Rasmus already answered the second part of your question. Let us construct a$\DeclareMathOperator{\KK}{KK}$
stable quasi-category and a triangulated equivalence from its homotopy category$\DeclareMathOperator{\Ob}{Ob}$
to KK. (Ob(KK) shall be the class of separable C*-algebras.) I know a$\DeclareMathOperator{\C}{\mathcal{C}}$
different, more abstract, construction, but you have heard me describe it$\DeclareMathOperator{\top}{top}$
before.$\DeclareMathOperator{\id}{id}$
$\DeclareMathOperator{\pr}{pr}$
$\DeclareMathOperator{\N}{N}$
$\DeclareMathOperator{\HKK}{\mathbf{KK}}$
$\DeclareMathOperator{\const}{const}$
$\DeclareMathOperator{\Hom}{Hom}$
Let 𝐂*Rasmus already answered the second part of your question. Let us construct a
stable quasi-category and a triangulated equivalence from its homotopy category
to $\KK$. ($\Ob(\KK)$ shall be the setclass of separable C*-algebras of set-rank ω⋅2. Let Δ be the set) I know a
of finite positive von Neumann ordinalsdifferent, more abstract, construction, but you have heard me describe it
before.
For two C-algebras A and B in 𝐂 and 𝑛 + 1 in Δ let 𝐊ⁿ(A, B)Let $𝐂^*$ be the set of all
Kasparov cycles from A to 𝐶(Δⁿₜₒₚ, B) – theseparable C*-algebraalgebras of continuous functionsset-rank $ω⋅2$. Let $Δ$ be
from the topological 𝑛–simplex to Bthe set of finite positive von Neumann ordinals.
Let R be the strict 2–category of sets, manyFor two C*-valued functionsalgebras $A$ and inclusions$B$ in $𝐂^*$ and $n + 1$ in $Δ$ let $𝐊^n(A, B)$ be
the set of all Kasparov cycles from $A$ to $\C(Δ^n_{\top}, B)$ – the
functionsC*-algebra of continuous functions from the topological $n$–simplex to $B$.
Choose 𝑛 and 𝑖 in Δ, γ: 𝑖 → 𝑛 order preservingLet $R$ be the strict $2$–category of sets, many-valued functions and 𝑓 in 𝐊ⁿ(A, B). Let γ* 𝑓 be
the set of preimagesinclusions of γ*[𝑓] in 𝐊ⁱ(A, B)functions.
The map 𝑛 + 1 ↦ 𝐊ⁿ(AChoose $n$ and $i$ in $Δ$, B) is thus contravariantly oplaxly functorial for $γ\colon i → n$ order
preserving maps preserving, and many-valued functions$f$ in
$𝐊^n(A, B)$. Let 𝐊(A, B)$γ^*𝑓$ be that functor – a
simplicial R–setthe set of preimages of $γ^*[f]$ in $𝐊^i(A, B)$.
LetThe map (ϕ, ℰ, T) be in 𝐊ⁿ(A, B)$n + 1 ↦ 𝐊^n(A, B)$ is thus contravariantly oplaxly functorial for
order preserving maps and many-valued functions. Let ℱ$𝐊(A, B)$ be the 𝐙/2–graded Hilbert
𝐶(Δⁿₜₒₚ, B)–module 𝐶(Δⁿₜₒₚ, ℰ) with the inner product ⟨ξ, η⟩(𝑠) ≔that functor
⟨ξ(𝑠), η(𝑠)⟩(𝑠) and the module structure (ξ⋅𝑏)(𝑠) ≔ ξ(𝑠)⋅𝑏. Define– a
representation ψ of C(Δⁿₜₒₚ, A) by even adjointable operators on ℱ by
ψ(𝒶)(ξ)(𝑠) ≔ ϕ(𝒶(𝑠))(ξ(𝑠)) and define an odd adjointable operator S on ℱ by
S(ξ)(𝑠) ≔ T(ξ(𝑠)). Put M(ϕ, ℰ, T) ≔ simplicial (ψ, ℱ, S)$R$–set.
