Bicategories of C*-correspondences as Dwyer-Kan localisations

TL;DR

This paper demonstrates that the bicategory of proper correspondences can be obtained by localizing the category of C*-algebras at specific *-homomorphisms, providing a new perspective on their structure.

Contribution

It establishes that the bicategory of proper correspondences is the Dwyer-Kan localization of the category of C*-algebras at a particular class of *-homomorphisms, linking bicategorical and categorical localizations.

Findings

Bicategory of proper correspondences is a Dwyer-Kan localization.

Identifies the class of *-homomorphisms used for localization.

Provides a new categorical framework for C*-algebras.

Abstract

We show that the bicategory of proper correspondences is the Dwyer-Kan localisation of the category of C*-algebras at a certain class of *-homomorphisms.

Equations

$${\mathcal{E}}_{ij}{\otimes }_{A_j}({\mathcal{E}}_{jk}{\otimes }_{A_k}{\mathcal{E}}_{kl})$$

$${\Gamma }_{\phi }{\otimes }_B{\Gamma }_{\psi }\cong {\Gamma }_{\psi \circ \phi },b\otimes c\mapsto \psi (b)c;$$

$$i_{\mathcal{E}}:B\cong \mathbb{K}(B)\to \mathbb{K}(\mathcal{E}\oplus B).$$

$$\mathbb{K}(\mathcal{E}\oplus B,B){\otimes }_B\mathbb{K}(B,\mathcal{E}\oplus B)\cong \mathbb{K}(\mathcal{E}\oplus B),\mathbb{K}(B,\mathcal{E}\oplus B){\otimes }_{\mathbb{K}(\mathcal{E}\oplus B)}\mathbb{K}(\mathcal{E}\oplus B,B)\cong B.$$

$$B\stackrel{i_1}{\to }\mathbb{K}(\mathcal{E}\oplus B)\stackrel{i_2}{\to }\mathbb{K}(\mathcal{E}\oplus B\oplus {\ell }^2({\mathbb{N}}_{\ge 1},B))\cong \mathbb{K}({\ell }^2(\mathbb{N},B)).$$

$$\mathbb{K}(\mathcal{E}\oplus B,\mathcal{E}\oplus B\oplus {\ell }^2({\mathbb{N}}_{\ge 1},B))$$

$${\Gamma }^{\ast }:\mathrm{Map}(N{\mathfrak{Corr}}_{\mathrm{prop}},\mathcal{D})\to {\mathrm{Map}}_{\mathbb{K}}(N{\mathcal{C}}_{+}^{\ast },\mathcal{D}),F\mapsto F\circ \Gamma .$$

$${\mathcal{E}}_S:={\mathcal{E}}_{i_0,i_k}\oplus {\mathcal{E}}_{i_1,i_k}\oplus \cdots \oplus {\mathcal{E}}_{i_k,i_k},$$

$${\mathcal{E}}_S{\otimes }_{A_{i_k}}{\mathcal{E}}_{i_k,j}\stackrel{\simeq }{\to }\underset{\ell =0}{\overset{k}{⨁}}{\mathcal{E}}_{i_{\ell },j}\subset \underset{i\in T}{⨁}{\mathcal{E}}_{i,j}={\mathcal{E}}_T.$$

$$u_{mjk}((u_{mij}(x\otimes 1_{{\mathcal{E}}_{ij}})u_{lij}^{\ast })\otimes 1_{{\mathcal{E}}_{jk}})u_{ljk}^{\ast }\in \mathbb{K}({\mathcal{E}}_{l,k},{\mathcal{E}}_{m,k}),$$

$$B\cdot \mathbb{K}(\mathcal{E}\oplus B)\cong \mathbb{K}(\mathcal{E}\oplus B,B).$$

$$A\stackrel{f_{\mathcal{E}}}{\to }\mathbb{K}(\mathcal{E}\oplus B)\stackrel{i_{\mathcal{E}}}{\leftarrow }B$$

$$F(A)$$

$$G=G(A_i,{\mathcal{E}}_{ij},u_{ijk}):\mathrm{Bar}{△}^n\to N{\mathcal{C}}_{+}^{\ast }\stackrel{F}{\to }\mathcal{D}.$$

$$(i_0,i_1,\dots ,i_k,S_{k+1},\dots ,S_{\ell })$$

$$\{i_0,\dots ,i_k\}\subseteq S_{k+1}⊊S_{k+2}⊊\cdots ⊊S_{\ell }\subseteq [n].$$

$$\bar{G}=\bar{G}(A_i,{\mathcal{E}}_{ij},u_{ijk}):\overline{\mathrm{Bar}}{△}^n\to \mathcal{D}.$$

$$(A_{\phi (i)},{\mathcal{E}}_{\phi (i),\phi (j)},u_{\phi (i),\phi (j),\phi (k)}),$$

$${\phi }_{\ast }:\overline{\mathrm{Bar}}{△}^m\to \overline{\mathrm{Bar}}{△}^n.$$

$$\bar{G}(A_{\phi (i)},{\mathcal{E}}_{\phi (i)\phi (j)},u_{\phi (i)\phi (j)\phi (k)})=\bar{G}(A_i,{\mathcal{E}}_{ij},u_{ijk})\circ {\phi }_{\ast }.$$

$$\bar{G}(i_0,\dots ,i_k,S_{k+1},\dots ,S_{\ell })$$

$$(i_0,\dots ,i_k,S_{k+1},\dots ,S_{\ell })$$

$$(i_0,\dots ,i_k,S_{k+2},\dots ,S_{\ell })$$

$$(i_0,\dots ,i_k,T_{k+1},\dots ,T_{\ell -1})$$

$$(i_0,\dots ,i_k,\{i_0,\dots ,i_k\},T_{k+1},\dots ,T_{\ell -1}).$$

$$\bar{F}(A_i,{\mathcal{E}}_{ij},u_{ijk}):=\bar{G}(A_i,{\mathcal{E}}_{ij},u_{ijk})(0,1,\dots ,n)$$

$$h:N{\mathcal{C}}_{+}^{\ast }\times {△}^m\to \mathcal{D},\partial H:N{\mathfrak{Corr}}_{\mathrm{prop}}\times \partial {△}^m\to \mathcal{D}$$

$${\Gamma }^{\ast }:\mathrm{Map}(N\mathfrak{Corr},\mathcal{D})\to {\mathrm{Map}}_{\mathbb{K}}(N{\mathcal{C}}_{\mathrm{str}}^{\ast },\mathcal{D}),F\mapsto F\circ \Gamma .$$

$${\Gamma }^{\ast }:\mathrm{Map}(N{\mathfrak{Corr}}_{\mathrm{full}},\mathcal{D})\to {\mathrm{Map}}_{\mathbb{K}}(N{\mathcal{C}}_{\mathrm{full}}^{\ast },\mathcal{D}),F\mapsto F\circ \Gamma ,$$

