KK-theory as a stable infinity-category and KU Mod

Question

The category KK of bivariant operator K-theory (or possibly its E-theory variant) ought to be the homotopy category of something at least close to a stable infinity-category; notably in that it carries a by now well-known triangulated category structure.

What seems like a step in the direction of establishing such a stable $\infty$-category structure is in the note

• Michael Joachim, Stephan Stolz, An enrichment of KK-theory over the category of symmetric spectra Münster J. of Math. 2 (2009), 143–182 (pdf)

which produces

1. an enrichment $\mathbb{KK}$ of $KK$ in symmetric spectra, in fact in KU-module spectra;

2. a symmetric monoidal enriched functor $\mathbb{KK} \to \mathrm{KU} Mod$.

(A partial equivariant generalization of this is given by Mitchener in arXiv:0711.2152.)

This prompts some evident questions:

1. Does this enrichment exhibit a presentation of a stable $\infty$-category structure (or close)?

2. How far is that functor from being homotopy full and faithful?

Has anyone thought about this? What can one say?

(I see that Mahanta has a note arXiv:1211.6576 along these lines, but not sure yet if it helps with KK.)

Answer 1

There seems to be a mistake in the construction from "An enrichment..." See http://arxiv.org/pdf/1104.3441v1 page 3. That paper gives an alternative construction of a symmetric spectrum representing (equivariant) K-theory.

Concerning question 2: the induced functor $\mathrm{KK}\to\mathrm{Der}(\mathbf{K})$ is fully faithful and strongly monoidal on the bootstrap class of Rosenberg--Schochet (the localizing subcategory generated by $\mathbb C$). It cannot be fully faithful on all of $\mathrm{KK}$ because there are counterexamples to the Universal Coefficient Theorem in $\mathrm{KK}$ but not in $\mathrm{Der}(\mathbf{K})$.

Answer 2

$\DeclareMathOperator{\KK}{KK}$ $\DeclareMathOperator{\Ob}{Ob}$ $\DeclareMathOperator{\C}{\mathcal{C}}$ $\DeclareMathOperator{\top}{top}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\pr}{pr}$ $\DeclareMathOperator{\N}{N}$ $\DeclareMathOperator{\HKK}{\mathbf{KK}}$ $\DeclareMathOperator{\const}{const}$ $\DeclareMathOperator{\Hom}{Hom}$

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 class of separable C*-algebras.) I know a different, more abstract, construction, but you have heard me describe it before.

Let $𝐂^*$ be the set of separable C*-algebras of set-rank $ω⋅2$. Let $Δ$ be the set of finite positive von Neumann ordinals.

For two C*-algebras $A$ and $B$ 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 C*-algebra of continuous functions from the topological $n$–simplex to $B$.

Let $R$ be the strict $2$–category of sets, many-valued functions and inclusions of functions.

Choose $n$ and $i$ in $Δ$, $γ\colon i → n$ order preserving, and $f$ in $𝐊^n(A, B)$. Let $γ^*𝑓$ be the set of preimages of $γ^*[f]$ in $𝐊^i(A, B)$.

The map $n + 1 ↦ 𝐊^n(A, B)$ is thus contravariantly oplaxly functorial for order preserving maps and many-valued functions. Let $𝐊(A, B)$ be that functor – a simplicial $R$–set.

Let $(ϕ, ℰ, T)$ be in $𝐊^n(A, B)$. Let $ℱ$ be the $𝐙/2$–graded Hilbert $\C(Δ^n_{\top}, B)$–module $\C(Δ^n_{\top}, ℰ)$ with the inner product $⟨ξ, η⟩(s) ≔ ⟨ξ(s), η(s)⟩(s)$ and the module structure $(ξ⋅b)(s) ≔ ξ(s)⋅b$. Define a representation $ψ$ 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 $S(ξ)(s) ≔ T(ξ(s))$. Put $M(ϕ, ℰ, T) ≔ (ψ, ℱ, S)$. For three C*-algebras $A$, $B$ 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 $A ∈ 𝐂^*$ and $n ∈ Δ$. Consider $\C(Δ^n_{\top}, A)$ as a $𝐙/2$–graded Hilbert module over itself in the usual way and let $ρ$ be the representation of $A$ by the even operators of pointwise left-multiplication on $\C(Δ^n_{\top}, A)$. We write $η^n(A)$ for the many-valued function defined on $\{0\}$ that takes as values all the Kasparov cycles equivalent to $(ρ, \C(Δ^n_{\top}, A), 0) ∈ 𝐊^n(A, A)$.