For three C-algebras A, B and C in 𝐂 and 𝑛 + 1Let $(ϕ, ℰ, T)$ be in Δ let μⁿ(A, B, C)$𝐊^n(A, B)$. Let $ℱ$ be the $𝐙/2$–graded Hilbert
many-valued function from 𝐊ⁿ$\C(Δ^n_{\top}, B)–module $\C(BΔ^n_{\top}, Cℰ) × 𝐊ⁿ(A$ with the inner product $⟨ξ, B) to 𝐊ⁿη⟩(A, Cs) that associates ℎ ∈
𝐊ⁿ≔ ⟨ξ(A, Cs) with (𝑓, 𝑔η(s) ∈ 𝐊ⁿ⟩(B, Cs) × 𝐊ⁿ$ and the module structure $(A, Bξ⋅b) if [ℎ] = [M(𝑓s)] ∘ [𝑔] ∈ ≔ $ξ(s)⋅b$. Define a
KK(A, 𝐶(Δⁿₜₒₚrepresentation $ψ$ of $\C(Δ^n_{\top}, A)$ by even adjointable operators on $ℱ$ by
$ψ(a)(ξ)(s) ≔ ϕ(a(s))(ξ(s))$ and define an odd adjointable operator $S$ on $ℱ$
by $S(ξ)() ≔ T(ξ(s))$. Put $M(ϕ, ℰ, T) ≔ (ψ, ℱ, S)$.
For three C*-algebras $A$, C)) $B$ and $C$ in $𝐂^*$ and $n + 1$ in $Δ$ let
$μ^n(A, B, C)$ be the many-valued function from $𝐊^n(B, C) × 𝐊^n(A, B)$ to
$𝐊^n(A, C)$ that associates $h ∈ 𝐊^n(A, C)$ with $(f, g) ∈
𝐊^n(B, C) × 𝐊^n(A, B)$ if $[h] = [M(f)] ∘ [g] ∈ \KK(A, \C(Δ^n_{\top}, C))$.
Choose A ∈ 𝐂*$A ∈ 𝐂^*$ and 𝑛 ∈ Δ$n ∈ Δ$. Consider 𝐶(Δⁿₜₒₚ, A)$\C(Δ^n_{\top}, A)$ as a 𝐙/2–graded Hilbert module$𝐙/2$–graded
overHilbert module over itself in the usual way and let ρ$ρ$ be the representation of A
of $A$ by the even
operators operators of pointwise left-multiplication on 𝐶(Δⁿₜₒₚ, A)
$\C(Δ^n_{\top}, A)$. We write ηⁿ(A)$η^n(A)$ for
the the many-valued function defined on {0}
$\{0\}$ that takes as values all the Kasparov
cycles cycles equivalent to (ρ, 𝐶(Δⁿₜₒₚ, A), 0) ∈ 𝐊ⁿ(A, A)
$(ρ, \C(Δ^n_{\top}, A), 0) ∈ 𝐊^n(A, A)$.
