{D}}$ are additive, in both
$\infty$-categories coproducts and products coincide. Therefore it suffices to
show that $\bar{F}$ preserves binary products. Since $F$ is split-exact and
reduced, it sends a split exact square of the form

$\textstyle{A\oplus
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

to a cartesian square with zero in the lower right corner. We conclude that
$F$ preserves products. Now by Proposition 2.17.2 any product in
$\mathrm{KK}_{\mathrm{sep}}^{G}$ is equivalent to the image of a product in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. In view
of the equivalence $\bar{F}\circ\mathrm{kk}_{\mathrm{sep}}^{G}\simeq F$ we can
conclude that $\bar{F}$ preserves binary products.

Hence $\bar{F}$ preserves binary coproducts. Since $F$ preserves empty and
binary coproducts it preserves all finite coproducts. ∎

Theorem 2.23 says that if ${\mathcal{D}}$ is an object of
${\mathbf{Cat}}^{\mathrm{add}}_{\infty}$ and
$F\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to{\mathcal{D}}$
is a reduced, $\mathbb{K}^{G}$-stable, and split exact functor, then there
exists an essentially unique, finite coproduct-preserving factorization
$\bar{F}$ as indicated in the diagram

---
$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\mathrm{kk}_{\mathrm{sep}}^{G}}$$\textstyle{{\mathcal{D}}\,.}$$\textstyle{\mathrm{KK}_{\mathrm{sep}}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{F}}$

Since ${\mathcal{D}}$ and $\mathrm{KK}_{\mathrm{sep}}^{G}$ are additive, in
both $\infty$-categories products and coproducts coincide. Hence $\bar{F}$
also preserves all finite products. If ${\mathcal{D}}$ admits finite limits,
e.g. if ${\mathcal{D}}$ is stable, one might wonder whether the functor
$\bar{F}$ in addition preserves finite limits. In general it does not, see
Remark 2.25 for an example, but we have the following characterization.

Let ${\mathcal{D}}$ be an additive $\infty$-category, and let
$F\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\to{\mathcal{D}}$
be a functor which is reduced, $\mathbb{K}^{G}$-stable and split exact. Let
$\bar{F}\colon\mathrm{KK}_{\mathrm{sep}}^{G}\to{\mathcal{D}}$ be the
factorization as explained above. In the following statement we use the
Definition 2.13.1 of semiexactness for functors with additive targets.

######  Theorem 2.24.

If in addition ${\mathcal{D}}$ admits finite limits and $F$ is semiexact, then
$\bar{F}$ preserves finite limits.

###### Proof.

It suffices to prove that $\bar{F}$ is reduced and sends fibre sequences to
fibre sequences. We already know that $\bar{F}$ is reduced by Theorem 2.23.

By Proposition 2.17.3, any fibre sequence in $\mathrm{KK}_{\mathrm{sep}}^{G}$
is the image under $\mathrm{kk}_{\mathrm{sep}}^{G}$ of a semisplit exact
sequence in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. By
assumption, $F$ sends such a semisplit exact sequence to a fibre sequence. In
view of the relation $\bar{F}\circ\mathrm{kk}_{\mathrm{sep}}^{G}\simeq F$ we
conclude that $\bar{F}$ sends the image of this semisplit exact sequence under
$\mathrm{kk}_{\mathrm{sep}}^{G}$ to a fibre sequence. Consequently $\bar{F}$
preserves fibre sequences. ∎

###### Proof of Theorem 1.5.

The theorem immediately follows by specializing the Theorems 2.23 and 2.24 to
stable target categories. ∎

######  Remark 2.25.

A natural example of a functor
$C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}\to\mathbf{Sp}$ which is
reduced, $\mathbb{K}$-stable and split exact is the composite