Convince yourself of the following for all $A, B, C, D ∈ 𝐂^*$, $n + 1, i + 1 ∈ Δ$, $γ\colon i + 1 → i + 1$ order preserving:

1. $∃f ∈ 𝐊^n(B, D): ((f, g), h) ∈ μ^n(A, B, D) ∧ ((d, e), f) ∈ μ^n(B, C, D) ⇔ ∃j ∈ 𝐊^n(A, C): ((d, j), h) ∈ μ^n(A, C, D) ∧ ((e, g), j) ∈ μ^n(A, B, C);$

2. $μ^i(A, B, C) ∘ (γ^* × γ^*) = γ^* ∘ μ^n(A, B, C): 𝐊^n(B, C) × 𝐊^n(A, B) → 𝐊^i(A, B);$

3. $γ^* ∘ η^n(A) = η^i(A)$;

4. $μ^n(A, A, B)(\id × η^n(A)) ⊇ \pr_1 ∧ μ^n(A, B, B)(η^n(B) × \id) ⊇ \pr_2.$

Thus $(𝐂^*, 𝐊, μ, η)$ is a category enriched in the $2$–category of simplicial $R$–sets. Let us call such objects simplicial $R$–categories, as categories enriched in simplicial sets are – also in this answer – called simplicial categories.

Let us write $Δ^n$ for the $n$–dimensional simplicial ($R$–)simplex, $Λ^n(k)$ for its $k$–th horn and $ι(n, k)$ for the inclusion of $Λ^n(k)$ into $Δ^n$. We say that a simplicial $R$–set $X$ is a Kan complex if and only if for every $n + 1$ and $k + 1$ in $Δ$ and every single-valued map $σ\colon Λ^n(k) → X$ there is a map $τ\colon Δ^n → X$ such that $σ = τ ∘ ι(n, k)$.

Claim: The simplicial $R$–set $𝐊(A, B)$ is a Kan complex for all $A$ and $B$ in $𝐂^*$.

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Proof. Pick any $n + 1$ and $k + 1$ in $Δ$. Every single-valued map $σ\colon Λ^n(k) → 𝐊(A, B)$ determines some $f ∈ 𝐊^0(A, \C(Λ^n(k)_{\top}, B))$, where $Λ^n(k)_{\top}$ is the corresponding $(n − 1)$–dimensional topological horn. The map $B ↦ 𝐊^0(A, B)$ is functorial for ${}^*$–homomorphisms and many-valued functions; thus $f$ lifts to some $g ∈ 𝐊^0(A, \C(Δ^n_{\top}, B))$. Use $g$ to fill in the missing values and thereby extend $σ$ to $Δ^n$.

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Let $ℭ(Δ^n)$ be the canonical realisation of Δⁿ as cofibrant simplicial category.

Let $D$ be a simplicial $R$–category. We define the homotopy coherent nerve $\N(D)$ of $D$ by

$$ \N(D)(n + 1) ≔ \Hom_{\operatorname{single-valued}}(ℭ(Δ^n), D). $$

Claim: The homotopy-coherent nerve of a simplicial $R$–category $D$ is a quasi-category if all the simplicial arrow $R$–sets are Kan complexes.

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Proof. Choose $n ≥ 2$ and $i$ with $1 ≤ i ≤ n − 1$. Put $S ≔ ℭ(Δ^n)$ and let $Q_k$ be the set of degenerate $k$–simplices in $S(0, n)$. Let $L$ be the largest simplicial subcategory of $S$ with $\Ob(L) = \Ob(S)$ and

$$ ∀k ≥ 1\colon \{σ ∈ L_k(0, n) ∣ σ_0 = \{0, n\} ∧ σ_k ∪ \{i\} = \{0, \dotsc, n\}\} ⊆ Q_k. $$

$L$ realises the horn $Λ^n(i)$ as cofibrant simplicial category.

Let $F\colon L → D$ be a functor. We need to extend $F$ to $S$. Because $L(j, k)$ and $S(j, k)$ only differ for $j = 0$ and $k = n$ we only need to extend the map $F(0; n)\colon L(0, n) → D(F(0), F(n))$ to $S(0, n)$. The inclusion of $L(0, n)$ into $S(0, n)$ is a cofibration. It is also a weak homotopy equivalence, because the geometric realisation of $S(0, n)$ is homeomorphic to $[0, 1]^{n − 1}$, and the image of $L(0, n)$ is the contractible subset

$$ \{x ∈ [0, 1]^{n − 1} ∣ x_i = 1 ∨ ∃j ≠ i: (x_j = 0 ∨ x_j = 1)\} $$

under this homeomorphism. Therefore it is anodyne. It has the extension property with respect to all simplicial $R$–sets that are Kan complexes because the inclusion of simplicial sets into simplicial $R$–sets and single-valued maps preserves colimits.

