# Noncommutative CW-spectra as enriched presheaves on matrix algebras

Gregory Arone
Stockholm University
<EMAIL_ADDRESS>Supported in part by the Swedish Research Council,
grant number 2016-05440 Ilan Barnea
Haifa University
<EMAIL_ADDRESS>Supported by ISF 786/19 Tomer M. Schlank
Hebrew University
<EMAIL_ADDRESS>Supported by ISF 1588/18 and BSF 2018389

###### Abstract

Motivated by the philosophy that $C^{*}$-algebras reflect noncommutative
topology, we investigate the stable homotopy theory of the (opposite) category
of $C^{*}$-algebras. We focus on $\operatorname{C}^{*}$-algebras which are
non-commutative CW-complexes in the sense of [ELP]. We construct the stable
$\infty$-category of noncommutative CW-spectra, which we denote by
$\mathtt{NSp}$. Let $\mathcal{M}$ be the full spectral subcategory of
$\mathtt{NSp}$ spanned by ``noncommutative suspension spectra'' of matrix
algebras. Our main result is that $\mathtt{NSp}$ is equivalent to the
$\infty$-category of spectral presheaves on $\mathcal{M}$.

To prove this we first prove a general result which states that any compactly
generated stable $\infty$-category is naturally equivalent to the
$\infty$-category of spectral presheaves on a full spectral subcategory
spanned by a set of compact generators. This is an $\infty$-categorical
version of a result by Schwede and Shipley [ScSh1]. In proving this we use the
language of enriched $\infty$-categories as developed by Hinich [Hin2, Hin3].

We end by presenting a ``strict'' model for $\mathcal{M}$. That is, we define
a category $\mathcal{M}_{s}$ strictly enriched in a certain monoidal model
category of spectra $\mathtt{Sp^{M}}$. We give a direct proof that the
category of $\mathtt{Sp^{M}}$-enriched presheaves
$\mathcal{M}_{s}^{\operatorname{op}}\to\mathtt{Sp^{M}}$ with the projective
model structure models $\mathtt{NSp}$ and conclude that $\mathcal{M}_{s}$ is a
strict model for $\mathcal{M}$.

###### Contents

  1. 1 Introduction
  2. 2 The $\infty$-category of noncommutative CW-complexes
  3. 3 Enriched infinity categories
  4. 4 Stable $\infty$-categories and spectral presheaves
  5. 5 Stabilization of categories
  6. 6 The $\infty$-category of noncommutative CW-spectra

##  1 Introduction

The celebrated Gelfand theorem gives a contravariant equivalence between the
categories of locally compact Hausdorff spaces and commutative
$C^{*}$-algebras. This correspondence led to the point of view that
$C^{*}$-algebras are noncommutative generalizations of topological spaces. The
study of $C^{*}$-algebras from this perspective is the subject of
noncommutative geometry and topology. In this paper we study noncommutative
stable homotopy theory, i.e., the stable homotopy category of the opposite of
the category of $C^{*}$-algebras. In doing this we are continuing the
investigations of Østvær [Ost] and Mahanta [Mah1], among others.

To be a little more specific, in this paper we construct the $\infty$-category
of noncommutative CW-spectra, which we denote $\mathtt{NSp}$, and show that
$\mathtt{NSp}$ is equivalent to the category of spectral presheaves over a
spectrally enriched category $\mathcal{M}$. The objects of $\mathcal{M}$ are
noncommutative suspension spectra of matrix algebras, and its morphisms are
mapping spectra between matrix algebras. In a companion paper [ABS2] we
analyze the category $\mathcal{M}$ in considerable detail. In that paper we
introduce a rank filtration of $\mathcal{M}$, describe the subquotients of the
rank filtration, and use it, in particular, to give an explicit model for the
rationalization of $\mathtt{NSp}$.

We construct the $\infty$-category of noncommutative CW-spectra, as the
stabilization of the $\infty$-category of noncommutative CW-complexes. Our
construction of the $\infty$-category of noncommutative CW-complexes mimics
Lurie's construction of the $\infty$-category of ``ordinary'' CW-complexes as
the ind-completion of the $\infty$-category of finite CW-complexes [Lur1]. The
latter is considered an $\infty$-category by first viewing it as a topological
category in the obvious way and then taking the topological nerve [Lur1,
Section 1.1.5]. (By a topological category in this paper we mean a category
enriched in the category of compactly generated weak Hausdorff spaces.)

In more detail, consider the class of $C^{*}$-algebras called ``noncommutative
CW-complexes'' in [ELP, Section 2.4]. These are algebras generated from the
finite dimensional matrix algebras by a finite inductive procedure,
generalizing the construction of finite CW-complexes from $S^{0}$ in the
commutative case. We will therefore call them _finite noncommutative CW-
complexes_ in this paper.

Finite noncommutative CW-complexes have been studied in several places (for
instance [Ped] and [Die]). In Section 2 we define the _topological category of
finite noncommutative CW-complexes_ to be the opposite of the topological
category whose objects are the $C^{*}$-algebras which are noncommutative CW-
complexes and whose hom-spaces are given by taking the topology of pointwise
norm convergence on the sets of $*$-homomorphisms. We define the _$\infty$
-category of finite noncommutative CW-complexes_ by taking the topological
nerve of this topological category. We denote both versions of this category
by $\mathtt{NCW}^{f}$. We now define the _$\infty$ -category of noncommutative
CW-complexes_ to be the ind-completion of $\mathtt{NCW}^{f}$. This will be our
generalization of the $\infty$-category of spaces and we denote it by
$\mathtt{NCW}$. It can be shown that the $\infty$-category $\mathtt{NCW}^{f}$
is pointed, essentially small and admits finite colimits, so $\mathtt{NCW}$ is
a pointed compactly generated $\infty$-category.

Let $\mathtt{NSp}:=\mathtt{Sp}(\mathtt{NCW})$ be the _$\infty$ -category of
noncommutative CW-spectra_, i.e., the stabilization of $\mathtt{NCW}$. By
construction $\mathtt{NSp}$ is a stable $\infty$-category. In particular, it
is enriched and tensored over the $\infty$-category of ``ordinary'' spectra
$\mathtt{Sp}$. There is a suspension-spectrum functor from noncommutative
spaces to noncommutative spectra, which we denote
$\Sigma^{\infty}_{\mathtt{NC}}\colon\mathtt{NCW}\to\mathtt{NSp}$. It can be
shown (see Section 2) that the (maximal) tensor product of $C^{*}$-algebras
induces a closed symmetric monoidal structure on both $\mathtt{NCW}$ and
$\mathtt{NSp}$, such that
$\Sigma_{\mathtt{NC}}^{\infty}:\mathtt{NCW}\longrightarrow\mathtt{NSp}$ is
symmetric monoidal. Our main result is a presentation of $\mathtt{NSp}$ as a
category of spectral presheaves over a full spectral subcategory spanned by an
explicit set of generators.

In order to prove this we first prove a general result about presenting a
compactly generated stable $\infty$-category as a category of spectral
presheaves over a full spectral subcategory spanned by a set of compact
generators. Such a result was proven by Schwede and Shipley [ScSh1] using
model categories (see also [GM] for the same result under more general
hypotheses). However, in this paper we need a more general result formed in
the language of $\infty$-categories. To obtain this we use the formalism of
enriched $\infty$-categories developed by Hinich [Hin2, Hin3] and reviewed in
Section 3. We can formulate our result as follows:

######  Theorem 1.1 (Theorem 4.1).

Let $\mathcal{D}$ be a cocomplete stable $\infty$-category. Suppose that there
is a small set $C$ of compact objects in $\mathcal{D}$, that generates
$\mathcal{D}$ under colimits and desuspensions. Thinking of $\mathcal{D}$ as
left-tensored over the $\infty$-category of spectra $\mathtt{Sp}$, we let
$\mathcal{C}$ be the full $\mathtt{Sp}$-enriched subcategory of $\mathcal{D}$
spanned by $C$. Then $\mathcal{D}$ is naturally equivalent to the
$\infty$-category of spectral presheaves on $\mathcal{C}$, denoted
$P_{\mathtt{Sp}}(\mathcal{C})$.

There is also a monoidal version of the Theorem 1.1, given in Theorem 4.3.

The classical stable infinity category of spectra $\mathtt{Sp}$ is generated
by a single object, the sphere spectrum ${\mathbb{S}}$, and this is closely
related to the fact that $\mathtt{Sp}$ can be identified with the category of
${\mathbb{S}}$-modules. By contrast $\mathtt{NSp}$ requires infinitely many
generators. Let $M_{n}$ be the algebra of $n\times n$ matrices over
$\mathbb{C}$. The algebras $\\{M_{n}\mid n=1,2,\ldots\\}$ are the finite-
dimensional simple $C^{*}$-algebras. Collectively, they play the same role in
$\mathtt{NCW}$ as $S^{0}$ in the usual category of CW-complexes. The
suspension spectra $\\{\Sigma^{\infty}_{\mathtt{NC}}M_{n}\mid n=1,2,\ldots\\}$
are compact objects of $\mathtt{NSp}$, and they generate $\mathtt{NSp}$ under
$\infty$-colimits and desuspensions. Let $\mathcal{M}$ be the full
$\mathtt{Sp}$-enriched subcategory of $\mathtt{NSp}$ spanned by
$\\{\Sigma^{\infty}_{\mathtt{NC}}M_{n}\mid n=1,2,\ldots\\}$.