$C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}\longrightarrow\mathrm{Rings}_{\mathrm{inv}}^{\mathrm{nu}}\stackrel{{\scriptstyle
L}}{{\longrightarrow}}\mathbf{Sp}$

where the first functor is the forgetful functor taking the underlying ring
with involution of a $C^{*}$-algebra and the second takes the (projective,
symmetric) L-theory spectrum of a ring with involution. This functor descends
to a functor ${\mathrm{KK}_{\mathrm{sep}}}\to\mathbf{Sp}$ which preserves
finite products, but is not exact. Indeed, the failure of exactness in this
case can be described explicitly, we refer to [LN18, Thm. 4.2] for a general
treatment. ∎

##  3 The s-finitary extension

Let
$F\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to{\mathcal{D}}$ be
a functor to a target category ${\mathcal{D}}$ admitting all small filtered
colimits. Then we have a canonical natural transformation $\hat{F}\to F$,
where $\hat{F}$ is the left Kan extension of $F_{s}\coloneqq
F_{|{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})}$ as
indicated in the following diagram:

$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{s}}$$\scriptstyle{\mathrm{incl}}$$\textstyle{\mathcal{D}}$$\textstyle{{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Downarrow\
}$$\scriptstyle{\hat{F}}$$\scriptstyle{\>\Downarrow}$$\scriptstyle{F\quad}$

######  Lemma 3.1.

The functor $F$ is s-finitary if and only if the natural transformation
$\hat{F}\to F$ is an equivalence.

###### Proof.

The pointwise formula for the left Kan-extension shows that

$\operatorname*{\mathtt{colim}}_{(A^{\prime}\to
A)\in(C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})_{/A}}F(A^{\prime})\simeq\hat{F}(A)\,.$

If we compare this formula with the morphism (1.14) appearing in the condition
for being s-finitary it becomes clear that we must show that the poset of
separable subalgebras of $A$ is cofinal in
$(C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})_{/A}$. This follows from the
following observation.

If $f\colon A^{\prime}\to A$ is any morphism in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$, then we have a
factorization

$A^{\prime}\to f(A^{\prime})\to A\,,$

where the image $f(A^{\prime})$ is a $G$-$C^{*}$-subalgebra of $A$. If
$A^{\prime}$ is separable, then $f(A^{\prime})$ is separable, too. ∎

The next lemma is the essential step for the derivation of Theorem 1.18 from
Theorem 1.4. Let $F$ and $F_{s}$ be as above and assume in addition that
${\mathcal{D}}$ is stable.

######  Lemma 3.2.

Assume that $F$ is s-finitary.

  1. 1.

If $F_{s}$ is reduced, then so is $F$.

  2. 2.

If $F_{s}$ is homotopy invariant, then so is $F$.

  3. 3.

If $F_{s}$ is $\mathbb{K}^{G}$-stable, then so is $F$.

  4. 4.

If $F_{s}$ is semiexact, then so is $F$.

  5. 5.

If $F_{s}$ sends exact sequences to fibre sequences, then so does $F$.

###### Proof.

Assertion 1 is obvious since the zero algebra is separable.

We show Assertion 2. For every $t$ in $[0,1]$ let
${\mathtt{ev}}_{t}:C([0,1])\to{\mathbb{C}}$ be the evaluation at $t$. If $B$
is an invariant separable subalgebra of $C([0,1])\otimes A$, then the values
$({\mathtt{ev}}_{t}\otimes{\mathtt{id}}_{A})(b)$ for all $b$ in $B$ and $t$ in
$[0,1]$ generate an invariant separable subalgebra $A^{\prime}$ of $A$ such
that $B\subseteq C([0,1])\otimes A^{\prime}$. Hence the invariant subalgebras
of $C([0,1])\otimes A$ the form $C([0,1])\otimes A^{\prime}$ with $A^{\prime}$
an invariant separable subalgebra of $A$ is cofinal in all invariant separable
subalgebras of $C([0,1])\otimes A$. Since we assume that $F$ is s-finitary we
have the chain of equivalences