Full text

1. Introduction

The correspondence bicategory is a bicategory that has $\mathrm{C}^{*}$-algebras as objects, correspondences as arrows, and isomorphisms of correspondences as $2$-arrows. In this article, we use the conventions in [Buss-Meyer-Zhu:Higher_twisted], that is, an $A,B$-correspondence is an arrow from $A$ to $B$ and the composition is the balanced tensor product of $\mathrm{C}^{*}$-correspondences in reverse order. The correspondence bicategory was first introduced to study Morita–Rieffel equivalences of $\mathrm{C}^{*}$-algebras because the latter are just the equivalences in this bicategory (see [Brouwer:Bicategorical, Landsman:Bicategories]). It allows to formulate what it means for a group to act on a $\mathrm{C}^{*}$-algebra by Morita–Rieffel equivalences, and such an action turns out to be the same as a saturated Fell bundle (see [Buss-Meyer-Zhu:Higher_twisted]). There is a bicategorical variant of the usual concept of a (co)limit in a category. The universal property that defines a bicategorical colimit in the correspondence bicategory is identical to the universal property of the crossed product of a group action and equivalent to the universal property of the Cuntz–Pimsner algebra of a proper $\mathrm{C}^{*}$-correspondence or a proper product system (see [Albandik-Meyer:Colimits]). In this way, many important $\mathrm{C}^{*}$-algebraic constructions are unified by viewing them as colimits of different diagrams in the bicategory of proper correspondences. This point of view allows to clarify the functoriality properties of relative Cuntz–Pimsner algebras (see [Meyer-Sehnem:Bicategorical_Pimsner]). The correspondence bicategory has a groupoid analogue, and this allows to study groupoid models for important examples of Cuntz–Pimsner algebras such as the $\mathrm{C}^{*}$-algebras of (self-similar, topological, higher-rank) graphs (see [Albandik:Thesis, Albandik-Meyer:Product, Antunes-Ko-Meyer:Groupoid_correspondences, Meyer:Groupoid_models_relative]).

In higher category theory, a bicategory is only the first step from a category to an $\infty$-category. Many constructions in homotopy theory and homological algebra lead to $\infty$-categories right away. Thus one may wonder whether the correspondence bicategory may be enriched further by some even higher arrows, giving a true $\infty$-category. There is, however, no obvious way to do this. One aim of this article is to explain why this is so. Namely, we are going to show that the quasi-category defined by the proper correspondences is the Dwyer–Kan localisation of the category of $\mathrm{C}^{*}$-algebras and ∗-homomorphisms at the class of corner embeddings. This is remarkable because Dwyer–Kan localisation is a construction whose output is usually an $\infty$-category. In this case, however, this $\infty$-category is just a $2$-category, that is, all higher arrows are trivial. Our result gives a universal property of the proper correspondence bicategory: a functor from the category of $\mathrm{C}^{*}$-algebras and ∗-homomorphisms to a quasi-category or even to an $\infty$-category factors through the proper correspondence bicategory if and only if it is $\mathrm{C}^{*}$-stable, and this factorisation is unique up to equivalence if it exists. This universal property improves upon a known universal property of the truncation of the proper correspondence bicategory to a category (see [Meyer:KK-survey]*Proposition 39).

The description of the proper correspondence bicategory as a Dwyer–Kan localisation also clarifies its relationship to recent constructions of $\infty$-category enrichments of Kasparov’s bivariant KK-theory, which use Dwyer–Kan localisation as well. Recall that Kasparov’s bivariant KK-theory, viewed as a category, is characterised uniquely by a universal property. Namely, the canonical functor from the category of separable $\mathrm{C}^{*}$-algebras and ∗-homomorphisms to the KK-category is the universal $\mathrm{C}^{*}$-stable, split exact, homotopy invariant functor. A functor is $\mathrm{C}^{*}$-stable, split exact, or homotopy invariant if and only if it maps certain ∗-homomorphisms to isomorphisms. Thus KK-theory is a localisation of the category of $\mathrm{C}^{*}$-algebras. Recently, $\infty$-categorical enrichments of KK-theory have been constructed by Bunke, Engel and Land [Bunke-Engel-Land:Stable_infinity-category_KK] using Dwyer–Kan localisation. This leads to a stable $\infty$-category whose homotopy category is equivalent to KK-theory as a triangulated category.

In this article, we describe $\infty$-categories as quasi-categories, that is, simplicial sets in which all inner horns may be filled (see [Joyal:Quasi-categories]*Definition 1.1). The nerve of a category or of a bicategory with invertible $2$-arrows is a quasi-category in this sense. Thus we view the category of $\mathrm{C}^{*}$-algebras and the correspondence bicategory as quasi-categories. A functor of categories or a homomorphism of bicategories induces a simplicial map between their nerves, and an equivalence of functors or bicategory homomorphisms induces a homotopy between these maps. Our main theorem describes homotopy classes of simplicial maps from the nerve of the proper correspondence bicategory to any other quasi-category $\mathcal{D}$ as homotopy classes of simplicial maps from the category of $\mathrm{C}^{*}$-algebras to $\mathcal{D}$ that map all corner embeddings to equivalences in $\mathcal{D}$. This universal property is also used to define the Dwyer–Kan localisation of a quasi-category at a family of morphisms.

Our main theorem treats only proper correspondences because it starts with ∗-homomorphisms. In the final section, we briefly discuss variants of it that lead to the bicategories of all correspondences or all full correspondences. These start with (full) strictly continuous ∗-homomorphisms on multiplier algebras. In the following, we state our results in the setting of $\mathrm{C}^{*}$-algebras without extra structure for simplicity. We may, however, generalise all our arguments to $\mathrm{C}^{*}$-algebras with extra structure such as an action of a locally compact groupoid, a quantum group, or various other kinds of extra structure. For these variants, we merely have to ask all ∗-homomorphisms and $\mathrm{C}^{*}$-correspondences to be equivariant with respect to the chosen extra structure.

This article is based on joint work with Rohan Lean a couple of years ago, which was intended for a book project which is taking an awfully long time to complete. I decided that the results in this article deserve to be published now because of recent work on Dwyer–Kan localisations of the category of $\mathrm{C}^{*}$-algebras.