For every $n,m\geq 0$ we have $M_{n}\otimes M_{m}\simeq M_{n\times m}$, so the
set $\\{M_{n}\mid n=1,2,\ldots\\}$ is closed under the tensor product. Since
$\Sigma^{\infty}_{\mathtt{NC}}$ is monoidal, we see that
$\\{\Sigma^{\infty}_{\mathtt{NC}}M_{n}\mid n=1,2,\ldots\\}$ is also closed
under the tensor product. It follows that the $\mathtt{Sp}$-enriched category
$\mathcal{M}$ acquires a symmetric monoidal structure from $\mathtt{NSp}$.
This monoidal structure induces a symmetric monoidal structure on the
$\infty$-category of spectral presheaves $P_{\mathtt{Sp}}(\mathcal{M})$ (Day
convolution). Using the monoidal version of Theorem 1.1 we obtain

######  Theorem 1.2 (Theorem 6.2).

The symmetric monoidal $\infty$-category $\mathtt{NSp}$ is naturally
equivalent to the symmetric monoidal $\infty$-category
$P_{\mathtt{Sp}}(\mathcal{M})$ of spectral presheaves on $\mathcal{M}$.

Thus, understanding the spectral $\infty$-category $\mathcal{M}$ should help
us understand the $\infty$-category $\mathtt{NSp}$. The objects of
$\mathcal{M}$ are in one-one correspondence with natural numbers and the
monoidal product acts as multiplication. Given natural numbers $k,l$, we
denote the corresponding mapping spectrum by $\mathbb{S}^{k,l}$

$\mathbb{S}^{k,l}:=\operatorname{Hom}_{\mathtt{NSp}}(\Sigma^{\infty}_{\mathtt{NC}}M_{k},\Sigma^{\infty}_{\mathtt{NC}}M_{l}).$

One can describe $\mathbb{S}^{k,l}$ explicitly as follows. First, let us
define a functor $G_{k,l}$ from finite pointed spaces to pointed spaces by the
formula

$G_{k,l}(X)=\operatorname{Map}_{\mathtt{NCW}^{f}}(M_{k},X\wedge M_{l}).$

Since the pointed $\infty$-category $\mathtt{NCW}^{f}$ has finite colimits, it
is tensored over finite spaces and enriched over spaces. The spectrum
$\mathbb{S}^{k,l}$ is the stabilization of $G_{k,l}$, i.e., $\mathbb{S}^{k,l}$
is the spectrum given by the sequence
$\\{G_{k,l}(S^{0}),G_{k,l}(S^{1}),\ldots$}. In the companion paper [ABS2] we
undertake a detailed study of the spectra $\mathbb{S}^{k,l}$ and the structure
of $\mathcal{M}$.

We end the paper by constructing a ``strict" version of $\mathcal{M}$. Namely,
let $\mathtt{Sp^{M}}$ be the category of continuous pointed functors from
finite pointed CW-complexes to topological spaces, endowed with the stable
model structure. This is a symmetric monoidal model category, that models the
$\infty$-category of spectra [Lyd, MMSS]. In Definition 6.3, we define a
symmetric monoidal category, strictly enriched in $\mathtt{Sp^{M}}$, denoted
$\mathcal{M}_{s}$. We give a direct proof of the following, which can be
considered a strict version of Theorem 1.2:

######  Theorem 1.3 (Theorem 6.7).

The category of $\mathtt{Sp^{M}}$-enriched functors
$\mathcal{M}_{s}^{\operatorname{op}}\to\mathtt{Sp^{M}}$ with the projective
model structure and Day convolution is a symmetric monoidal model category
that models the symmetric monoidal $\infty$-category $\mathtt{NSp}$.

In Definition 3.6 we define the notion of enriched $\infty$-localization. This
takes a category strictly enriched in a monoidal model category, and produces
an $\infty$-category enriched in the $\infty$-localization of this model
category (see also Remark 2.1). A consequence of Theorem 1.3 is that the
enriched $\infty$-localization of $\mathcal{M}_{s}$ is equivalent to
$\mathcal{M}$.

######  Remark 1.4.

Using the work of Blom and Moerdijk [BM], it is possible to define a model
category structure on the opposite of the pro-category of separable
$C^{*}$-algebras, that models $\mathtt{NCW}$. This model structure is a right
Bousfield localization of the model structure presented in [BJM]. We might
then be able to use known results on stable model categories to prove a
similar result to Theorem 1.3. We did not pursue this approach in this paper.

### An alternative definition, via nonabelian derived categories

We will now digress to describe another natural way of defining a
noncommutative analogue to the $\infty$-category of pointed spaces of Lurie.
It relies even more on $\infty$-categorical constructions, and we do not
develop it in this paper. Namely, we can do so using the concept of a
nonabelian derived category (see [Lur1, Section 5.5.8]). If $\mathcal{C}$ is a
small $\infty$-category with finite coproducts, Lurie defines the nonabelian
derived category of $\mathcal{C}$, denoted
$\mathcal{P}_{\Sigma}(\mathcal{C})$, as the $\infty$-category obtained from
$\mathcal{C}$ by formally adjoining colimits of sifted diagrams. Loosely
speaking sifted diagrams are generated by filtered diagrams and the simplicial
diagram $\Delta^{\operatorname{op}}$. Taking $\mathcal{C}=\mathtt{Fin}_{*}$ to
be the category of finite pointed sets, we obtain the $\infty$-category of
pointed spaces, that is, we have a natural equivalence
$\mathcal{P}_{\Sigma}(\mathtt{Fin}_{*})\simeq\operatorname{Top}$.

Under the Gelfand correspondence, the finite pointed sets correspond to the
finite dimensional commutative $C^{*}$-algebras. We thus denote by
$\mathtt{NFin_{*}}$ the full subcategory of $\mathtt{NCW}^{f}$ spanned by the
finite dimensional $C^{*}$-algebras (which are just finite products of matrix
algebras). We can now define a noncommutative analogue to the
$\infty$-category of pointed spaces to be the nonabelian derived category of
$\mathtt{NFin_{*}}$:

$\overline{\mathtt{NCW}}:=\mathcal{P}_{\Sigma}(\mathtt{NFin_{*}}).$

We have natural inclusions

$\mathtt{NFin_{*}}\hookrightarrow\mathtt{NCW}^{f}\hookrightarrow\operatorname{Ind}(\mathtt{NCW}^{f})=\mathtt{NCW}$

and the $\infty$-category $\mathtt{NCW}$ admits sifted colimits, so by the
universal property we have an induced functor

$\overline{\mathtt{NCW}}=\mathcal{P}_{\Sigma}(\mathtt{NFin_{*}})\to\mathtt{NCW},$

that commutes with sifted colimits. We do not know if this functor is an
equivalence. This is true iff for every $n\geq 0$ and every simplicial object
$X$ in $\mathtt{NCW}^{f}$ the natural map

${\mathop{\operatorname{colim}}}_{\Delta^{\operatorname{op}}}\operatorname{Map}_{\mathtt{NCW}}(M_{n},X)\to\operatorname{Map}_{\mathtt{NCW}}(M_{n},{\mathop{\operatorname{colim}}}_{\Delta^{\operatorname{op}}}^{\mathtt{NCW}}X)$

is an equivalence. We know how to prove this last assertion when $X$ is a
simplicial object in $\mathtt{NFin_{*}}$ or $X$ has the form $Y\otimes M_{k}$
for $k\geq 1$ and $Y$ is a simplicial object in $\mathtt{NCW}^{f}$ composed of
commutative algebras.

What we do know is that the induced map on stabilizations:

$\mathtt{Sp}(\overline{\mathtt{NCW}})\to\mathtt{Sp}(\mathtt{NCW})=\mathtt{NSp}$

is an equivalence. To see this note that, by a similar reasoning as in the
beginning of Section 6, we have that
$\overline{M}:=\\{\Sigma^{\infty}M_{i}\mid i\in\mathbb{N}\\}$ generates
$\mathtt{Sp}(\overline{\mathtt{NCW}})$ under small colimits. Thus, by Theorem
6.2, it is enough to show that for every $k,l\geq 1$ the induced map

$\operatorname{Hom}_{\mathtt{Sp}(\overline{\mathtt{NCW}})}(\Sigma^{\infty}M_{k},\Sigma^{\infty}M_{l})\to\operatorname{Hom}_{\mathtt{NSp}}(\Sigma^{\infty}M_{k},\Sigma^{\infty}M_{l})$

is an equivalence. We can define the functor $\overline{G}_{k,l}$ from finite
pointed spaces to pointed spaces by

$\overline{G}_{k,l}(X):=\operatorname{Map}_{\overline{\mathtt{NCW}}}(M_{k},X\wedge
M_{l}).$

The stabilization of $\overline{G}_{k,l}$ is the mapping spectrum

$\operatorname{Map}_{\mathtt{Sp}(\overline{\mathtt{NCW}})}(\Sigma^{\infty}M_{k},\Sigma^{\infty}M_{l}).$

It is thus enough to show that the induced natural transformation
$\overline{G}_{k,l}\to G_{k,l}$ is an equivalence. The functor
$\overline{G}_{k,l}$ clearly commutes with sifted colimits and therefore it is
equivalent to the (derived) left Kan extension of
$\overline{G}_{k,l}|_{\mathtt{Fin}_{*}}$ along the inclusion
$\mathtt{Fin}_{*}\subseteq\mathcal{S}_{*}$. We prove in [ABS2] that the same
holds for the functor $G_{k,l}$. Therefore it is enough to show that the
restriction $\overline{G}_{k,l}|_{\mathtt{Fin}_{*}}\to
G_{k,l}|_{\mathtt{Fin}_{*}}$ is an equivalence. But for every
$[t]\in\mathtt{Fin}_{*}$ we have

$\overline{G}_{k,l}([t])=\operatorname{Map}_{\mathcal{P}_{\Sigma}(\mathtt{NFin_{*}})}(M_{k},[t]\wedge
M_{l})\simeq\operatorname{Map}_{\mathtt{NFin_{*}}}(M_{k},M_{l}^{t})=$
$\operatorname{Map}_{\mathtt{NCW}^{f}}(M_{k},M_{l}^{t})\simeq\operatorname{Map}_{\operatorname{Ind}(\mathtt{NCW}^{f})}(M_{k},[t]\wedge
M_{l})=G_{k,l}([t]),$

so we are done. Note that since the main results in this paper (and in [ABS2])
concern the stabilization of $\mathtt{NCW}$, they apply equally well to the
stabilization of the alternative model $\overline{\mathtt{NCW}}$.