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We now know that $\HKK ≔ \N(𝐂^*, 𝐊, μ, η)$ is a quasi-category. Its objects are elements $A$ of $𝐂^*$ paired with a choice of one element from $η^n(A)$ for every $n + 1$ in $Δ$. We can choose such elements canonically (see above); we write $ι$ for this inclusion of $𝐂^*$ into $\Ob(\HKK)$. The restriction of $\HKK$ to $𝐂^*$ is equivalent to $\HKK$ via the inclusion. On the other hand also $\KK$ restricted to $𝐂^*$ is equivalent to $\KK$ via the inclusion. For $A$ and $B$ in $𝐂^*$, the cycles for $π(\HKK)(ι(A), ι(B))$ map onto $\KK(A, B)$. That map $θ$ is a bijection modulo equivalence because $θ(f) = θ(g)$ in $\KK(A, B)$ only if $[M(h)] ∘ [f((s_0)^n(\const \{0, 1\}))] = [g((s_0)^n(\const \{0, 1\}))]$ in $\KK(A, \C(Δ^n_{\top}, B))$ for all $n + 1$ in $Δ$ and some $h$ in $η^n(B)$.

We now only need to see that $\HKK$ is stable in a way that is compatible with the triangulation on $\KK$.

For $0 ∈ 𝐂^*$, $ι(0)$ is a final object of $\HKK$, because $0$ is a final object of $\KK$.

Claim: For all matrix-stable $A, B, D$ in $𝐂^*$, $f = (ϕ, D, 0)$ in $𝐊^0(A, D)$ and $g = (ψ, D, 0)$ in $𝐊^0(B, D)$, a map $F: Δ^1 × Δ^1 → \HKK$ is a Cartesian square if

1. • $F_0(0; 0) = \{(a, b, d, h_1, h_2) ∈ A ⊕ B ⊕ D ⊕ \C([0, 1], D) ⊕ \C([0, 1], D) ∣ ∀d_2 ∈ D: h_1(0)⋅d_2 = ϕ(a)(d_2) ∧ h_1(1) = d = h_2(1) ∧ h_2(0)⋅d_2 = ψ(b)(d₂)\}$,

     • $F_0(0; 1) = A$,
     • $F_0(1; 0) = B$,
     • $F_0(1; 1) = D$;
2. • $F_1(0, 0; 0, 1) = ((a, b, d, h_1, h_2) ↦ a_2 ↦ aa_2, A, 0)$,
     • $F_1(0, 1; 0, 0) = ((a, b, d, h_1, h_2) ↦ b_2 ↦ bb_2, B, 0)$,
     • $F_1(0, 1; 1, 1) = f$,
     • $F_1(1, 1; 0, 1) = g$,
     • $F_1(0, 1; 0, 1) = ((a, b, d, h_1, h_2) ↦ d_2 ↦ dd_2, D, 0)$;
3. • $F_2(0, 0, 1; 0, 1, 1) = ((a, b, d, h_1, h_2) ↦ ξ ↦ h_1ξ, \C([0, 1], D), 0)$,
     • $F_2(0, 1, 1; 0, 0, 1) = ((a, b, d, h_1, h_2) ↦ ξ ↦ h_2ξ, \C([0, 1], D), 0)$;

Remark: Every cospan in $\HKK$ is equivalent to one as in the claim.

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Proof. Choose $n ≥ 1$ and a map $G\colon ∂Δ^n × Δ^1 × Δ^1 → \HKK$ of the appropriate kind. Without loss of generality we can adjust your choice of $G$ so that all the Kasparov cycles in the image come from ${}^*$–homomorphisms (because $F_0(0; 0)$ is matrix-stable). $G$ extends to $Δ^n × Δ^1 × Δ^1$ because $F$ determines a Cartesian square in the homotopy coherent nerve of the topologically enriched category with $𝐂^*$ as set of objects and ${}^*$–homomorphisms with the topology of pointwise norm-convergence as mapping spaces.

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Bott periodicity therefore tells us that a suspension functor on $\HKK$ is an equivalence. Therefore $\HKK$ is stable. The description of Cartesian squares makes it clear that the triangulation on the homotopy category of $\HKK$ is compatible with the equivalence to $\KK|𝐂^*$.

Answer 3

Answering your first question in more generality: an $\infty$-category that is enriched over spectra is a stable $\infty$-category if it is closed under finite limits.