Convince yourself of the following for all A, B, C, D ∈ 𝐂*, 𝑛 + 1$A, B, C, D ∈ 𝐂^*$, 𝑖 + 1 ∈ Δ$n + 1, i + 1 ∈
Δ$,
γ: 𝑖 + 1 → 𝑛 + 1 $γ\colon i + 1 → i + 1$ order preserving:
∃𝑓 ∈ 𝐊ⁿ(B, D): ((𝑓, 𝑔), ℎ) ∈ μⁿ(A, B, D) ∧ ((𝑑, 𝑒), 𝑓) ∈ μⁿ(B, C, D)
⇔ ∃𝑗 ∈ 𝐊ⁿ(A, C): ((𝑑, 𝑗), ℎ) ∈ μⁿ(A, C, D) ∧ ((𝑒, 𝑔), 𝑗) ∈ μⁿ(A, B, C);$∃f ∈ 𝐊^n(B, D): ((f, g), h) ∈ μ^n(A, B, D) ∧ ((d, e), f) ∈ μ^n(B, C, D) ⇔
∃j ∈ 𝐊^n(A, C): ((d, j), h) ∈ μ^n(A, C, D) ∧ ((e, g), j) ∈ μ^n(A, B, C);$
μⁱ(A, B, C) ∘ (γ* × γ*) = γ* ∘ μⁿ(A, B, C):
𝐊ⁿ(B, C) × 𝐊ⁿ(A, B) → 𝐊ⁱ(A, B);$μ^i(A, B, C) ∘ (γ^* × γ^*) = γ^* ∘ μ^n(A, B, C):
𝐊^n(B, C) × 𝐊^n(A, B) → 𝐊^i(A, B);$
γ* ∘ ηⁿ(A) = ηⁱ(A)$γ^* ∘ η^n(A) = η^i(A)$;
μⁿ(A, A, B)(id × ηⁿ(A)) ⊇ pr₁ ∧ μⁿ(A, B, B)(ηⁿ(B) × id) ⊇ pr₂.$μ^n(A, A, B)(\id × η^n(A)) ⊇ \pr_1 ∧ μ^n(A, B, B)(η^n(B) × \id) ⊇ \pr_2.$
Thus (𝐂*, 𝐊, μ, η)$(𝐂^*, 𝐊, μ, η)$ is a category enriched in the 2–category$2$–category of simplicial
R–sets$R$–sets. Let us call such objects simplicial R–categories$R$–categories, as categories
enriched in simplicial sets are – also in this answer – called simplicial
categories.
Let us write Δⁿ$Δ^n$ for the 𝑛–dimensional$n$–dimensional simplicial (R–$R$–)simplex, Λⁿ(k) for its$Λ^n(k)$
𝑘–thfor its $k$–th horn and ι(n, k)$ι(n, k)$ for the inclusion of Λⁿ(k)$Λ^n(k)$ into Δⁿ$Δ^n$. We say
say that a
simplicial R–set X simplicial $R$–set $X$ is a Kan complex if and only if for every 𝑛 + 1
$n + 1$ and 𝑘 + 1$k + 1$ in
Δ $Δ$ and every single-valued map σ: Λⁿ(k) → X there$σ\colon Λ^n(k) → X$
there is a map τ: Δⁿ → X$τ\colon Δ^n → X$ such that σ
= τ ∘ ι(n, k)$σ = τ ∘ ι(n, k)$.
Claim: The simplicial R–set 𝐊(A, B)$R$–set $𝐊(A, B)$ is a Kan complex for all A$A$ and B in
𝐂*$B$ in $𝐂^*$.
Proof. Pick any 𝑛 + 1$n + 1$ and 𝑘 + 1$k + 1$ in Δ$Δ$. Every single-valued map σ: Λⁿ(k) →
𝐊(A, B)$σ\colon Λ^n(k) → 𝐊(A, B)$ determines some 𝑓 ∈ 𝐊⁰(A, 𝐶(Λⁿ(k)ₜₒₚ, B))$f ∈ 𝐊^0(A, \C(Λ^n(k)_{\top}, B))$, where Λⁿ(k)ₜₒₚ
where $Λ^n(k)_{\top}$ is the
corresponding corresponding (𝑛 − 1)$(n − 1)$–dimensional topological horn
horn. The map B ↦ 𝐊⁰(A, B)$B ↦ 𝐊^0(A, B)$ is
functorial functorial for *–homomorphisms${}^*$–homomorphisms and many
many-valued functions; thus 𝑓$f$ lifts to some
𝑔 ∈ 𝐊⁰(A, 𝐶(Δⁿₜₒₚ, B)) $g ∈ 𝐊^0(A, \C(Δ^n_{\top}, B))$. Use 𝑔
Use $g$ to fill in the missing values and thereby extend
σ $σ$ to Δⁿ$Δ^n$.