### Comparison with previous work

We end the introduction by relating the $\infty$-categories $\mathtt{NCW}^{f}$
and $\mathtt{NCW}$ constructed here with different $\infty$-categories
constructed in [Mah1]. For more detail see Section 2. In [Mah1], Mahanta
constructed the $\infty$-category $\mathtt{SC_{\infty}^{*}}$ as the
topological nerve of the topological category of all separable
$C^{*}$-algebras, with the mapping spaces given by the topology of pointwise
norm convergence on the sets of $*$-homomorphisms. He called
$(\mathtt{SC_{\infty}^{*}})^{\operatorname{op}}$ the $\infty$-category of
_pointed compact metrizable noncommutatives spaces_. He then defined the
$\infty$-category $\mathtt{N}{\mathcal{S}}_{*}$ as the ind-completion of
$(\mathtt{SC_{\infty}^{*}})^{\operatorname{op}}$, and called it the
$\infty$-category of _pointed noncommutative spaces_.

It can be shown that our $\infty$-category $\mathtt{NCW}^{f}$ is a full
subcategory of $(\mathtt{SC_{\infty}^{*}})^{\operatorname{op}}$ and the
inclusion commutes with finite colimits. It follows that our $\infty$-category
$\mathtt{NCW}$ is a coreflective full subcategory of
$\mathtt{N}\mathcal{S}_{*}$, and thus the inclusion admits a right adjoint:

$i\colon\mathtt{NCW}\rightleftarrows\mathtt{N}\mathcal{S}_{*}\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}R.$

We call a morphism $g\colon X\to Y$ in $\mathtt{N}\mathcal{S}_{*}$ a _weak
homotopy equivalence_ if for every $n\geq 1$ the induced map

$g_{*}\colon\operatorname{Map}_{\mathtt{N}\mathcal{S}_{*}}(M_{n},X)\to\operatorname{Map}_{\mathtt{N}\mathcal{S}_{*}}(M_{n},Y)$

is an equivalence of spaces. This is analogous to weak homotopy equivalences
between topological spaces. The counit of the adjunction above $i\circ
R\to\operatorname{Id}_{\mathtt{N}\mathcal{S}_{*}}$ is a levelwise weak
equivalence, and thus can be thought of as a CW approximation to elements in
$\mathtt{N}\mathcal{S}_{*}$. If $X$ and $Y$ are noncommutative CW-complexes
then $g$ is a weak equivalence iff it is an equivalence in
$\mathtt{N}{\mathcal{S}}_{*}$.

Informally speaking, since the equivalences in
$(\mathtt{SC_{\infty}^{*}})^{\operatorname{op}}$ are homotopy equivalences of
$C^{*}$-algebras, the category
$\mathtt{N}\mathcal{S}_{*}=\operatorname{Ind}((\mathtt{SC_{\infty}^{*}})^{\operatorname{op}})$
is somewhat analogous to the infinity category modeled by the Strøm model
structure on topological spaces [Str], in which the weak equivalences are the
homotopy equivalences. The category $\mathtt{NCW}$ constructed here is
analogous to the infinity category modeled by the Quillen model structure on
topological spaces, in which the weak equivalences are the weak homotopy
equivalences.

### Section by section outline of the paper

In Section 2 we define the $\infty$-category $\mathtt{NCW}$: a noncommutative
analogue of the $\infty$-category of pointed spaces.

In Section 3 we review the theory of enriched $\infty$-categories, as
developed by Hinich [Hin2, Hin3]. In particular we state the enriched Yoneda
lemma for $\infty$-categories. We also present a way to pass from model
categories to $\infty$-categories is the enriched setting.

In Section 4 we review the notion of a stable $\infty$-category and show that
a compactly generated stable $\infty$-category is equivalent to the category
of spectral presheaves on a full subcategory spanned by a set of compact
generators.

In Section 5 we review the process of stabilizing an $\infty$-category. We use
the framework established by Lurie in [Lur2, Section 1.4]. We also review how
a similar procedure can be applied to an ordinary topologically enriched
category, and compare the strict and the $\infty$-categorical versions of
stabilization.

In Section 6 we define the category of non-commutative CW-spectra
$\mathtt{NSp}$ as the stabilization of the category $\mathtt{NCW}$. We
identify the suspension spectra of matrix algebras as an explicit set of
generators of $\mathtt{NSp}$. Letting $\mathcal{M}$ be the full subcategory of
$\mathtt{NSp}$ spanned by matrix algebras, we conclude that $\mathtt{NSp}$ is
(monoidally) equivalent to $P_{\mathtt{Sp}}(\mathcal{M})$, the category of
spectral presheaves on $\mathcal{M}$. We give a strict model for
$\mathcal{M}$, denoted $\mathcal{M}_{s}$, as a category enriched over a
Quillen model category of spectra $\mathtt{Sp^{M}}$. We also show that the
category of $\mathtt{Sp^{M}}$-enriched functors
$\mathcal{M}_{s}^{\operatorname{op}}\to\mathtt{Sp^{M}}$ with the projective
model structure models the $\infty$-category $\mathtt{NSp}$ and conclude that
$\mathcal{M}$ is equivalent to the enriched $\infty$-localization of
$\mathcal{M}_{s}$.

### Acknowledgements

We are grateful to Vladimir Hinich for explaining to us his theory of enriched
infinity categories and its relevance to our work.

##  2 The $\infty$-category of noncommutative CW-complexes

In this section we define a noncommutative analogue of the $\infty$-category
of pointed spaces defined by Lurie [Lur1].

Let $\mathtt{SC^{*}}$ (resp. $\mathtt{CSC^{*}}$) denote the category of all
(resp. commutative) separable $C^{*}$-algebras and $*$-homomorphisms.
Following the common convention in the field, the term $C^{*}$-algebra or
$*$-homomorphism will always mean _non-unital_. The Gelfand correspondence
implies that the functor

$X\mapsto\mathrm{C}_{0}(X):\mathtt{CM_{*}}\longrightarrow\mathtt{CSC^{*}}^{\operatorname{op}}$

that assigns to every pointed compact metrizable space $X$ the commutative
separable $C^{*}$-algebra of continuous complex valued functions on $X$ that
vanish at the basepoint, is an equivalence of categories. It is thus natural
to regard $\mathtt{SC^{*}}^{\operatorname{op}}$ as the category of
_noncommutative_ pointed compact metrizable spaces.

Consider $\mathtt{CM_{*}}$ as a topologically enriched category, where for
every $X,Y\in\mathtt{CM_{*}}$ we endow the set of pointed continuous maps
$\mathtt{CM_{*}}(X,Y)$ with the _compact open topology_. Now we take the
topological nerve [Lur1, Section 1.1.5] of this topological category and
obtain the $\infty$-category $(\mathtt{CM_{*}})_{\infty}$. It is well-known
that $(\mathtt{CM_{*}})_{\infty}$ admits finite $\infty$-colimits and that
$\infty$-pushouts can be calculated using the standard cylinder object.

Let us construct the $\infty$-category of pointed spaces in a way that admits
a natural generalization to the noncommutative case. We denote by
$\mathtt{CW}^{f}_{*}$ the smallest full subcategory of $\mathtt{CM_{*}}$ that
contains $S^{0}$ and is closed under finite homotopy-colimits using the
standard cylinder object. Thus $\mathtt{CW}^{f}_{*}$ is the topological
category of finite pointed CW-complexes. We will also consider
$\mathtt{CW}^{f}_{*}$ as an $\infty$-category, by applying the coherent nerve
functor to it. We will use the same notation $\mathtt{CW}^{f}_{*}$ to indicate
both the ordinary (topologically enriched) and the $\infty$-categorical
incarnation of the category, trusting that it is clear from the context which
is meant. The $\infty$-category of pointed spaces can be defined as the
$\infty$-categorical $\operatorname{Ind}$ construction of
$\mathtt{CW}^{f}_{*}$. Note that under the Gelfand correspondence, $S^{0}$
corresponds to $\mathbb{C}$, which is the only nonzero finite dimensional
simple commutative $C^{*}$-algebra.