$F(A)\simeq\operatorname*{\mathtt{colim}}_{A^{\prime}{\subseteq_{\mathrm{sep}}A}}F_{s}(A^{\prime})\simeq\operatorname*{\mathtt{colim}}_{A^{\prime}{\subseteq_{\mathrm{sep}}A}}F_{s}(C([0,1])\otimes
A^{\prime})\simeq F(C([0,1])\otimes A)\,,$

where the colimit runs over the poset of all invariant separable subalgebras
$A^{\prime}$ of $A$ and the middle equivalence is a consequence of the
assumption on $F_{s}$. Since this morphism is induced by the canonical map
(1.5) this shows that $F$ is homotopy invariant

We show Assertion 3. Let $A$ be in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$, let $H\to H^{\prime}$ be
an equivariant isometric inclusion of separable $G$-Hilbert spaces such that
$H\not=0$, and $K(H)\to K(H^{\prime})$ be the corresponding inclusion of the
algebras of compact operators. We note that these algebras are separable. It
follows that the family of $G$-invariant subalgebras $A^{\prime}\otimes K(H)$
for all $G$-invariant separable subalgebras $A^{\prime}$ of $A$ is cofinal in
all separable $G$-invariant subalgebras of $A\otimes K(H)$, and similar for
$H^{\prime}$. Using that $F$ is s-finitary we conclude that the morphism
obtained by applying $F$ to (1.4) has a factorization over the chain of
equivalences

$\displaystyle F(A\otimes K(H))$ $\displaystyle\simeq$
$\displaystyle\operatorname*{\mathtt{colim}}_{A^{\prime}}F_{s}(A^{\prime}\otimes
K(H))$ $\displaystyle\simeq$
$\displaystyle\operatorname*{\mathtt{colim}}_{A^{\prime}}F_{s}(A^{\prime}\otimes
K(H^{\prime}))$ $\displaystyle\simeq$ $\displaystyle F(A\otimes K(H))\,,$

where the colimit runs over the poset of all invariant separable subalgebras
$A^{\prime}$ of $A$ and the middle equivalence follows from the assumption on
$F_{s}$.

We show Assertion 4. We now consider a semisplit exact sequence

$0\to A^{\prime}\stackrel{{\scriptstyle
i}}{{\to}}A\stackrel{{\scriptstyle\pi}}{{\to}}A^{\prime\prime}\to 0$ (3.1)

${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$. Let
$s:A^{\prime\prime}\to A$ denote the equivariant cpc split of $\pi$. We
consider the family ${\mathcal{S}}$ of exact sequences

${\mathcal{C}}\quad:\quad 0\to C^{\prime}\to C\to C^{\prime\prime}\to 0$

of separable $G$-invariant subalgebras of (3.1) such that
$s(C^{\prime\prime})\subseteq C$. Such a sequence is again equivariantly
semisplit since we can use the restriction of $s$. Since $F$ is s-finitary it
is enough to show that the families of constituents $C^{\prime}$, $C$, and
$C^{\prime\prime}$ for ${\mathcal{C}}$ running in ${\mathcal{S}}$ are cofinal
families of $G$-invariant separable $C^{*}$-algebras in $A^{\prime}$, $A$ and
$A^{\prime\prime}$, respectively. Indeed, then using the assumption on $F_{s}$
the fibre sequence

$F(A^{\prime})\to F(A)\to F(A^{\prime\prime})\to\Sigma F(A^{\prime})$

is the colimit over ${\mathcal{S}}$ of the family of fibre sequences

$F_{s}(C^{\prime})\to F_{s}(C)\to F_{s}(C^{\prime\prime})\to\Sigma
F_{s}(C^{\prime})\,.$

Let $B^{\prime}$, $B$ and $B^{\prime\prime}$ be separable $C^{*}$-subalgebras
of $A^{\prime}$, $A$, and $A^{\prime\prime}$, respectively.