Let ℭ(Δⁿ)$ℭ(Δ^n)$ be the canonical realisation of Δⁿ as cofibrant simplicial category
category.
Let D$D$ be a simplicial R–category$R$–category. We define the homotopy-coherent coherent nerve N(D)
of D$\N(D)$ of $D$ by
$$
\operatorname{N}(D)(𝑛 + 1)
≔ \operatorname{Hom}_{\operatorname{single-valued}}(ℭ(Δⁿ), D).
$$$$
\N(D)(n + 1) ≔ \Hom_{\operatorname{single-valued}}(ℭ(Δ^n), D).
$$
Claim: The homotopy-coherent nerve of a simplicial R–category D$R$–category $D$ is a
quasi-category if all the simplicial arrow R–sets$R$–sets are Kan complexes.
Proof. Choose 𝑛 ≥ 2$n ≥ 2$ and 𝑖$i$ with 1 ≤ 𝑖 ≤ 𝑛 − 1$1 ≤ i ≤ n − 1$. Put S ≔ ℭ(Δⁿ)$S ≔ ℭ(Δ^n)$ and let Qₖ be
thelet $Q_k$ be the set of degenerate 𝑘–simplices$k$–simplices in S(0, 𝑛)$S(0, n)$. Let L$L$ be the largest simplicial
subcategorylargest simplicial subcategory of S$S$ with Ob(L) = Ob(S)$\Ob(L) = \Ob(S)$ and
$$
∀𝑘 ≥ 1: \{σ ∈ Lₖ(0, 𝑛) ∣ σ₀ = \{0, 𝑛\} ∧ σₖ ∪ \{𝑖\} = \{0, \dotsc, 𝑛\}\} ⊆ Qₖ.
$$$$
∀k ≥ 1\colon
\{σ ∈ L_k(0, n) ∣ σ_0 = \{0, n\} ∧ σ_k ∪ \{i\} = \{0, \dotsc, n\}\} ⊆ Q_k.
$$
L$L$ realises the horn Λⁿ(𝑖)$Λ^n(i)$ as cofibrant simplicial category.
Let F: L → D$F\colon L → D$ be a functor. We need to extend F$F$ to S$S$. Because L(𝑗, 𝑘) and
S(𝑗, 𝑘)$L(j, k)$ and $S(j, k)$ only differ for 𝑗 = 0$j = 0$ and 𝑘 = 𝑛$k = n$ we only need to extend
extend the map F(0; 𝑛):
L(0, 𝑛) → D(F(0), F(𝑛))$F(0; n)\colon L(0, n) → D(F(0), F(n))$ to S(0, 𝑛)$S(0, n)$. The inclusion
inclusion of L(0, 𝑛)$L(0, n)$ into S(0, 𝑛)$S(0, n)$ is a
cofibration cofibration. It is also a weak homotopy
homotopy equivalence, because the geometric
realisation realisation of S(0, 𝑛)$S(0, n)$ is homeomorphic
homeomorphic to $[0, 1]^{𝑛 − 1}$, and the image of L$[0, 1]^{n − 1}, and the image of $L(0, 𝑛n)
is$ is the contractible subset
subset
$$
\{𝑥 ∈ [0, 1]^{𝑛 − 1} ∣ xᵢ = 1 ∨ ∃j ≠ i: (xⱼ = 0 ∨ xⱼ = 1)\}
$$$$
\{x ∈ [0, 1]^{n − 1} ∣ x_i = 1 ∨ ∃j ≠ i: (x_j = 0 ∨ x_j = 1)\}
$$
under this homeomorphism. Therefore it is anodyne. It has the extension
property with respect to all simplicial R–sets$R$–sets that are Kan complexes because
the inclusion of simplicial sets into simplicial R–sets$R$–sets and single-valued maps
preservesmaps preserves colimits.