We now turn to the noncommutative analogue. We first recall from [Mah1,
Section 2.1] the construction of the $\infty$-category
$\mathtt{SC_{\infty}^{*}}$. Consider $\mathtt{SC^{*}}$ as a topologically
enriched category, where for every $A,B\in\mathtt{SC^{*}}$ we endow the set of
$*$-homomorphisms $\mathtt{SC^{*}}(A,B)$ with the topology of pointwise norm
convergence. Now we take the topological nerve of this topological category
and obtain the $\infty$-category $\mathtt{SC_{\infty}^{*}}$. It is shown in
[Mah1, Section 2.1] that $\mathtt{SC_{\infty}^{*}}$ is (essentially) small,
pointed, and finitely complete.

######  Remark 2.1.

Recall that any relative category, that is a pair $(\mathcal{C},\mathcal{W})$
consisting of a category $\mathcal{C}$ an a subcategory
$\mathcal{W}\subseteq\mathcal{C}$, has a canonically associated
$\infty$-category $\mathcal{C}_{\infty}$, obtained by formally inverting the
morphisms in $\mathcal{W}$, in the infinity categorical sense. There is also a
canonical localization functor $\mathcal{C}\to\mathcal{C}_{\infty}$ satisfying
a universal property. We refer the reader to [Hin1] for a thorough account,
and also to the discussion in [BHH, Section 2.2]. We refer to
$\mathcal{C}_{\infty}$ as the $\infty$-localization of $\mathcal{C}$ (with
respect to $\mathcal{W}$). If $\mathcal{C}$ is a model category or a
(co)fibration category, we always take $\mathcal{W}$ to be the set of weak
equivalences in $\mathcal{C}$.

Using $\infty$-localization, there is another natural way of considering
separable $C^{*}$-algebras as an $\infty$-category. There is a well known
notion of homotopy equivalence between $C^{*}$-algebras. We can consider
$\mathtt{SC^{*}}$ as a relative category, with the weak equivalences given by
the homotopy equivalences, and take its $\infty$-localization. This is the
point of view taken, for instance, in [AG, Uuy]. It follows from [BJM,
Proposition 3.17] that we obtain an $\infty$-category equivalent to
$\mathtt{SC_{\infty}^{*}}$.

######  Remark 2.2.

It is well-known that $\mathtt{SC^{*}}$ is cotensored over the category of
pointed finite CW-complexes [AG]. If $K$ is a finite pointed CW-complex and
$A\in\mathtt{SC^{*}}$ then the cotensoring of $A$ and $K$ is given by the
$C^{*}$-algebra of pointed continuous functions from $K$ to $A$. One can
define finite homotopy limits in $\mathtt{SC^{*}}$ using this cotensoring.
Consequently, the $\infty$-pullbacks in $\mathtt{SC_{\infty}^{*}}$ can be
calculated as homotopy pullbacks using the standard path object [Mah1,
Proposition 2.7].

######  Definition 2.3.

We denote by $\mathtt{NCW}^{f}$ the opposite of the smallest full subcategory
of $\mathtt{SC^{*}}$ that contains the nonzero finite dimensional simple
algebras in $\mathtt{SC^{*}}$ (which are just the matrix algebras over
$\mathbb{C}$) and is closed under finite homotopy-limits using the standard
path object. We call $\mathtt{NCW}^{f}$ the category of _finite pointed
noncommutative CW-complexes_. The category $\mathtt{NCW}^{f}$ is an
``ordinary'' topologically enriched category. We will also consider
$\mathtt{NCW}^{f}$ as an $\infty$-category, by applying the coherent nerve
functor to it. Like in the commutative case, will use the same notation
$\mathtt{NCW}^{f}$ to indicate both the ordinary (topologically enriched) and
the $\infty$-categorical incarnation of the category, trusting that it is
clear from the context which is meant.

The topological category $\mathtt{NCW}^{f}$ is the category of _noncommutative
CW-complexes_ as defined in [ELP, Section 2.4]. Using [Ped, Theorem 11.14],
the same proof as in [Mah2, Proposition 1.1] gives that the (maximal) tensor
product of $C^{*}$-algebras induces a symmetric monoidal structure on the
$\infty$-category $\mathtt{NCW}^{f}$, that preserves finite colimits in each
variable separately. Note also that since the topological category
$\mathtt{SC^{*}}$ is cotensored over pointed finite CW-complexes, the
topological category $\mathtt{NCW}^{f}$ is tensored over pointed finite CW-
complexes. We will denote the tensoring of a finite CW-complex $K$ and a
noncommutative finite complex $X$ by $K\wedge X$.

We now define the $\infty$-category of _noncommutative pointed CW-complexes_
to be the $\infty$-categorical ind-completion of $\mathtt{NCW}^{f}$,

$\mathtt{NCW}:=\operatorname{Ind}(\mathtt{NCW}^{f}).$

The $\infty$-category $\mathtt{NCW}^{f}$ is (essentially) small, pointed and
finitely cocomplete so $\mathtt{NCW}$ is a compactly generated pointed
$\infty$-category. By [Lur2, Corollary 4.8.1.14] there is an induced closed
symmetric monoidal structure on the $\infty$-category
$\mathtt{NCW}\simeq\operatorname{Ind}(\mathtt{NCW}^{f})$ such that the natural
embedding $j\colon\mathtt{NCW}^{f}\longrightarrow\mathtt{NCW}$ is symmetric
monoidal.

In [Mah1], Mahanta defined the $\infty$-category $\mathtt{N}\mathcal{S}_{*}$
as the ind-completion of $(\mathtt{SC_{\infty}^{*}})^{\operatorname{op}}$, and
called it the $\infty$-category of _pointed noncommutative spaces_. By
definition our category $\mathtt{NCW}^{f}$ is a full subcategory of
$(\mathtt{SC_{\infty}^{*}})^{\operatorname{op}}$. Since $\infty$-pushouts in
both $\mathtt{NCW}^{f}$ and $(\mathtt{SC_{\infty}^{*}})^{\operatorname{op}}$
can be calculated as homotopy pushouts using the standard cylinder object, we
see that the inclusion
$\mathtt{NCW}^{f}\hookrightarrow(\mathtt{SC_{\infty}^{*}})^{\operatorname{op}}$
commutes with finite colimits. Passing to ind-completions we get that the
induced inclusion $\mathtt{NCW}\hookrightarrow\mathtt{N}\mathcal{S}_{*}$
admits a right adjoint (or in other words, $\mathtt{NCW}$ is a coreflective
full subcategory of $\mathtt{N}\mathcal{S}_{*}$)

$i\colon\mathtt{NCW}\rightleftarrows\mathtt{N}\mathcal{S}_{*}\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}R.$

We call a morphism $g\colon X\to Y$ in $\mathtt{N}\mathcal{S}_{*}$ a _weak
homotopy equivalence_ if $R(g)$ is an equivalence in $\mathtt{NCW}$, or
equivalently, if for every object $W$ in $\mathtt{NCW}$ the induced map

$g_{*}\colon\operatorname{Map}_{\mathtt{N}\mathcal{S}_{*}}(i(W),X)\to\operatorname{Map}_{\mathtt{N}\mathcal{S}_{*}}(i(W),Y)$

is an equivalence in $\mathcal{S}$. Since every object in $\mathtt{NCW}$ is a
small colimit of matrix algebras and $i$ commutes with small colimits, we see
that $g$ is a weak equivalence iff for every $n\geq 1$ the induced map

$g_{*}\colon\operatorname{Map}_{\mathtt{N}\mathcal{S}_{*}}(M_{n},X)\to\operatorname{Map}_{\mathtt{N}\mathcal{S}_{*}}(M_{n},Y)$

is an equivalence in $\mathcal{S}$. This is analogous to weak homotopy
equivalences between topological spaces. The counit of the adjunction above
$i\circ R\to\operatorname{Id}_{\mathtt{N}\mathcal{S}_{*}}$ is a levelwise weak
equivalence, and thus can be thought of as a CW approximation to elements in
$\mathtt{N}\mathcal{S}_{*}$. If $X$ and $Y$ are noncommutative CW-complexes
then $g$ is a weak equivalence iff it is an equivalence in
$\mathtt{N}\mathcal{S}_{*}$. Since the weak equivalences in
$(\mathtt{SC_{\infty}^{*}})^{\operatorname{op}}$ are the homotopy
equivalences, the category $\mathtt{N}\mathcal{S}_{*}$ is somewhat analogous
to the infinity category modeled by the Strøm model category on topological
spaces [Str].

##  3 Enriched infinity categories

In Theorem 1.1 we make use of enriched infinity categories. There are a few
approaches to this theory (see, for example, [GH, Lur2]) but so far only in
[Hin2] the Yoneda embedding is defined and its basic properties are shown.
Since we need these results, we chose to follow Hinich's approach in this
paper. In this section we give an overview of the basic definitions and
constructions needed for later on. We also present some new material in
subsection 3.0.1, concerning the connection between model categories and
$\infty$-categories in the enriched setting.

Let $\mathtt{Cat}$ denote the $\infty$-category of $\infty$-categories, and
let $\mathtt{Cat^{L}}$ denote the $\infty$-subcategory of $\mathtt{Cat}$ whose
objects are $\infty$-categories having small colimits and whose morphisms
preserve these colimits. The category $\mathtt{Cat}$ is symmetric monoidal
under the cartesian product, while $\mathtt{Cat^{L}}$ has a symmetric monoidal
structure induced from the cartesian structure on $\mathtt{Cat}$ (see [Lur2,
4.8.1.3, 4.8.1.4]). With this structure on $\mathtt{Cat^{L}}$,
$\operatorname{Map}_{\mathtt{Cat^{L}}}(P\otimes L,M)$ is the subspace of
$\operatorname{Map}_{\mathtt{Cat}}(P\times L,M)$ consisting of functors
preserving small colimits along each argument. Note that a monoidal
$\infty$-category is equivalent to an associative algebra object in
$\mathtt{Cat}$, while an associative algebra in $\mathtt{Cat^{L}}$ is
equivalent to a monoidal $\infty$-category with colimits, whose monoidal
product commutes with colimits in each variable. We define a _closed monoidal
$\infty$-category_ to be an associative algebra in $\mathtt{Cat^{L}}$.