Then the invariant $C^{*}$-subalgebra
$C^{\prime\prime}:=A^{\prime\prime}(\pi(B),B^{\prime\prime})$ of
$A^{\prime\prime}$ generated by the subsets $\pi(B)$ and $B^{\prime\prime}$ is
separable and contains $B^{\prime\prime}$. Then
$C:=A(B,s(C^{\prime\prime}),i(B^{\prime}))$ is a separable subalgebra of $A$
containing $B$ such that the restriction of $\pi$ induces a surjection $C\to
C^{\prime\prime}$ and $s(C^{\prime\prime})\subseteq C$. We finally define
$C^{\prime}:={\mathtt{ker}}(C\to C^{\prime\prime})$ considered as a subalgebra
of $A^{\prime}$. Then $B^{\prime}\subseteq C$. Since $B^{\prime}$ is a closed
subspace of the separable $C^{*}$-algebra it is again separable.

By construction we have an equivariantly semisplit exact sequence

$0\to C^{\prime}\to C\to C^{\prime\prime}\to 0$

in ${\mathcal{S}}$ and

$B^{\prime}\subseteq C^{\prime}\,,\quad B\subseteq C\,,\quad
B^{\prime\prime}\subseteq C^{\prime\prime}\,.$

The proof of Assertion 5 is again a small modification of of the proof of
Assertion 4. Since we do not have a split at all, we define
$C:=A(B,G\hat{C}^{\prime\prime},i(B^{\prime}))$, where
$\hat{C}^{\prime\prime}$ is a set of choices of preimages of a countable dense
subset of $C^{\prime\prime}$. Otherwise the argument is the same. ∎

###### Proof of Theorem 1.18.

Since the functor $\mathrm{kk}^{G}$ is equivalent by Definition 1.10 to the
left Kan-extension of its restriction $\mathrm{kk}_{\mathrm{sep}}^{G}$ to
separable $G$-$C^{*}$-algebras it is s-finitary by Lemma 3.1. This is
Assertion 1.18.1 which verifies the assumption in Lemma 3.2 for
$F=\mathrm{kk}^{G}$ and $F_{s}=\mathrm{kk}_{\mathrm{sep}}^{G}$.

The remaining assertions of Theorem 1.18 now follow by applying Lemma 3.2 and
Theorem 1.4 asserting the corresponding properties for
$\mathrm{kk}_{\mathrm{sep}}^{G}$. ∎

Our next theorem settles the universal property of $\mathrm{kk}^{G}$ stated as
Theorem 1.19 in the introduction.

######  Theorem 3.3.

The functor $\mathrm{kk}^{G}$ is initial among functors from
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ to objects of
$\mathbf{CAT}_{\infty}^{\mathrm{ccpl}\cap{\mathrm{ex}}}$ which are s-finitary,
reduced, $\mathbb{K}^{G}$-stable and semiexact.

###### Proof.

Let ${\mathcal{D}}$ be a cocomplete stable $\infty$-category. We then have the
following chain of equivalences

$\displaystyle{\mathbf{Fun}}^{\operatorname*{\mathtt{colim}}}(\mathrm{KK}^{G},{\mathcal{D}})$
$\displaystyle\stackrel{{\scriptstyle y^{G,*},\simeq}}{{\to}}$
$\displaystyle{\mathbf{Fun}}^{\mathrm{ex}}(\mathrm{KK}_{\mathrm{sep}}^{G},{\mathcal{D}})$
$\displaystyle\stackrel{{\scriptstyle(\mathrm{kk}_{\mathrm{sep}}^{G})^{*},\simeq}}{{\to}}$
$\displaystyle{\mathbf{Fun}}^{rse}({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}),{\mathcal{D}})$
$\displaystyle\stackrel{{\scriptstyle\mathrm{incl}^{*},\simeq}}{{\leftarrow}}$
$\displaystyle{\mathbf{Fun}}^{frse}({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}),{\mathcal{D}})\,,$

where the superscript $\operatorname*{\mathtt{colim}}$ stands for colimit
preserving functors. Furthermore the superscript ${(f)rse}$ stands the full
subcategory of functors which are (s-finitary,) reduced,
$\mathbb{K}^{G}$-stable and semiexact, respectively. 333Note that in the proof
of Theorem 2.23, the $e$ in the superscript $rse$ referred to split exact as
opposed to semiexact in the present situation. We apologise for the notational
clash..