2. The nerve of the proper correspondence bicategory

Let $\mathcal{C}^{*}_{+}$ be the category of $\mathrm{C}^{*}$-algebras and ∗-homomorphisms. Let $N\mathcal{C}^{*}_{+}$ be its nerve. This is $\mathcal{C}^{*}_{+}$ viewed as a quasi-category. Let $\mathfrak{Corr}_{\mathrm{prop}}$ denote the bicategory of proper correspondences (see [Buss-Meyer-Zhu:Higher_twisted], where bicategories are called weak $2$-categories). It has $\mathrm{C}^{*}$-algebras as objects, proper $A,B$-correspondences as arrows $A\to B$, and isomorphisms of $A,B$-correspondences as $2$-arrows (called bigons in [Buss-Meyer-Zhu:Higher_twisted]). The composition of arrows is the balanced tensor product of $\mathrm{C}^{*}$-correspondences in reverse order. Beware that I am using the conventions of [Buss-Meyer-Zhu:Higher_twisted] here, although in more recent articles, I have advocated to view an $A,B$-correspondence as an arrow from $B$ to $A$. It is important to allow only isomorphisms of $\mathrm{C}^{*}$-correspondences as $2$-arrows as in [Buss-Meyer-Zhu:Higher_twisted] in order to turn this bicategory into a quasi-category.

The nerve $N\mathcal{C}$ of a bicategory $\mathcal{C}$ is described by Duskin [Duskin:Simplicial], generalising the nerve of a category. Duskin also proves that it is a quasi-category (see [Duskin:Simplicial]). Duskin’s definition leads to the following description of the nerve:

Definition 2.1**.**

Let $N\mathfrak{Corr}_{\mathrm{prop}}$ be the nerve of the bicategory of proper correspondences. An $n$-simplex in $N\mathfrak{Corr}_{\mathrm{prop}}$ is given by $\mathrm{C}^{*}$-algebras $A_{i}$ for $0\leq i\leq n$, $\mathrm{C}^{*}$-correspondences $\mathcal{E}_{ij}\colon A_{i}\to A_{j}$ for $0\leq i\leq j\leq n$, and isomorphisms of $\mathrm{C}^{*}$-correspondences $u_{ijk}\colon\mathcal{E}_{ij}\otimes_{A_{j}}\mathcal{E}_{jk}\to\mathcal{E}_{ik}$ for $0\leq i\leq j\leq k\leq n$, such that

1. (1)

$\mathcal{E}_{ii}$ is the identity correspondence $A_{i}$ and $u_{iik}$ and $u_{ikk}$ are the canonical isomorphisms $A_{i}\otimes_{A_{i}}\mathcal{E}_{ik}\xrightarrow{\simeq}\mathcal{E}_{ik}$ and $\mathcal{E}_{ik}\otimes_{A_{k}}A_{k}\xrightarrow{\simeq}\mathcal{E}_{ik}$ for $0\leq i\leq k\leq n$; 2. (2)

if $0\leq i\leq j\leq k\leq l\leq n$, then the following diagram of isomorphisms of $\mathrm{C}^{*}$-correspondences commutes:

[TABLE]

An increasing map $\varphi\colon[n]\to[m]$ induces a map from the set of $m$-simplices to the set of $n$-simplices by mapping $(A_{i},\mathcal{E}_{ij},u_{ijk})$ to $(A_{\varphi(i)},\mathcal{E}_{\varphi(i)\varphi(j)},u_{\varphi(i)\varphi(j)\varphi(k)})$.

A ∗-homomorphism $\varphi\colon A\to B$ induces a proper $A,B$-correspondence $\Gamma_{\varphi}$ as follows: its underlying Hilbert $B$-module is the right ideal $\Gamma_{\varphi}\mathrel{\vcentcolon=}\varphi(A)\cdot B\subseteq B$, with $A$ acting nondegenerately on the left by $a\cdot b\mathrel{\vcentcolon=}\varphi(a)b$ for $a\in A$, $b\in\Gamma_{\varphi}$. For two composable ∗-homomorphisms $\varphi\colon A\to B$ and $\psi\colon B\to C$, there is a canonical isomorphism

[TABLE]

this map is isometric, and $\psi\circ\varphi(A)\psi(B)C=\psi\circ\varphi(A)C$ implies that it is surjective. The isomorphisms in (2) make $\varphi\mapsto\Gamma_{\varphi}$ into a homomorphism of bicategories $\mathcal{C}^{*}_{+}\to\mathfrak{Corr}_{\mathrm{prop}}$. So it induces a simplicial map $\Gamma\colon N\mathcal{C}^{*}_{+}\to N\mathfrak{Corr}_{\mathrm{prop}}$.

3. $\mathrm{C}^{*}$-stable functors

Let $\mathcal{D}$ be a quasi-category. Let $F\colon N\mathcal{C}^{*}_{+}\to\mathcal{D}$ be a simplicial map, which we consider as a functor $\mathcal{C}^{*}_{+}\to\mathcal{D}$.

Definition 3.1**.**

For a $\mathrm{C}^{*}$-algebra $B$ and a Hilbert $B$-module $\mathcal{E}$, let $\mathcal{E}\oplus B$ be the direct sum of Hilbert modules. We define the corner embedding for $\mathcal{E}$ to be the canonical inclusion map

[TABLE]

We call a functor $F\colon N\mathcal{C}^{*}_{+}\to\mathcal{D}$ $\mathrm{C}^{*}$-stable if $F(i_{\mathcal{E}})$ is an equivalence for each Hilbert module $\mathcal{E}$.

Example 3.2*.*

The canonical functor $\Gamma\colon N\mathcal{C}^{*}_{+}\to N\mathfrak{Corr}_{\mathrm{prop}}$ is $\mathrm{C}^{*}$-stable. To see this, we first identify $\Gamma(i_{\mathcal{E}})$ with the proper correspondence $\mathbb{K}(\mathcal{E}\oplus B,B)$. This is an equivalence, its quasi-inverse is $\mathbb{K}(B,\mathcal{E}\oplus B)$. That is,

[TABLE]

We may also speak of $\mathrm{C}^{*}$-stable functors on subcategories of $\mathcal{C}^{*}_{+}$ such as the subcategory of $\sigma$-unital or of separable $\mathrm{C}^{*}$-algebras. In these cases, we may simplify the definition, replacing the corner embeddings for general Hilbert modules by a smaller class of “special” corner embeddings:

Lemma 3.3**.**

A functor on the category of $\sigma$-unital $\mathrm{C}^{*}$-algebras is $\mathrm{C}^{*}$-stable if and only if it maps the special corner embeddings $B\to\mathbb{K}(\ell^{2}\mathbb{N})\otimes B$, $b\mapsto E_{00}\otimes b$, to equivalences.