If $\mathcal{M}$ is a closed monoidal $\infty$-category, then a category left-
tensored over $\mathcal{M}$ is by definition a left module over $\mathcal{M}$
in $\mathtt{Cat^{L}}$. More generally, if $\mathcal{O}$ is an $\infty$-operad,
an $\mathcal{O}$-monoidal category is an algebra over $\mathcal{O}$ in
$\mathtt{Cat^{L}}$. If $\mathcal{M}$ is an $\mathcal{O}$-monoidal category one
can define an $\mathcal{O}$-algebra in $\mathcal{M}$.

Let $\mathtt{Ass}$ be the associative operad and $\mathtt{LM}$ be the two
colored operad of left modules. Algebras over $\mathtt{Ass}$ are associative
algebras and algebras over $\mathtt{LM}$ consist of an associative algebra and
a left module over it. Thus, an $\mathtt{Ass}$-monoidal category is just a
closed monoidal $\infty$-category and an $\mathtt{LM}$-monoidal category is a
pair consisting of a closed monoidal $\infty$-category and a category left-
tensored over it.

Let $\mathcal{M}$ be a closed monoidal $\infty$-category. For every space
(i.e., an $\infty$-groupoid) $X$, Hinich constructs (see [Hin2, Sections 3 and
4]) a closed monoidal structure on the $\infty$-category of Quivers

$\operatorname{Quiv}_{X}(\mathcal{M}):=\operatorname{Fun}(X^{\operatorname{op}}\times
X,\mathcal{M}).$

Hinich's monoidal structure is an $\infty$-categorical version of the usual
convolution product that one uses to define ordinary enriched categories. For
$\mathcal{B}$ a category left-tensored over $\mathcal{M}$, Hinich constructs a
left action of the closed monoidal $\infty$-category
$\operatorname{Quiv}_{X}(\mathcal{M})$ on the $\infty$-category
$\operatorname{Fun}(X,\mathcal{B})$. In his notation we obtain an
$\mathtt{LM}$-monoidal category

$\operatorname{Quiv}^{\mathtt{LM}}_{X}(\mathcal{M},\mathcal{B}):=(\operatorname{Quiv}_{X}(\mathcal{M}),\operatorname{Fun}(X,\mathcal{B})).$

######  Definition 3.1.

A $\mathcal{M}$-enriched category, with space of objects $X$ is an associative
algebra in $\operatorname{Quiv}_{X}(\mathcal{M})$.

######  Remark 3.2.

Hinich uses the term $\mathcal{M}$-enriched precategory for an associative
algebra in $\operatorname{Quiv}_{X}(\mathcal{M})$. He reserves the term
$\mathcal{M}$-enriched category for precategories satisfying a version of
Segal completeness condition (see [Hin2, Definition 7.1.1]). In this paper we
are not concerned with Segal completeness, so we will not distinguish between
enriched categories and precategories. We will just say ``enriched category''
where Hinich might have said ``enriched precategory''.

######  Remark 3.3.

If $\mathcal{M}$ is the $\infty$-category of spaces, then the category of
$\mathcal{M}$-enriched categories with space of objects $X$ is equivalent to
the category of simplicial spaces satisfying the Segal condition and equalling
$X$ in simplicial degree zero. See [Hin2, Corollary 5.6.1], where a more
general statement is proved. In other words, a category enriched in spaces is
the same thing as an ordinary $\infty$-category. For a general closed monoidal
$\infty$-category $\mathcal{M}$, there is a monoidal ``forgetful'' functor
from $\mathcal{M}$ to spaces, given by
$\operatorname{Map}_{\mathcal{M}}(\mathtt{1},-)$. In this way we obtain a
forgetful functor from the category of $\mathcal{M}$-enriched categories to
ordinary infinity categories (compare with [Hin2, Definition 7.1.1]).

######  Remark 3.4.

As we show in next subsection 3.0.1, the theory of monoidal model categories
and categories enriched or tensored over them extends nicely to the theory
presented above upon application of $\infty$-localization.

In [Hin2, Section 6] Hinich defines the notion of an $\mathcal{M}$-functor
from an $\mathcal{M}$-enriched category to a category left-tensored over
$\mathcal{M}$. Let $\mathcal{A}$ be an $\mathcal{M}$-enriched category with
space of objects $X$ and $\mathcal{B}$ a category left-tensored over
$\mathcal{M}$. Then $\mathcal{A}$ is an associative algebra in
$\operatorname{Quiv}_{X}(\mathcal{M})$ and $\operatorname{Fun}(X,\mathcal{B})$
is left tensored over $\operatorname{Quiv}_{X}(\mathcal{M})$. An
$\mathcal{M}$-functor $\mathcal{A}\to\mathcal{B}$ is defined to be a left
module over $\mathcal{A}$ in $\operatorname{Fun}(X,\mathcal{B})$, and the
$\infty$-category of $\mathcal{M}$-functors $\mathcal{A}\to\mathcal{B}$ is
defined to be

$\operatorname{Fun}_{\mathcal{M}}(\mathcal{A},\mathcal{B}):=\mathtt{LMod}_{\mathcal{A}}(\operatorname{Fun}(X,\mathcal{B})).$

For $x,y$ objects of $\mathcal{B}$, we define the presheaf
$\operatorname{Hom}_{\mathcal{B}}(x,y)\in P(\mathcal{M})$ by

$\operatorname{Hom}_{\mathcal{B}}(x,y)(K):=\operatorname{Map}_{\mathcal{B}}(K\otimes
x,y).$

Clearly
$\operatorname{Hom}_{\mathcal{B}}(x,y):\mathcal{M}^{\operatorname{op}}\to\mathcal{S}$
preserves limits, but it is not necessarily representable. If it happens to be
representable, then the representing object serves as an internal mapping
object from $b$ to $c$. Every $\mathcal{M}$-functor
$F:\mathcal{A}\to\mathcal{B}$ induces maps in $P(\mathcal{M})$

$h_{\mathcal{A}(x,y)}\to\operatorname{Hom}_{\mathcal{B}}(F(x),F(y))$

for $x,y\in X$. The $\mathcal{M}$-functor $F$ is called $\mathcal{M}$-fully
faithful if all these maps are equivalences.

If
$\operatorname{Hom}_{\mathcal{B}}(b,c):\mathcal{M}^{\operatorname{op}}\to\mathcal{S}$
is representable for all objects $b,c$ of $\mathcal{B}$, then $\mathcal{B}$ is
enriched as well as left tensored. More generally and more precisely, Hinich
proves the following proposition (see [Hin2, Proposition 6.3.1 and Corollary
6.3.4]). We note that if $\mathcal{M}$ is presentable, then any functor
$\mathcal{M}^{\operatorname{op}}\to\mathcal{S}$ that preserves limits is
representable (see [Lur1, Proposition 5.5.2.2]).

######  Proposition 3.5.

Let $\mathcal{M}$ be a closed monoidal $\infty$-category and $\mathcal{B}$
left tensored over $\mathcal{M}$. Let $C$ be a class of objects of
$\mathcal{B}$. If for all $x,y\in C$, the presheaf
$\operatorname{Hom}_{\mathcal{B}}(x,y)$ is representable then there exists an
$\mathcal{M}$-enriched category $\mathcal{C}$ whose class of objects is $C$,
such that for any two objects $x,y$ of $C$, the morphism object
$\mathcal{C}(x,y)$ is a representing object for the functor
$\operatorname{Hom}_{\mathcal{B}}(x,y)$. There is a fully-faithful
$\mathcal{M}$-enriched functor $\mathcal{C}\to\mathcal{B}$, extending the
inclusion of $C$ into $\mathcal{B}$.

Using [Hin2, Lemma 6.3.3] it is not hard to see that given the conditions of
Theorem 3.5 all the categories $\mathcal{C}$ that can be obtained are
canonically equivalent (via the choice of $X$ as the full subspace of
$\mathcal{B}$ spanned by $C$ in [Hin2, Corollary 6.3.4]). We can thus call
$\mathcal{C}$ the full enriched subcategory of $\mathcal{B}$ spanned by $C$.
Taking $C$ to be the class of all objects in $\mathcal{B}$ we see that if
$\operatorname{Hom}_{\mathcal{B}}(x,y)$ is representable for all $x,y$, then
$\mathcal{B}$ is enriched as well as tensored over $\mathcal{M}$.

###  3.0.1 From enriched model categories to enriched infinity categories

Let $\mathtt{Cat}_{1}$ denote the category of small categories and functors
between them and let $\mathcal{S}_{\mathtt{M}}$ denote the category of
simplicial sets. We have the usual nerve functor

$\mathtt{N}:\mathtt{Cat}_{1}\to\mathcal{S}_{\mathtt{M}}.$

The functor $\mathtt{N}$ is limit preserving and in particular it is a
(cartesian) monoidal functor.