Here for the first two steps we use the universal properties of $y^{G}$ in
(1.6) and $\mathrm{kk}_{\mathrm{sep}}^{G}$ (Theorem 1.5). The last equivalence
is a consequence of the Lemmas 3.1 and 3.2 which imply that if we left Kan-
extend a functor in
${\mathbf{Fun}}^{{rse}}({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}),{\mathcal{D}})$
along the inclusion $\mathrm{incl}$ in (1.7), then the result is in
${\mathbf{Fun}}^{{frse}}({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}),{\mathcal{D}})$.
Since $\mathrm{kk}^{G}\circ\mathrm{incl}\simeq
y^{G}\circ\mathrm{kk}_{\mathrm{sep}}^{G}$ by (1.7) we conclude

${\mathbf{Fun}}^{\operatorname*{\mathtt{colim}}}(\mathrm{KK}^{G},{\mathcal{D}})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathbf{Fun}}^{frse}({\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}),{\mathcal{D}})$

which is the desired equivalence. ∎

######  Remark 3.4.

Note that the functor
$y^{G}\colon\mathrm{KK}_{\mathrm{sep}}^{G}\to\mathrm{KK}^{G}$ in (1.6) only
preserves finite colimits. Therefore, in contrast to Proposition 2.19.2, the
functor $\mathrm{kk}^{G}$ only sends finite sums to coproducts. One could
improve this situation by forming the Bousfield localization

$L:\mathrm{KK}^{G}\rightleftarrows\mathrm{KK}^{G,\prime}:\mathrm{incl}$

of $\mathrm{kk}^{G}$ at the set of maps $\bigoplus_{n\in{\rm
Nature}}\mathrm{kk}^{G}(A_{n})\to\mathrm{kk}^{G}(\bigoplus_{n\in{\rm
Nature}}A_{n})$ for all families $(A_{n})_{n\in{\rm Nature}}$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ and
setting

$\mathrm{kk}^{G,\prime}:=L\circ\mathrm{kk}^{G}:{\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to\mathrm{KK}^{G,\prime}\
.$

We restrict to families of separable algebras in order have a set of
generators.

The functor $\mathrm{kk}^{G,\prime}$ is s-finitary. In addition, it preserves
countable sums. Indeed, it preserves countable sums of families of separable
$G$-$C^{*}$-algebras by construction. But using that $\mathrm{kk}^{G,\prime}$
is s-finitary we can remove the assumption on separability.

As an immediate consequence of Theorem 3.3 and the definitions the universal
property of $\mathrm{kk}^{G,\prime}$ is as follows: The functor
$\mathrm{kk}^{G,\prime}$ is initial among functors from
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ to objects of
$\mathbf{CAT}_{\infty}^{\mathrm{ccpl}\cap\mathrm{ex}}$ which are s-finitary,
reduced, $\mathbb{K}^{G}$-stable, semiexact, and which send countable sums to
coproducts.∎

The next proposition is Proposition 1.20 from the introduction.

######  Proposition 3.5.

If $A$ is in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ and $B$ is
in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\sigma})$, then the
functor $\mathrm{ho}$ induces an isomorphism of ${\mathbb{Z}}$-graded groups

$\pi_{*}\mathrm{KK}^{G}(A,B)\cong K\\!K^{G}_{*}(A,B)\,.$ (3.2)

###### Proof.

Let $A$ be in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$. We
consider the functors