Proof.

The special corner embeddings are the corner embeddings for the Hilbert modules $\mathbb{K}(\ell^{2}(\mathbb{N}_{\geq 1})\otimes B)$, and so a $\mathrm{C}^{*}$-stable functor maps them to equivalences. Conversely, let us assume that the functor $F$ maps the special corner embeddings to equivalences. Let $B$ be a $\sigma$-unital $\mathrm{C}^{*}$-algebra and let $\mathcal{E}$ be a Hilbert module such that $\mathbb{K}(\mathcal{E}\oplus B)$ is also $\sigma$-unital. This is equivalent to $\mathcal{E}\oplus B$ being countably generated as a Hilbert module over $B$ (see [Lance:Hilbert_modules]*Proposition 6.7). Then Kasparov’s Stabilisation Theorem applies and shows that $\mathcal{E}\oplus\ell^{2}(\mathbb{N}_{\geq 1},B)\cong\ell^{2}(\mathbb{N}_{\geq 1},B)$ (see [Lance:Hilbert_modules]*Theorem 6.2). We consider the extension by zero maps

[TABLE]

The composite $i_{3}\mathrel{\vcentcolon=}i_{2}\circ i_{1}$ is a special corner embedding, so that $F(i_{3})$ is an equivalence. Then $F(i_{2})\circ F(i_{1})$ is an equivalence as well. We may turn

[TABLE]

into a Hilbert module over $\mathbb{K}(\mathcal{E}\oplus B)$ that contains $\mathbb{K}(\mathcal{E}\oplus B)$ as a direct summand, and the compact operators on this Hilbert module are $\mathbb{K}(\mathcal{E}\oplus B\oplus\ell^{2}(\mathbb{N}_{\geq 1},B))$. Thus $i_{2}$ is also a corner embedding. The argument above provides another extension-by-zero map $i_{4}\colon\mathbb{K}(\mathcal{E}\oplus B\oplus\ell^{2}(\mathbb{N}_{\geq 1},B))\to\mathbb{K}((\mathcal{E}\oplus B)\otimes\ell^{2}(\mathbb{N}))$ so that $i_{5}\mathrel{\vcentcolon=}i_{4}\circ i_{2}$ is a special corner embedding, and then $F(i_{4})\circ F(i_{2})$ is an equivalence. Now the Two-Out-Of-Six Lemma for the composition of $F(i_{4})$, $F(i_{2})$ and $F(i_{1})$ implies that $F(i_{1})$ is an equivalence. ∎

For two simplicial sets $X$ and $Y$, let $\operatorname{Map}(X,Y)$ denote the simplicial set of simplicial maps $X\times{\vartriangle^{\!n}}\to Y$. This is a quasi-category if $Y$ is one (see [Lurie:Higher_Topos]*Proposition 1.2.7.3.(1), proven in [Lurie:Higher_Topos]*Section 2.2.5). In particular, $\operatorname{Map}(N\mathcal{C}^{*}_{+},\mathcal{D})$ and $\operatorname{Map}(N\mathfrak{Corr}_{\mathrm{prop}},\mathcal{D})$ are quasi-categories for any quasi-category $\mathcal{D}$; their $n$-simplices are simplicial maps $N\mathcal{C}^{*}_{+}\times{\vartriangle^{\!n}}\to\mathcal{D}$ and $N\mathfrak{Corr}_{\mathrm{prop}}\times{\vartriangle^{\!n}}\to\mathcal{D}$, respectively. We are interested in the full subcategory $\operatorname{Map}_{\mathbb{K}}(N\mathcal{C}^{*}_{+},\mathcal{D})$ of $\mathrm{C}^{*}$-stable simplicial maps $N\mathcal{C}^{*}_{+}\to\mathcal{D}$; its $n$-simplices are those simplicial maps $N\mathcal{C}^{*}_{+}\times{\vartriangle^{\!n}}\to\mathcal{D}$ for which the adjunct simplicial map $N\mathcal{C}^{*}_{+}\to\operatorname{Map}({\vartriangle^{\!n}},\mathcal{D})$ is $\mathrm{C}^{*}$-stable.

Theorem 3.4**.**

Let $\mathcal{D}$ be a quasi-category. The following simplicial map is a homotopy equivalence:

[TABLE]

This theorem is our main theorem. It says in elementary language that the proper correspondence quasi-category $N\mathfrak{Corr}_{\mathrm{prop}}$ is the Dwyer–Kan localisation of the category of $\mathrm{C}^{*}$-algebras $\mathcal{C}^{*}_{+}$ at the class of corner embeddings.

Since $\Gamma$ is $\mathrm{C}^{*}$-stable, $F\circ\Gamma$ is $\mathrm{C}^{*}$-stable for any simplicial map $F\colon N\mathfrak{Corr}_{\mathrm{prop}}\to\mathcal{D}$. Hence $\Gamma^{*}$ is a simplicial map to $\operatorname{Map}_{\mathbb{K}}(N\mathcal{C}^{*}_{+},\mathcal{D})$ as asserted.

4. Proof of the main theorem

The following proposition is a key first step in the proof:

Proposition 4.1**.**

For any $F\in\operatorname{Map}_{\mathbb{K}}(N\mathcal{C}^{*}_{+},\mathcal{D})$ there is $\bar{F}\in\operatorname{Map}(N\mathfrak{Corr}_{\mathrm{prop}},\mathcal{D})$ with $\bar{F}\circ\Gamma=F$.

To prove this, we need the barycentric subdivision $\operatorname{Bar}{\vartriangle^{\!n}}$ of ${\vartriangle^{\!n}}$. This simplicial set has nonempty subsets $S\subseteq[n]$ as vertices and chains of inclusions $S_{0}\subseteq S_{1}\subseteq S_{2}\subseteq S_{3}\subseteq\cdots\subseteq S_{n}$ as $n$-simplices. Thus $\operatorname{Bar}{\vartriangle^{\!n}}$ is the nerve of the partially ordered set $\mathrm{Sub}{[n]}$ of nonempty subsets of $[n]$, viewed as a category by putting a unique arrow $S\to T$ for $S\subseteq T$ and no arrow otherwise.