In [Hor, Section 2.1], Horel constructs a (large, coloured)
$\mathtt{Cat}_{1}$-operad denoted $\mathtt{ModCat}$. The colours in
$\mathtt{ModCat}$ are model categories, while the category of multilinear
operations
$\operatorname{Map}_{\mathtt{ModCat}}(\mathcal{M}_{1},\cdots,\mathcal{M}_{n},\mathcal{N})$
is the category of left Quillen $n$-functors

$\mathcal{M}_{1}\times\dots\times\mathcal{M}_{n}\to\mathcal{N}$

and natural weak equivalences (on cofibrant objects) between them. Since
$\mathtt{N}$ is a monoidal functor, we obtain a simplicial operad from
$\mathtt{ModCat}$ by composing with $\mathtt{N}$. We denote this simplicial
operad also by $\mathtt{ModCat}$.

Let $\mathcal{S}_{\mathtt{M}}^{\Delta^{\operatorname{op}}}$ denote the
category of simplicial objects in $\mathcal{S}_{\mathtt{M}}$ with Rezk's model
structure. This is a combinatorial simplicial cartesian closed symmetric
monoidal model category with all objects cofibrant. Let $\mathtt{CSS}$ denote
the full simplicial subcategory of
$\mathcal{S}_{\mathtt{M}}^{\Delta^{\operatorname{op}}}$ spanned by the fibrant
objects. Then $\mathtt{CSS}$ is a monoidal simplicial category (under the
cartesian product) whose simplicial nerve is naturally equivalent to the
monoidal $\infty$-category $\mathtt{Cat}$. Horel also constructs in [loc.
cit.] another full simplicial subcategory
$\mathtt{Cat}_{\infty}\subseteq\mathcal{S}_{\mathtt{M}}^{\Delta^{\operatorname{op}}}$
containing $\mathtt{CSS}$ and closed under the cartesian product. He shows
that the inclusion $\mathtt{CSS}\to\mathtt{Cat}_{\infty}$ induces an
equivalence of $\infty$-categories after application of simplicial nerve.

Let $\mathtt{ModCat}^{c}$ be the full sub simplicial operad of
$\mathtt{ModCat}$ on model categories that are Quillen equivalent to a
combinatorial model category. For $\mathcal{M}\in\mathtt{ModCat}^{c}$, let
$\mathtt{N}_{\mathcal{R}}(\mathcal{M})$ denote the Resk nerve construction on
the cofibrant objects in $\mathcal{M}$ and weak equivalences between them. By
[Hor, Theorem 2.16], $\mathtt{N}_{\mathcal{R}}$ extends to a map of simplicial
operads $\mathtt{ModCat}^{c}\to\mathtt{Cat}_{\infty}$. Applying simplicial
nerve, we obtain a map of $\infty$-operads
$\mathtt{N}_{\mathcal{S}}(\mathtt{ModCat}^{c})\to\mathtt{Cat}$. By [Hor,
Remark 2.17], this map factors through $\mathtt{Cat^{L}}$. Since Resk's nerve
is one of the models for $\infty$-localization (see, for example, [BHH,
Section 2.2]), we obtain a map of $\infty$-operads

$(-)_{\infty}:\mathtt{N}_{\mathcal{S}}(\mathtt{ModCat}^{c})\to\mathtt{Cat^{L}}$

which acts as $\infty$-localization on objects.

Let $\mathtt{M}$ be the nonsymmetric operad (in $\operatorname{Set}$) freely
generated by an operation in degree 0 and 2. An algebra over $\mathtt{M}$ in
$\operatorname{Set}$ is a set with a binary multiplication and a base point.
Let $\mathtt{P}$ be the operad in $\mathtt{Cat}_{1}$ which is given in degree
$n$ by the groupoid whose objects are points of $\mathtt{M}(n)$ and a with a
unique morphism between any two objects. Then an algebra over $\mathtt{P}$ in
$\mathtt{Cat}_{1}$ is a monoidal category. The nerve of $\mathtt{P}$ is a
simplicial operad which we also denote by $\mathtt{P}$. Clearly, we have an
equivalence of $\infty$-operads
$\mathtt{N}_{\mathcal{S}}\mathtt{P}\simeq\mathtt{Ass}$.

Let $\mathcal{M}$ be a monoidal model category, Quillen equivalent to a
combinatorial model category. Then $\mathcal{M}$ is an algebra over
$\mathtt{P}$ in $\mathtt{ModCat}^{c}$ (as operads in $\mathtt{Cat}_{1}$ and
thus in $\mathcal{S}_{\mathtt{M}}$). Applying the simplicial nerve we get that
$\mathcal{M}$ is an algebra over
$\mathtt{N}_{\mathcal{S}}\mathtt{P}\simeq\mathtt{Ass}$ in
$\mathtt{N}_{\mathcal{S}}(\mathtt{ModCat}^{c})$. It follows that
$\mathcal{M}_{\infty}$ is an algebra over $\mathtt{Ass}$ in
$\mathtt{Cat^{L}}$, so $\mathcal{M}_{\infty}$ is a presentable closed monoidal
$\infty$-category. Furthermore, the localization functor

$\mathcal{M}\to\mathcal{M}_{\infty}$

is lax monoidal.

Now let $\mathcal{C}$ be a model category, Quillen equivalent to a
combinatorial one. Suppose that $\mathcal{C}$ is an $\mathcal{M}$-model
category, in the sense that $\mathcal{C}$ is tensored closed over
$\mathcal{M}$ and satisfies the Quillen SM7 axiom. As above, we can construct
a simplicial operad $\mathtt{Q}$ whose simplicial nerve is equivalent to
$\mathtt{LM}$ and such that $(\mathcal{M},\mathcal{C})$ is an algebra over
$\mathtt{Q}$ in $\mathtt{ModCat}^{c}$. Applying the simplicial nerve we get
that $(\mathcal{M},\mathcal{C})$ is an algebra over
$\mathtt{N}_{\mathcal{S}}\mathtt{Q}\simeq\mathtt{LM}$ in
$\mathtt{N}_{\mathcal{S}}(\mathtt{ModCat}^{c})$. It follows that
$(\mathcal{M}_{\infty},\mathcal{C}_{\infty})$ is an algebra over $\mathtt{LM}$
in $\mathtt{Cat^{L}}$, so $\mathcal{C}_{\infty}$ is a presentable
$\infty$-category left tensored over $\mathcal{M}_{\infty}$. Furthermore, the
localization functor

$(\mathcal{M},\mathcal{C})\to(\mathcal{M}_{\infty},\mathcal{C}_{\infty})$

is $\mathtt{LM}$-lax monoidal.

Let $(-)^{f}$ and $(-)^{c}$ denote fibrant and cofibrant replacement functors
in a model category. The model category $\mathcal{C}$ is an
$\mathcal{M}$-model category so for every $A\in\mathcal{C}$ we have a Quillen
pair

$(-)\otimes
A^{c}:\mathcal{M}\rightleftarrows\mathcal{C}:\operatorname{Hom}_{\mathcal{C}}(A^{c},-).$

By [Maz] we have an induced adjunction of $\infty$-categories

$\mathbb{L}(-)\otimes
A^{c}:\mathcal{M}_{\infty}\rightleftarrows\mathcal{C}_{\infty}:\mathbb{R}\operatorname{Hom}_{\mathcal{C}}(A^{c},-).$

Thus we have equivalences natural in $A,B\in\mathcal{C}$:

$\operatorname{Map}_{\mathcal{M}_{\infty}}(K,\mathbb{R}\operatorname{Hom}_{\mathcal{C}}(A^{c},B))\simeq\operatorname{Map}_{\mathcal{C}_{\infty}}(\mathbb{L}K\otimes
A^{c},B)$

Clearly $\mathbb{L}(-)\otimes
A^{c}:\mathcal{M}_{\infty}\to\mathcal{C}_{\infty}$ represents the tensor
product $(-)\otimes A$ of $\mathcal{C}_{\infty}$ as tensored over
$\mathcal{M}_{\infty}$ so we have natural equivalences

$\operatorname{Map}_{\mathcal{M}_{\infty}}(K,\operatorname{Hom}_{\mathcal{C}}(A^{c},B^{f}))\simeq\operatorname{Map}_{\mathcal{C}_{\infty}}(K\otimes
A,B).$

We see that
$\operatorname{Hom}_{\mathcal{C}}(A^{c},B^{f})\in\mathcal{M}_{\infty}$ is a
representing object for $\operatorname{Hom}_{\mathcal{C}_{\infty}}(A,B)\in
P(\mathcal{M}_{\infty})$ and thus we have

$\operatorname{Hom}_{\mathcal{C}}(A^{c},B^{f})\simeq\operatorname{Hom}_{\mathcal{C}_{\infty}}(A,B).$

######  Definition 3.6.

Let $\mathcal{A}$ be a strictly enriched category over $\mathcal{M}$. Let $X$
denote the discrete space on the set of objects of $\mathcal{A}$. Then
$\mathcal{A}$ is an algebra over $\mathtt{Ass}$ in the monoidal category
$\operatorname{Quiv}_{X}(\mathcal{M})$ (see [Hin4]). The localization functor

$\mathcal{M}\to\mathcal{M}_{\infty}$

is lax monoidal, so the induced functor

$\operatorname{Quiv}_{X}(\mathcal{M})\to\operatorname{Quiv}_{X}(\mathcal{M}_{\infty})$

is also lax monoidal. We define the _enriched $\infty$-localization_ functor
to be composition with this last functor:

$(-)_{\infty}:\operatorname{Alg}_{\mathtt{Ass}}(\operatorname{Quiv}_{X}(\mathcal{M}))\to\operatorname{Alg}_{\mathtt{Ass}}(\operatorname{Quiv}_{X}(\mathcal{M}_{\infty})).$

Thus, $\mathcal{A}_{\infty}$ is an enriched $\infty$-category over
$\mathcal{M}_{\infty}$.