We now know that 𝐊𝐊 ≔ N(𝐂* , 𝐊, μ, η)$\HKK ≔ \N(𝐂^*, 𝐊, μ, η)$ is a quasi-category. Its objects are
elements Aare elements $A$ of 𝐂*$𝐂^*$ paired with a choice of one element from ηⁿ(A)$η^n(A)$ for every 𝑛 + 1
in Δevery $n + 1$ in $Δ$. We can choose such elements canonically (see above); we write ι
write $ι$ for this
inclusion inclusion of 𝐂*$𝐂^*$ into Ob(𝐊𝐊)$\Ob(\HKK)$. The restriction of 𝐊𝐊
$\HKK$ to 𝐂*$𝐂^*$ is equivalent to 𝐊𝐊
via$\HKK$ via the inclusion. On the other hand also KK
also $\KK$ restricted to 𝐂*$𝐂^*$ is equivalent to
KK $\KK$ via the inclusion. For A
$A$ and B$B$ in 𝐂*$𝐂^*$, the cycles for π(𝐊𝐊)(ι(A), ι(B))$π(\HKK)(ι(A), ι(B))$ map onto
onto KK(A, B)$\KK(A, B)$. That map θ$θ$ is a bijection modulo equivalence because θ(𝑓) =
θ(𝑔)$θ(f) =
θ(g)$ in KK(A, B)$\KK(A, B)$ only if [M(ℎ)] ∘ [𝑓((𝑠₀)ⁿ(const {0, 1}))] =
[𝑔((𝑠₀)ⁿ(const {0, 1}))]$[M(h)] ∘ [f((s_0)^n(\const \{0, 1\}))] =
[g((s_0)^n(\const \{0, 1\}))]$ in KK(A, 𝐶(Δⁿₜₒₚ, B))$\KK(A, \C(Δ^n_{\top}, B))$ for all 𝑛 + 1 in Δ$n + 1$
in $Δ$ and some ℎ$h$ in
ηⁿ(B) $η^n(B)$.
We now only need to see that 𝐊𝐊$\HKK$ is stable in a way that is compatible with the
triangulationthe triangulation on KK$\KK$.
For 0 ∈ 𝐂*$0 ∈ 𝐂^*$, ι(0)$ι(0)$ is a final object of 𝐊𝐊$\HKK$, because 0$0$ is a final object
object of KK$\KK$.
Claim: For all matrix-stable A, B, D$A, B, D$ in 𝐂*, 𝑓 = (ϕ, D$𝐂^*$, 0) $f = (ϕ, D, 0)$ in 𝐊⁰(A, D) and
𝑔 =$𝐊^0(A, D)$ and (ψ, D, 0)$g = (ψ, D, 0)$ in 𝐊⁰(B, D)$𝐊^0(B, D)$, a map F: Δ¹ × Δ¹ → 𝐊𝐊$F: Δ^1 × Δ^1 → \HKK$ is a
a Cartesian square if
1.