Let $(A_{i},\mathcal{E}_{ij},u_{ijk})$ be an $n$-simplex in the nerve $N\mathfrak{Corr}_{\mathrm{prop}}$ as in Definition 2.1. We are going to construct a functor $\operatorname{Bar}{\vartriangle^{\!n}}\to N\mathcal{C}^{*}_{+}$ from it. Let $S=\{i_{0},\dotsc,i_{k}\}\subseteq[n]$ with $i_{0}<\dotsb<i_{k}$. Then we define a Hilbert $A_{i_{k}}$-module by

[TABLE]

and we let $A_{S}\mathrel{\vcentcolon=}\mathbb{K}(\mathcal{E}_{S})$.

First let $S\subseteq T$ satisfy $\max S=\max T$. Then both $\mathcal{E}_{S}$ and $\mathcal{E}_{T}$ are Hilbert modules over the same $\mathrm{C}^{*}$-algebra. There is an obvious adjointable inclusion map $\mathcal{E}_{S}\hookrightarrow\mathcal{E}_{T}$, which induces a ∗-homomorphism $f_{ST}\colon A_{S}\to A_{T}$ by letting an operator on $\mathcal{E}_{S}$ act as zero on the orthogonal complement of $\mathcal{E}_{S}$ in $\mathcal{E}_{T}$. By the definition of a simplex in $N\mathfrak{Corr}_{\mathrm{prop}}$, $\mathcal{E}_{i_{k},i_{k}}$ is the identity correspondence on $A_{i_{k}}$; thus the embedding $A_{i_{k}}\to A_{S}$ is a corner embedding as in Definition 3.1.

Now consider $S\subseteq T$ with $\max S\neq\max T$. Set $j\mathrel{\vcentcolon=}\max T$. The unitaries $u_{i_{\ell},i_{k},j}\colon\mathcal{E}_{i_{\ell},i_{k}}\otimes_{A_{i_{k}}}\mathcal{E}_{i_{k},j}\to\mathcal{E}_{i_{\ell},j}$ for $\ell=0,\dotsc,k$ provide an adjointable embedding

[TABLE]

We define $f_{ST}\colon A_{S}\to A_{T}$ by first mapping $x\mapsto x\otimes\mathrm{id}_{\mathcal{E}_{i_{k},j}}$ and then extending by zero on the orthogonal complement of the direct summand $\mathcal{E}_{S}\otimes_{A_{i_{k}}}\mathcal{E}_{i_{k},j}$ in $\mathcal{E}_{T}$.

Lemma 4.2**.**

The maps $f_{ST}\colon A_{S}\to A_{T}$ for $S\subseteq T$ form a functor $\operatorname{Bar}{\vartriangle^{\!n}}\to N\mathcal{C}^{*}_{+}$.

Proof.

We must show that $f_{TU}\circ f_{ST}=f_{SU}$ for $S\subseteq T\subseteq U$. Let $i$, $j$ and $k$ be the maxima of $S$, $T$ and $U$, respectively. The two maps $f_{TU}\circ f_{ST}$ and $f_{SU}$ map the $\mathrm{C}^{*}$-algebra of compact operators on the direct sum of $\mathcal{E}_{l,i}$ for $l\in S$ to the $\mathrm{C}^{*}$-algebra of compact operators on the direct sum of $\mathcal{E}_{l,k}$ for $l\in U$. It suffices to check $f_{TU}\circ f_{ST}(x)=f_{SU}(x)$ when $x$ is an operator in $\mathbb{K}(\mathcal{E}_{l,i},\mathcal{E}_{m,i})$ extended by zero on the other summands. By definition, $f_{SU}$ maps this to the operator $u_{mik}(x\otimes 1_{\mathcal{E}_{ik}})u_{lik}^{*}$ in $\mathbb{K}(\mathcal{E}_{l,k},\mathcal{E}_{m,k})$, again extended by zero on the other summands, whereas $f_{TU}\circ f_{ST}$ maps it to

[TABLE]

extended by zero. These operators are equal by (1). ∎

Lemma 4.3**.**

Let $\mathcal{E}$ be a proper $A,B$-correspondence. Define a ∗-homomorphism $f_{\mathcal{E}}\colon A\to\mathbb{K}(\mathcal{E})\subset\mathbb{K}(\mathcal{E}\oplus B)$ from the left $A$-action on $\mathcal{E}$ and let $i_{\mathcal{E}}\colon B\to\mathbb{K}(\mathcal{E}\oplus B)$ be the corner embedding. Let $\Gamma(f_{\mathcal{E}})$ and $\Gamma(i_{\mathcal{E}})$ denote the associated proper correspondences $A\to\mathbb{K}(\mathcal{E}\oplus B)$ and $B\to\mathbb{K}(\mathcal{E}\oplus B)$. There is a canonical isomorphism of correspondences $u_{\mathcal{E}}\colon\mathcal{E}\otimes_{B}\Gamma(i_{\mathcal{E}})\cong\Gamma(f_{\mathcal{E}})$.

Proof.

We may identify the underlying Hilbert $\mathbb{K}(\mathcal{E}\oplus B)$-module of $\Gamma(i_{\mathcal{E}})$ with

[TABLE]

This becomes a correspondence isomorphism when $B$ and $\mathbb{K}(\mathcal{E}\oplus B)$ act on $\mathbb{K}(\mathcal{E}\oplus B,B)$ by left and right multiplication, respectively. Similarly, $\Gamma(f_{\mathcal{E}})\cong\mathbb{K}(\mathcal{E}\oplus B,\mathcal{E})$ with the obvious Hilbert module structure over $\mathbb{K}(\mathcal{E}\oplus B)$ and $A$ acting on the left by multiplication with the operator in $\mathbb{K}(\mathcal{E})$ from the left multiplication in the proper $A,B$-correspondence $\mathcal{E}$.

There is an isometry $\mathcal{E}\otimes_{B}\mathbb{K}(\mathcal{E}\oplus B,B)\cong\mathbb{K}(\mathcal{E}\oplus B,\mathcal{E})$, mapping $\xi\otimes\eta$ to the rank-one operator $\lvert\xi\rangle\circ\eta\colon\mathcal{E}\oplus B\to B\to\mathcal{E}$; here $\lvert\xi\rangle(b)\mathrel{\vcentcolon=}\xi\cdot b$. A quick calculation shows that this is a correspondence isomorphism $\mathcal{E}\otimes_{B}\Gamma(i_{\mathcal{E}})\cong\Gamma(f_{\mathcal{E}})$. ∎

The diagram

[TABLE]

in Lemma 4.3 is exactly the functor $\operatorname{Bar}{\vartriangle^{\!1}}\to N\mathcal{C}^{*}_{+}$ constructed from the arrow $\mathcal{E}$ in $\mathfrak{Corr}_{\mathrm{prop}}$.