Let $F:\mathcal{A}\to\mathcal{C}$ be a strict $\mathcal{M}$-functor. We have
an $\mathtt{LM}$-monoidal category

$\operatorname{Quiv}^{\mathtt{LM}}_{X}(\mathcal{M},\mathcal{C}):=(\operatorname{Quiv}_{X}(\mathcal{M}),\operatorname{Fun}(X,\mathcal{C}))$

and $F$ is just a module in $\operatorname{Fun}(X,\mathcal{C})$ over
$\mathcal{A}$ (see [Hin4]). In this case $(\mathcal{A},F)$ is an algebra over
$\mathtt{LM}$ in the $\mathtt{LM}$-monoidal category
$\operatorname{Quiv}^{\mathtt{LM}}_{X}(\mathcal{M},\mathcal{C})$. The
localization functor

$(\mathcal{M},\mathcal{C})\to(\mathcal{M}_{\infty},\mathcal{C}_{\infty})$

is $\mathtt{LM}$-lax monoidal, so the induced functor

$\operatorname{Quiv}^{\mathtt{LM}}_{X}(\mathcal{M},\mathcal{C})\to\operatorname{Quiv}^{\mathtt{LM}}_{X}(\mathcal{M}_{\infty},\mathcal{C}_{\infty})$

is also $\mathtt{LM}$-lax monoidal. It follows that we obtain a functor that
we denote

$(-)_{\infty}:\operatorname{Alg}_{\mathtt{LM}}(\operatorname{Quiv}^{\mathtt{LM}}_{X}(\mathcal{M},\mathcal{C}))\to\operatorname{Alg}_{\mathtt{LM}}(\operatorname{Quiv}^{\mathtt{LM}}_{X}(\mathcal{M}_{\infty},\mathcal{C}_{\infty})).$

Clearly this functor lifts the functor above so we have

$(\mathcal{A},\mathcal{F})_{\infty}=(\mathcal{A}_{\infty},F_{\infty}).$

Thus, $F_{\infty}$ is an $\mathcal{M}_{\infty}$-functor from
$\mathcal{A}_{\infty}$ to $\mathcal{C}_{\infty}$ and we call it the _enriched
$\infty$-localization_ of $F$.

We call $F$ _homotopy fully faithful_ if for every $x,y\in X$ the composition

$\mathcal{A}(x,y)\xrightarrow{F}\operatorname{Hom}_{\mathcal{C}}(F(x),F(y))\to\operatorname{Hom}_{\mathcal{C}}(F(x)^{c},F(y)^{f})$

is an equivalences in the model category $\mathcal{M}$.

######  Theorem 3.7.

Let $\mathcal{A}$ be a strictly enriched category over $\mathcal{M}$ and let
$F:\mathcal{A}\to\mathcal{C}$ be a strict $\mathcal{M}$-functor which is
homotopy fully faithful. Then the $\mathcal{M}_{\infty}$-functor
$F_{\infty}:\mathcal{A}_{\infty}\to\mathcal{C}_{\infty}$ is
$\mathcal{M}_{\infty}$-fully faithful (in the sense described after Remark
3.4).

###### Proof.

Lat $L:\mathcal{M}\to\mathcal{M}_{\infty}$ denote the localization functor.
Then for every $x,y\in X$ we have a commutative square in
$\mathcal{M}_{\infty}$

$\textstyle{L(\mathcal{A}(x,y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{LF}$$\textstyle{L(\operatorname{Hom}_{\mathcal{C}}(F(x),F(y)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}_{\infty}(x,y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{\infty}}$$\textstyle{\operatorname{Hom}_{\mathcal{C}_{\infty}}(F_{\infty}(x),F_{\infty}(y)).}$

The left map is an equivalence by definition of $\mathcal{A}_{\infty}$ and the
right map is equivalent to $L$ applied to the map
$\operatorname{Hom}_{\mathcal{C}}(F(x),F(y))\to\operatorname{Hom}_{\mathcal{C}}(F(x)^{c},F(y)^{f})$.
Since $F$ is homotopy fully faithful, the diagonal map is an equivalence, and
thus also the bottom map. ∎

######  Corollary 3.8.

Let $\mathcal{A}$ be a strictly enriched category over $\mathcal{M}$ and let
$F:\mathcal{A}\to\mathcal{C}$ be a strict fully faithful $\mathcal{M}$-functor
that lands in the fibrant cofibrant objects. Then the
$\mathcal{M}_{\infty}$-functor
$F_{\infty}:\mathcal{A}_{\infty}\to\mathcal{C}_{\infty}$ is
$\mathcal{M}_{\infty}$-fully faithful.

### Enriched Yoneda Lemma

Hinich formulates and proves a version of enriched Yoneda lemma, which is of
key importance to us. We will review this part of Hinich's work next.

Let $\mathcal{M}$ be a closed monoidal $\infty$-category and let $\mathcal{A}$
be an $\mathcal{M}$-enriched category with space of objects $X$. Hinich
defines the opposite category $\mathcal{A}^{\operatorname{op}}$, which is an
$\mathcal{M}^{rev}$-enriched category with space of objects
$X^{\operatorname{op}}$, and constructs a structure of a category left-
tensored over $\mathcal{M}$ on the $\infty$-category of
$\mathcal{M}$-presheaves

$P_{\mathcal{M}}(\mathcal{A}):=\operatorname{Fun}_{\mathcal{M}^{rev}}(\mathcal{A}^{\operatorname{op}},\mathcal{M}).$

Here, $\mathcal{M}$ is considered as a right $\mathcal{M}$-module which is the
same as a left $\mathcal{M}^{rev}$-module.

######  Remark 3.9.

In the case of interest to us, $\mathcal{M}$ is the category of spectra, which
is a symmetric monoidal category. This means that there is a canonical
equivalence of monoidal categories $\mathcal{M}\simeq\mathcal{M}^{rev}$.

Hinich also constructs an $\mathcal{M}$-fully faithful functor called the
enriched Yoneda embedding

$Y:\mathcal{A}\to P_{\mathcal{M}}(\mathcal{A}).$

In [Hin3] it is shown that this construction has the following universal
property: If $\mathcal{B}$ is any category left-tensored over $\mathcal{M}$
then precomposition with $Y$ induces an equivalence

$\operatorname{Map}_{\mathtt{LMod}_{\mathcal{M}}}(P_{\mathcal{M}}(\mathcal{A}),\mathcal{B})\simeq\operatorname{Map}_{\mathcal{M}}(\mathcal{A},\mathcal{B}).$

In [Hin3], all the above is done more generally relative to an $\infty$-operad
$\mathcal{O}$. Taking $\mathcal{O}=\mathtt{Com}$ to be the terminal
$\infty$-operad and noting that
$\mathtt{Com}\otimes\mathtt{Ass}\simeq\mathtt{Com}$, we obtain the following.
Suppose $\mathcal{M}$ is a closed symmetric monoidal $\infty$-category. Then
the category of $\mathcal{M}$-left-tensored categories is symmetric monoidal
and we define a symmetric monoidal $\mathcal{M}$-left-tensored category to be
a commutative algebra in the category of $\mathcal{M}$-left-tensored
categories. Similarly, the category of $\mathcal{M}$-enriched categories is
symmetric monoidal and we define a symmetric monoidal $\mathcal{M}$-enriched
category to be to be a commutative algebra in the category of
$\mathcal{M}$-enriched categories. Moreover, one can define the notion of a
symmetric monoidal $\mathcal{M}$-functor from a symmetric monoidal
$\mathcal{M}$-enriched category to a symmetric monoidal $\mathcal{M}$-left-
tensored category.

If $\mathcal{A}$ is a symmetric monoidal $\mathcal{M}$-enriched category, the
category of presheaves $P_{\mathcal{M}}(\mathcal{A})$ acquires a canonical
symmetric monoidal $\mathcal{M}$-left-tensored structure (Day convolution),
and the Yoneda embedding $Y:\mathcal{A}\to P_{\mathcal{M}}(\mathcal{A})$
acquires a structure of a symmetric monoidal $\mathcal{M}$-functor. Moreover,
this construction has the following universal property: If $\mathcal{B}$ is
any symmetric monoidal $\mathcal{M}$-left-tensored category then
precomposition with $Y$ induces an equivalence

$\operatorname{Map}^{\mathtt{Com}}_{\mathtt{LMod}_{\mathcal{M}}}(P_{\mathcal{M}}(\mathcal{A}),\mathcal{B})\simeq\operatorname{Map}^{\mathtt{Com}}_{\mathcal{M}}(\mathcal{A},\mathcal{B}).$

##  4 Stable $\infty$-categories and spectral presheaves

In this section we consider the notion of stable $\infty$-categories. We show
that a compactly generated stable $\infty$-category is equivalent to
$P_{\mathtt{Sp}}(\mathcal{A})$ for some small $\mathtt{Sp}$-enriched category
$\mathcal{A}$.

Let $\mathcal{D}$ be a pointed finitely cocomplete $\infty$-category. We
define the _suspension functor_ on $\mathcal{D}$

$\Sigma_{\mathcal{D}}\colon\mathcal{D}\to\mathcal{D}$

by the formula

$\Sigma_{\mathcal{D}}(X):=*\coprod_{X}*.$

Alternatively, the suspension functor can be defined as the smash product
$S^{1}\wedge X$, using the fact that a pointed finitely cocomplete
$\infty$-category is tensored over pointed spaces.