* F₀(0; 0)
= {(𝑎, 𝑏, 𝑑, ℎ₁, ℎ₂) ∈ A ⊕ B ⊕ D ⊕ 𝐶([0, 1], D) ⊕ 𝐶([0, 1], D)
∣ ∀𝑑₂ ∈ D: ℎ₁(0)⋅𝑑₂ = ϕ(𝑎)(𝑑₂)
∧ ℎ₁(1) = 𝑑 = ℎ₂(1)
∧ ℎ₂(0)⋅𝑑₂ = ψ(𝑏)(𝑑₂)},
* F₀(0; 1) = A,
* F₀(1; 0) = B,
* F₀(1; 1) = D;
-
- F₁(0, 0; 0, 1) = ((𝑎, 𝑏, 𝑑, ℎ₁, ℎ₂) ↦ 𝑎₂ ↦ 𝑎𝑎₂, A, 0),
- F₁(0, 1; 0, 0) = ((𝑎, 𝑏, 𝑑, ℎ₁, ℎ₂) ↦ 𝑏₂ ↦ 𝑏𝑏₂, B, 0),
- F₁(0, 1; 1, 1) = 𝑓,
- F₁(1, 1; 0, 1) = 𝑔,
- F₁(0, 1; 0, 1) = ((𝑎, 𝑏, 𝑑, ℎ₁, ℎ₂) ↦ 𝑑₂ ↦ 𝑑𝑑₂, D, 0);
-
- F₂(0, 0, 1; 0, 1, 1) = ((𝑎, 𝑏, 𝑑, ℎ₁, ℎ₂) ↦ ξ ↦ ℎ₁ξ, 𝐶([0, 1], D), 0),
- F₂(0, 1, 1; 0, 0, 1) = ((𝑎, 𝑏, 𝑑, ℎ₁, ℎ₂) ↦ ξ ↦ ℎ₂ξ, 𝐶([0, 1], D), 0);
-
$F_0(0; 0)
= \{(a, b, d, h_1, h_2) ∈ A ⊕ B ⊕ D ⊕ \C([0, 1], D) ⊕ \C([0, 1], D)
∣ ∀d_2 ∈ D: h_1(0)⋅d_2 = ϕ(a)(d_2)
∧ h_1(1) = d = h_2(1)
∧ h_2(0)⋅d_2 = ψ(b)(d₂)\}$,
$F_0(0; 1) = A$,
$F_0(1; 0) = B$,
$F_0(1; 1) = D$;
-
- $F_1(0, 0; 0, 1) = ((a, b, d, h_1, h_2) ↦ a_2 ↦ aa_2, A, 0)$,
- $F_1(0, 1; 0, 0) = ((a, b, d, h_1, h_2) ↦ b_2 ↦ bb_2, B, 0)$,
- $F_1(0, 1; 1, 1) = f$,
- $F_1(1, 1; 0, 1) = g$,
- $F_1(0, 1; 0, 1) = ((a, b, d, h_1, h_2) ↦ d_2 ↦ dd_2, D, 0)$;
-
- $F_2(0, 0, 1; 0, 1, 1)
= ((a, b, d, h_1, h_2) ↦ ξ ↦ h_1ξ, \C([0, 1], D), 0)$,
- $F_2(0, 1, 1; 0, 0, 1)
= ((a, b, d, h_1, h_2) ↦ ξ ↦ h_2ξ, \C([0, 1], D), 0)$;
Remark: Every cospan in 𝐊𝐊$\HKK$ is equivalent to one as in the claim.
Proof. Choose 𝑛 ≥ 1$n ≥ 1$ and a map G: ∂Δⁿ × Δ¹ × Δ¹ → 𝐊𝐊$G\colon ∂Δ^n × Δ^1 × Δ^1 → \HKK$ of the appropriate
kindappropriate kind. Without loss of generality we can adjust your choice of G so$G$
so that all
the the Kasparov cycles in the image come from ${}^*$–homomorphisms
(because F₀(0; 0)$F_0(0; 0)$ is
matrix matrix-stable). G $G$ extends to Δⁿ × Δ¹ × Δ¹ because F$Δ^n × Δ^1 × Δ^1$
because $F$ determines a Cartesian
square square in the homotopy-coherent coherent nerve of the topologically
topologically enriched category
with 𝐂* with $𝐂^*$ as set of objects and *-homomorphisms
${}^*$–homomorphisms with the topology of pointwise
norm norm-convergence as mapping spaces
spaces.
Bott periodicity therefore tells us that a suspension functor on 𝐊𝐊$\HKK$ is an
equivalence. Therefore 𝐊𝐊$\HKK$ is stable. The description of Cartesian squares
makes it clear that the triangulation on the homotopy category of 𝐊𝐊$\HKK$ is
compatible with the equivalence to KK|𝐂*$\KK|𝐂^*$.