Let $\bar{F}\colon N\mathfrak{Corr}_{\mathrm{prop}}\to\mathcal{D}$ be a simplicial map to a quasi-category that extends a $\mathrm{C}^{*}$-stable simplicial map $F\colon N\mathcal{C}^{*}_{+}\to\mathcal{D}$. Then it maps the triangle for the isomorphism $u_{\mathcal{E}}$ in Lemma 4.3 to a triangle

[TABLE]

in $\mathcal{D}$. This gives a recipe to construct $\bar{F}(\mathcal{E})$. If $F$ is $\mathrm{C}^{*}$-stable, then $F(i_{\mathcal{E}})$ is an equivalence in $\mathcal{D}$. We may use a quasi-inverse of it to complete the pair of maps $F(f_{\mathcal{E}})$, $F(i_{\mathcal{E}})$ to a triangle as in (4). Hence we get $\bar{F}(\mathcal{E})$ for all arrows in $\mathfrak{Corr}_{\mathrm{prop}}$, together with triangles $\bar{F}(u_{\mathcal{E}})$.

Next we use the functors $\operatorname{Bar}{\vartriangle^{\!n}}\to N\mathcal{C}^{*}_{+}$ to construct $\bar{F}$ on $n$-simplices in $N\mathfrak{Corr}_{\mathrm{prop}}$. This is a more complicated version of the triangle filling construction above with several intermediate steps, which require some bookkeeping to make sure that each step is possible and that they all taken together well-define a simplicial map.

Let $A_{i}$ for $0\leq i\leq n$, $\mathcal{E}_{ij}\colon A_{i}\to A_{j}$ for $0\leq i\leq j\leq n$ and $u_{ijk}\colon\mathcal{E}_{ij}\otimes_{A_{j}}\mathcal{E}_{jk}\to\mathcal{E}_{ik}$ for $0\leq i\leq j\leq k\leq n$ give an $n$-simplex in $N\mathfrak{Corr}_{\mathrm{prop}}$. Lemma 4.2 shows that the construction above the lemma yields a simplicial map

[TABLE]

In order to define $\bar{F}$ on our given $n$-simplex, we extend $G$ to a larger simplicial set $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$, namely, the nerve of the partially ordered set that we get from $\mathrm{Sub}{[n]}$ by adjoining the relations $\{i\}\leq\{j\}$ for all $0\leq i\leq j\leq n$. Thus $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$ has $\ell$-simplices

[TABLE]

for all $0\leq i_{0}\leq i_{1}\leq\cdots\leq i_{k}\leq n$ and $S_{k+1}\subseteq\cdots\subseteq S_{\ell}\subseteq[n]$ with $i_{0},\dotsc,i_{k}\in S_{k+1}$ and $\lvert S_{k+1}\rvert\geq 2$, $-1\leq k\leq\ell$; the cases $k=-1$ or $k=\ell$ are allowed. Those simplices with $k\leq 0$ already belong to $\operatorname{Bar}{\vartriangle^{\!n}}$. As usual, the $j$th face map leaves out the $j$th entry in the list, and the $j$th degeneracy map doubles the $j$th entry. Thus a simplex is nondegenerate if $0\leq i_{0}<i_{1}<\cdots<i_{k}\leq n$ and

[TABLE]

We declare $S_{k}\mathrel{\vcentcolon=}\{i_{0},\dotsc,i_{k}\}$, so that $S_{\ell}$ is defined also for $k=\ell$.

We denote the desired extension of $G$ by

[TABLE]

We will only use the value of $\bar{G}$ on the $n$-simplex $(0,\dotsc,n)$ in $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$ to define $\bar{F}$; the values on the other simplices of $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$ are needed to construct this value and to relate it to the given functor $F$.

We want the maps $\bar{G}$ for different simplices in $N\mathfrak{Corr}_{\mathrm{prop}}$ to be compatible with the face and degeneracy maps of $N\mathfrak{Corr}_{\mathrm{prop}}$ in the following way. Let $\varphi\colon[m]\to[n]$ be an order-preserving map. This maps the $n$-simplex $(A_{i},\mathcal{E}_{ij},u_{ijk})$ to an $m$-simplex

[TABLE]

and it induces a simplicial map

[TABLE]

We want

[TABLE]

If the given simplex $(A_{i},\mathcal{E}_{ij},u_{ijk})$ is degenerate, then (5) for surjective maps $\varphi$ determines $\bar{G}$ by its values on simplices in $N\mathfrak{Corr}_{\mathrm{prop}}$ of smaller dimension. Since we are going to construct $\bar{G}$ inductively over the dimension of the simplices, there is nothing to do for degenerate simplices and we may assume from now on that $(A_{i},\mathcal{E}_{ij},u_{ijk})$ is nondegenerate. In addition, it then suffices to check (5) for injective maps.

The condition (5) determines

[TABLE]

if $S_{\ell}\neq[n]$ because then there is $\varphi\colon[n-1]\to[n]$ with $(i_{0},\dotsc,S_{\ell})\in\varphi_{*}(\operatorname{\overline{Bar}}{\vartriangle^{\!n-1}})$. Since $\bar{G}$ must extend $G$, we already know (6) if $k\leq 0$. These simplices form a subcomplex of $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$. The prescription for $\bar{G}$ on this subcomplex is a simplicial map. We must extend it to $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$.

It remains to define (6) for simplices in $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$ with $k\geq 1$ and $S_{\ell}=[n]$. We do this by a recursion through $k=1,\dotsc,n$. When constructing (6) for $k\geq 1$, we assume that $\bar{G}$ is already given on simplices in $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$ with smaller value of $k$ (which is indeed so for $k=1$). For fixed $k$, we run another recursion on $\ell$. In each step, we fill a horn to construct (6) if $S_{k+1}=\{i_{0},\dotsc,i_{k}\}$ and $S_{k+1}\subsetneq S_{k+2}\subsetneq\cdots\subsetneq S_{\ell}=[n]$, assuming that (6) has been constructed for smaller values of $\ell$.

Assume first that $S_{k+1}=\{i_{0},\dotsc,i_{k}\}$. The $j$th face of

[TABLE]

for $0\leq j\leq k$ has a smaller value of $k$, so that $\bar{G}$ on this face has been constructed. The $j$th face of (7) for $k+2\leq j\leq\ell$ has a smaller value of $\ell$ and still contains $S_{k+1}=\{i_{0},\dotsc,i_{k}\}$, so that $\bar{G}$ on this face has been constructed. The values of $\bar{G}$ on these boundary faces give an $(\ell,k+1)$-horn in $\mathcal{D}$. If $\ell>k+1$, this is an inner horn and so it may be filled in the quasi-category $\mathcal{D}$. Let $\ell=k+1$. The arrow $i_{k}\to S_{k+1}=\{i_{0},\dotsc,i_{k}\}$ induces a corner embedding, and $G$ maps this to an equivalence in $\mathcal{D}$ by assumption; hence we have got a special $(\ell,\ell)$-horn, and these may be filled in all quasi-categories (see [Joyal:Quasi-categories]*Theorem 1.3). (By the way, the case $\ell=k+1$ only occurs if $k=n$ because $S_{\ell}=[n]$.)

