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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.)

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    $\begingroup$ The last paragraph of this answer might be relevant: mathoverflow.net/a/12480/100. Are there supposed to be equivalences in $\mathbb{KK}$ that are not isomorphisms? $\endgroup$ – Benjamin Antieau Aug 6 '13 at 22:44
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    $\begingroup$ Also, Section 10 of the paper arxiv.org/pdf/1112.5563v1.pdf by Dell'Ambrogio and Tabuada is relevent, but not exactly what you're looking for. $\endgroup$ – Benjamin Antieau Aug 6 '13 at 22:55
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    $\begingroup$ @Benjamin: yes, spectrum enrichment is necessary but not sufficient for stability. Therefore my question! :-) $\endgroup$ – Urs Schreiber Aug 7 '13 at 6:02
  • $\begingroup$ @Rasmus: ah, thanks for pointing that out! That article also addresses my second question, in parts. But I still have to read it in detail now... $\endgroup$ – Urs Schreiber Aug 7 '13 at 9:46
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    $\begingroup$ The later arxiv versions of Mahanta's paper appear to take care of KK-theory. $\endgroup$ – Rasmus Jan 8 '15 at 20:19
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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 $\mathrm{Der}(\mathbf{K})$.

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  • $\begingroup$ Thanks! That's most useful. Let me have a look at the article... $\endgroup$ – Urs Schreiber Aug 7 '13 at 9:51
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$\DeclareMathOperator{\KK}{KK}$ $\DeclareMathOperator{\Ob}{Ob}$ $\DeclareMathOperator{\C}{\mathcal{C}}$ $\DeclareMathOperator{\top}{top}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\pr}{pr}$ $\DeclareMathOperator{\N}{N}$ $\DeclareMathOperator{\HKK}{\mathbf{KK}}$ $\DeclareMathOperator{\const}{const}$ $\DeclareMathOperator{\Hom}{Hom}$

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 $𝐂^*$.


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 $Δ^n$.


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.


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.


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

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

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

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


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.


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 $\KK|𝐂^*$.

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  • $\begingroup$ I would suggest that this question be rewritten using latex code (i.e. mathjax) instead of using highly complicated, pre-formatted text. $\endgroup$ – Ricardo Andrade Aug 12 '13 at 22:49
  • $\begingroup$ Is there anything in particular that you would format differently, or do you think that everything that is usually written in mathjax should be wrapped in dollar signs so that it stands out more? There seem to have been some accidents involving asterisks; I have tried to fix them. $\endgroup$ – Rohan Lean Aug 12 '13 at 23:11
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    $\begingroup$ Dear @Rohan: Your answer appears to contain hard-coded bold symbols, subscripts, etc. It does not even render in an alternative but well developed browser which I use. Mathjax is not meant for highlighting math, it is meant to make sure that math will display correctly and not break randomly. I believe your current code for this answer is likely to cause several problems. I have created a meta thread at meta.mathoverflow.net/questions/624/… $\endgroup$ – Ricardo Andrade Aug 12 '13 at 23:15
  • $\begingroup$ By the way, asterisks are part of the markdown code. Putting text between asterisks italicizes the text. $\endgroup$ – Ricardo Andrade Aug 12 '13 at 23:17
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    $\begingroup$ I am not really aware of such subtleties. In any case, it appears that in *asdf ghjk* the asterisks are recognized as markdown, but not in *asdf *. It appears that in order to recognize the asterisks as markdown, perhaps there needs to be no space right after the first asterisk or right before the second asterisk. $\endgroup$ – Ricardo Andrade Aug 12 '13 at 23:57
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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.

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