The above recursion for fixed $k$ gives (6) for all nondegenerate simplices in $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$ with $S_{k+1}=\{i_{0},\dotsc,i_{k}\}$ and $S_{\ell}=[n]$. It also gives $\bar{G}$ on the missing $k+1$st face

[TABLE]

of (7). Any nondegenerate $\ell-1$-simplex

[TABLE]

in $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$ with $T_{\ell-1}=[n]$ and $T_{k+1}\neq\{i_{0},\dotsc,i_{k}\}$ is of this form for a unique $(i_{0},\dotsc,S_{\ell})$, namely,

[TABLE]

Hence our recursion defines $\bar{G}(i_{0},\dotsc,S_{\ell})$ for all nondegenerate simplices in $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$ with $S_{\ell}=[n]$. We extend the map to degenerate simplices in the unique way dictated by the degeneracy maps and take this together with the values already known for $S_{\ell}\subsetneq[n]$. This defines $\bar{G}$ on all simplices of $\operatorname{\overline{Bar}}{\vartriangle^{\!n}}$. Since $\bar{G}$ satisfies (5),

[TABLE]

defines a simplicial map $\bar{F}\colon N\mathfrak{Corr}_{\mathrm{prop}}\to\mathcal{D}$ with $\bar{F}\circ\Gamma=F$. This finishes the proof of Proposition 4.1.

Let us examine the construction for $n=1$ and $n=2$ to understand better what is happening.

For $n=1$, there is only one recursion step $k=1$, $\ell=2$, which requires one $(2,2)$-horn to be filled. This is exactly the construction of the triangle in (4). The value $\bar{F}(\mathcal{E})$ on the edge $\mathcal{E}$ in $N\mathfrak{Corr}_{\mathrm{prop}}$ is indicated in (4).

For $n=2$, Lemma 4.2 defines $\bar{G}$ on the six triangles that make up the barycentric subdivision $\operatorname{Bar}{\vartriangle^{\!n}}$ of a triangle, see the top diagram in Figure 1.

The construction for $1$-simplices ($n=1$) has already defined $\bar{G}$ on the three triangles $i_{0}\to i_{1}\to\{i_{0},i_{1}\}$ for $(i_{0},i_{1})\in\{(0,1),(0,2),(1,2)\}$ that are added in the bottom left diagram in Figure 1. The three 60∘ corners in this figure give the $(3,2)$-horns in the tetrahedra $i_{0}<i_{1}<\{i_{0},i_{2}\}<\{0,1,2\}$ for $(i_{0},i_{1})\in\{(0,1),(0,2),(1,2)\}$. Our construction for $n=2$, $k=1$ fills these horns. This also gives values for $\bar{G}$ on the $2$-boundary faces $i_{0}<i_{1}<\{0,1,2\}$ of these tetrahedra. These triangles are shown in the bottom right figure of Figure 1. They are mapped by $\bar{G}$ to a $(3,3)$-horn in $\mathcal{D}$. This horn is special because the ∗-homomorphism $A_{\{2\}}\to A_{\{0,1,2\}}$, which is associated to the map $2\to 012$, is a corner embedding. Therefore, this horn in $\mathcal{D}$ may be filled. This is done in the recursion step for $k=2$, which gives the value of $\bar{G}$ on the triangle $0\to 1\to 2$. This is the value of $\bar{F}$ on the given triangle in $N\mathfrak{Corr}_{\mathrm{prop}}$.

Let $m\in\mathbb{N}$. Since $\operatorname{Map}(\partial{\vartriangle^{\!m}},\mathcal{D})$ is a quasi-category as well and the internal hom is adjoint to the product of simplicial sets, Proposition 4.1 implies that any simplicial map $f\colon N\mathcal{C}^{*}_{+}\times\partial{\vartriangle^{\!m}}\to\mathcal{D}$ factors as $F\circ(\Gamma\times\mathrm{id}_{\partial{\vartriangle^{\!m}}})$ for a simplicial map $F\colon N\mathfrak{Corr}_{\mathrm{prop}}\times\partial{\vartriangle^{\!m}}\to\mathcal{D}$. That is, the map $\Gamma$ induces surjections on all homotopy groups. To prove that $\Gamma$ is a homotopy equivalence, it remains to prove that it also induces injective maps on all homotopy groups. To prove this, we show that, given simplicial maps

[TABLE]

with $h\circ(\mathrm{id}\times\iota)=\partial H\circ(\Gamma\times\mathrm{id})$, where $\iota\colon\partial{\vartriangle^{\!m}}\to{\vartriangle^{\!m}}$ is the inclusion map, there is a simplicial map $H\colon N\mathfrak{Corr}_{\mathrm{prop}}\times{\vartriangle^{\!m}}\to\mathcal{D}$ with $H\circ(\mathrm{id}\times\iota)=\partial H$ and $H\circ(\Gamma\times\mathrm{id})=h$.

The construction of $H$ is a variant of the one above. We have to extend a given map from $N\mathfrak{Corr}_{\mathrm{prop}}\times\partial{\vartriangle^{\!m}}\cup\Gamma(N\mathcal{C}^{*}_{+})\times{\vartriangle^{\!m}}$ to $\mathcal{D}$ to a map defined on $N\mathfrak{Corr}_{\mathrm{prop}}\times{\vartriangle^{\!m}}$; we do this by defining the values on $n$-simplices by induction over $n$. We only construct something new for a nondegenerate $n$-simplex $s$ in $N\mathfrak{Corr}_{\mathrm{prop}}\times{\vartriangle^{\!m}}$ that is not contained in $N\mathfrak{Corr}_{\mathrm{prop}}\times\partial{\vartriangle^{\!m}}\cup\Gamma(N\mathcal{C}^{*}_{+})\times{\vartriangle^{\!m}}$. Write $s=(s_{1},s_{2})$ with $n$-simplices $s_{1}$ and $s_{2}$ in $N\mathfrak{Corr}_{\mathrm{prop}}$ and ${\vartriangle^{\!m}}$. The simplex $s_{1}$ gives a simplicial map $G^{\prime}\colon\operatorname{Bar}{\vartriangle^{\!n}}\to N\mathcal{C}^{*}_{+}$ as above, and $(G^{\prime},s_{2})\colon\operatorname{Bar}{\vartriangle^{\!n}}\to N\mathcal{C}^{*}_{+}\times{\vartriangle^{\!m}}$ may be composed with $h$ to get a simplicial map $G\colon\operatorname{Bar}{\vartriangle^{\!n}}\to\mathcal{D}$. As above, we extend this to a map $\bar{G}\colon\operatorname{\overline{Bar}}{\vartriangle^{\!n}}\to\mathcal{D}$, such that $\bar{G}$ as a function of $s$ is compatible with the face maps. This construction is exactly the same extension problem as above. Having constructed $\bar{G}$, we let $H(s)\mathrel{\vcentcolon=}\bar{G}(0,\dotsc,n)$. This defines a simplicial map $H\colon N\mathfrak{Corr}_{\mathrm{prop}}\times{\vartriangle^{\!m}}\to\mathcal{D}$ with the required properties.

What we have just proved says that $\Gamma_{*}$ induces an isomorphism on all homotopy groups. Thus $\Gamma_{*}$ is a homotopy equivalence by general results in homotopy theory.

This finishes the proof of Theorem 3.4.

5. Some variants of the main theorem

We have interpreted the proper correspondence bicategory as a Dwyer–Kan localisation of the category of $\mathrm{C}^{*}$-algebras and ∗-homomorphisms. There is a similar interpretation for the bicategory of all correspondences, proper or not.

An $A,B$-correspondence with underlying Hilbert module $B$ is the same as a nondegenerate ∗-homomorphism from $A$ to the multiplier algebra of $B$, briefly called a morphism from $A$ to $B$. To formulate $\mathrm{C}^{*}$-stability, we need the corner embeddings $i_{\mathcal{E}}\colon B\to\mathbb{K}(\mathcal{E}\oplus B)$, which are degenerate. Hence $\mathrm{C}^{*}$-stability makes no sense for functors on the morphism category $\mathcal{C}^{*}$: we must allow certain degenerate morphisms.

A convenient choice is to take the strictly continuous ∗-homomorphisms $A\to\mathcal{M}(B)$. These are those ∗-homomorphisms $\varphi\colon A\to\mathcal{M}(B)$ that extend to a strictly continuous ∗-homomorphism $\bar{\varphi}\colon\mathcal{M}(A)\to\mathcal{M}(B)$; here we use that $\mathcal{M}(A)$ is the completion of $A$ in the strict topology (see [Busby:Double_centralizer]). Therefore, the strictly continuous ∗-homomorphisms $A\to\mathcal{M}(B)$ still form a category, which we denote by $\mathcal{C}^{*}_{\mathrm{str}}$.

The extension $\bar{\varphi}\colon\mathcal{M}(A)\to\mathcal{M}(B)$ maps $1\in\mathcal{M}(A)$ to a projection $p\in\mathcal{M}(B)$, and $\bar{\varphi}\colon\mathcal{M}(A)\to p\mathcal{M}(B)p=\mathcal{M}(pBp)$ is a unital, strictly continuous ∗-homomorphism. Thus $\varphi(A)\subseteq\mathcal{M}(pBp)$ and $\varphi$ is nondegenerate as a ∗-homomorphism to $\mathcal{M}(pBp)$. A $\mathrm{C}^{*}$-subalgebra of $B$ of the form $pBp$ is called a corner in $B$. The strictly continuous ∗-homomorphisms $A\to\mathcal{M}(B)$ are therefore the same as the morphisms from $A$ to corners in $B$.

An arrow $\varphi$ in $\mathcal{C}^{*}_{\mathrm{str}}$ gives a correspondence, namely, $pB\subseteq B$ with the nondegenerate left $A$-action given by $\varphi$. As for $\mathcal{C}^{*}_{+}$, this identifies the nerve $N\mathcal{C}^{*}_{\mathrm{str}}$ with a subcategory in $N\mathfrak{Corr}$. The homomorphisms $i_{\mathcal{E}}$ and $f_{\mathcal{E}}$ for a correspondence $\mathcal{E}$ in Lemma 4.3 and the maps $f_{ST}\colon A_{S}\to A_{T}$ for an $n$-simplex in $N\mathfrak{Corr}$ are strictly continuous. With this extra fact, the proofs above carry over and show:

Theorem 5.1**.**

Let $\mathcal{D}$ be a quasi-category. The following simplicial map is a homotopy equivalence:

[TABLE]

We also get a variant of Theorem 3.4, where we replace the category $\mathcal{C}^{*}_{+}$ by its subcategory of all strictly continuous ∗-homomorphisms $A\to B$.

An $A,B$-correspondence $\mathcal{E}$ is called full if the inner products $\langle\xi\,|\,\mathopen{}\eta\rangle$ for $\xi,\eta\in\mathcal{E}$ span $B$. The full correspondences form a subbicategory of the correspondence bicategory. We call a strictly continuous homomorphism $\varphi\colon A\to\mathcal{M}(B)$ full if the projection $p=\bar{\varphi}(1)$ is full, that is, the Hilbert module $pB$ is full. Equivalently, the closed linear span of $BpB$ is $B$. This is a subcategory of $\mathcal{C}^{*}_{\mathrm{str}}$, which we denote by $\mathcal{C}^{*}_{\mathrm{full}}$.

The homomorphisms $i_{\mathcal{E}}$ and $f_{\mathcal{E}}$ for a correspondence $\mathcal{E}$ in Lemma 4.3 are full if $\mathcal{E}$ is full as a right Hilbert module, and the maps $f_{ST}\colon A_{S}\to A_{T}$ for an $n$-simplex in $N\mathfrak{Corr}$ are full if all $\mathcal{E}_{ij}$ are full as right Hilbert modules. Using this, the same arguments as above show that the simplicial map

[TABLE]

is a homotopy equivalence.

For the bicategory $\mathfrak{Corr}_{\mathrm{full}}\cap\mathfrak{Corr}_{\mathrm{prop}}$ of full, proper correspondences, we get two versions of Theorem 3.4: it is a Dwyer–Kan localisation of the category of full ∗-homomorphisms $A\to B$ or of the category of full, strictly continuous ∗-homomorphisms $A\to B$. Here a ∗-homomorphism $\varphi\colon A\to B$ is called full if $B\varphi(A)B$ spans $B$; equivalently, the correspondence $\mathcal{E}_{\varphi}$ is full.