If the suspension functor is an equivalence of categories, then $\mathcal{D}$
is called _stable_. A stable presentable $\infty$-category is naturally left-
tensored over the closed monoidal $\infty$-category of spectra $\mathtt{Sp}$
(see [Lur2, Proposition 4.8.2.18]). Moreover, $\mathtt{Sp}$ is presentable, so
for every $b,c\in\mathcal{D}$ the presheaf
$\operatorname{Hom}_{\mathcal{D}}(b,c)\in P(\mathtt{Sp})$ is representable (we
will denote the representing object also by
$\operatorname{Hom}_{\mathcal{D}}(b,c)\in\mathtt{Sp}$). By Proposition 3.5 it
follows that a stable presentable $\infty$-category is canonically enriched
over $\mathtt{Sp}$ (this was observed by Gepner-Haugseng in [GH, Example
7.4.14], where they also pointed out that the presentability assumption is
unnecessary).

######  Theorem 4.1.

Let $\mathcal{D}$ be a cocomplete stable $\infty$-category. Suppose that there
is a small set $C$ of compact objects in $\mathcal{D}$, that generates
$\mathcal{D}$ under colimits and desuspensions. Thinking of $\mathcal{D}$ as
left-tensored over $\mathtt{Sp}$, we let $\mathcal{C}$ be the full
$\mathtt{Sp}$-enriched subcategory of $\mathcal{D}$ spanned by $C$. Then we
have a natural functor of categories left-tensored over $\mathtt{Sp}$

$P_{\mathtt{Sp}}(\mathcal{C})\xrightarrow{\sim}\mathcal{D},$

which is an equivalence of the underlying $\infty$-categories and sends each
representable presheaf $Y(c)\in P_{\mathtt{Sp}}(\mathcal{C})$ to $c\in C$.

######  Remark 4.2.

Theorem 4.1 appears in [Lur2, Theorem 7.1.2.1] for the case that $|C|=1$. The
general case, formulated in the language of model categories, can be found in
[ScSh1, Theorem 3.3.3]. In [GM] the last result can be found under more
general hypotheses

###### Proof.

By defintion of full enriched subcategory, there is a fully-faithful
$\mathtt{Sp}$-functor $i:\mathcal{C}\to\mathcal{D}$. By the universal property
of the Yoneda embedding, we have an induced functor of categories left-
tensored over $\mathtt{Sp}$

$I\colon P_{\mathtt{Sp}}(\mathcal{C})\to\mathcal{D},$

such that $I\circ Y\simeq i$. We thus get an equivalence $I(Y(c))\simeq
i(c)\simeq c$, for every $c\in\mathcal{C}$, and it remains to show that $I$ is
an equivalence.

The functor $I$ is a morphism of left modules over $\mathtt{Sp}$ in
$\mathtt{Cat^{L}}$, so in particular $I$ commutes with colimits. The
$\infty$-category $\mathcal{D}$ is presentable by [Lur1, Theorem 5.5.1.1]. Let
$X$ denote the full subspace of $\mathcal{D}$ generated by $C$ and recall that
the $\infty$-category $P_{\mathtt{Sp}}(\mathcal{C})$ is defined as the
category of left $\mathcal{C}^{\operatorname{op}}$-modules with values in
$\operatorname{Fun}(X^{\operatorname{op}},\mathtt{Sp})$. Since
$\operatorname{Fun}(X^{\operatorname{op}},\mathtt{Sp})$ is stable and
presentable, so is $P_{\mathtt{Sp}}(\mathcal{C})$ (see [Lur2, 1.1.3.1 and
4.2.3.5]). Thus, by the adjoint functor theorem $I$ has a right adjoint $J$:

$I\colon
P_{\mathtt{Sp}}(\mathcal{C})\rightleftarrows\mathcal{D}\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}J.$

We first show that the unit $Y(c)\to J(I(Y(c)))$ of the adjunction $I\dashv J$
is an equivalence for every $c\in\mathcal{C}$. It is not hard to show that $J$
preserves $\mathtt{Sp}$-enrichment, so both $Y$ and $J\circ I\circ Y$ are
$\mathtt{Sp}$-functors $\mathcal{C}\to P_{\mathtt{Sp}}(\mathcal{C})$, and that
the unit induces a map $Y\to J\circ I\circ Y$ of $\mathtt{Sp}$-functors. Note
that

$\operatorname{Fun}_{\mathtt{Sp}}(\mathcal{C},P_{\mathtt{Sp}}(\mathcal{C}))=\mathtt{LMod}_{\mathcal{C}}(\operatorname{Fun}(X,P_{\mathtt{Sp}}(\mathcal{C})))=$
$\mathtt{LMod}_{\mathcal{C}}(\operatorname{Fun}(X,\operatorname{Fun}_{\mathtt{Sp}^{rev}}(\mathcal{C}^{\operatorname{op}},\mathtt{Sp})))=\mathtt{LMod}_{\mathcal{C}}(\operatorname{Fun}(X,\mathtt{LMod}_{\mathcal{C}^{\operatorname{op}}}(\operatorname{Fun}(X^{\operatorname{op}},\mathtt{Sp}))))=$
$\mathtt{LMod}_{\mathcal{C}}(\mathtt{LMod}_{\mathcal{C}^{\operatorname{op}}}(\operatorname{Fun}(X,\operatorname{Fun}(X^{\operatorname{op}},\mathtt{Sp}))))=\mathtt{LMod}_{\mathcal{C}}(\mathtt{RMod}_{\mathcal{C}}(\operatorname{Fun}(X^{\operatorname{op}}\times
X,\mathtt{Sp}))),$

so an $\mathtt{Sp}$-functor $\mathcal{C}\to P_{\mathtt{Sp}}(\mathcal{C})$ is
the same as a $\mathcal{C}$-$\mathcal{C}$-bimodule in the category
$\operatorname{Fun}(X^{\operatorname{op}}\times X,\mathtt{Sp})$. Thus we can
view $Y\to J\circ I\circ Y$ as a map of $\mathcal{C}$-$\mathcal{C}$-bimodules
in $\operatorname{Fun}(X^{\operatorname{op}}\times X,\mathtt{Sp})$ and we need
to show that it is an equivalence. The forgetful functor to
$\operatorname{Fun}(X^{\operatorname{op}}\times X,\mathtt{Sp})$ reflects
equivalences, and an equivalence in
$\operatorname{Fun}(X^{\operatorname{op}}\times X,\mathtt{Sp})$ can be
verified objectwise, so we can fix two objects $c,d$ in $\mathcal{C}$ and show
that the induced map of spectra

$Y(c,d)\to(J\circ I\circ Y)(c,d)$

is an equivalence. But since $Y$ and $i$ are $\mathtt{Sp}$-fully faithful, we
have

$(J\circ I\circ Y)(c,d)=J(I(Y(d)))(c)\simeq$
$\operatorname{Hom}_{P_{\mathtt{Sp}}(\mathcal{C})}(Y(c),J(I(Y(d))))\simeq\operatorname{Hom}_{\mathcal{D}}(I(Y(c)),I(Y(d)))\simeq$
$\operatorname{Hom}_{\mathcal{D}}(i(c),i(d))\simeq{\mathcal{C}}(c,d)\simeq\operatorname{Hom}_{P_{\mathtt{Sp}}(\mathcal{C})}(Y(c),Y(d))\simeq
Y(d)(c)\simeq Y(c,d).$

Since $I(Y(c))\simeq c$ and $J(c)\simeq Y(c)$, the counit $I(J(c))\to c$ of
$I\dashv J$ is also an equivalences, for every $c\in C$. Note that $C$
generates $\mathcal{D}$ under colimits, $\\{Y(c)|c\in C\\}$ generates
$P_{\mathtt{Sp}}(\mathcal{C})$ under colimits and the functor $I$ commutes
with colimits. Thus, if we can show that $J$ also commutes with colimits, it
would follow that the unit and counit of $I\dashv J$ are equivalences, and we
are done.

Let us show first that $J$ commutes with filtered colimits. So let
$d=\mathop{\operatorname{colim}}_{i\in I}d_{i}$ be a filtered colimit diagram
in $\mathcal{D}$. We need to verify that the induced map
$\mathop{\operatorname{colim}}_{i\in I}J(d_{i})\to J(d)$ is an equivalence.
Recall that

$P_{\mathtt{Sp}}(\mathcal{C})=\operatorname{Fun}_{\mathtt{Sp}^{rev}}(\mathcal{C}^{\operatorname{op}},\mathtt{Sp})=\mathtt{LMod}_{\mathcal{C}^{\operatorname{op}}}(\operatorname{Fun}(X^{\operatorname{op}},\mathtt{Sp})).$

The forgetful functor $U$ from $P_{\mathtt{Sp}}(\mathcal{C})$ to
$\operatorname{Fun}(X^{\operatorname{op}},\mathtt{Sp})$ commutes with colimits
(see [Lur2, 4.2.3.5]) and reflects equivalences, so it is enough to verify
that

${\mathop{\operatorname{colim}}}_{i\in I}U(J(d_{i}))\to U(J(d))$

is an equivalence. Now colimits in
$\operatorname{Fun}(X^{\operatorname{op}},\mathtt{Sp})$ are pointwise, so we
can fix $c\in\mathcal{C}$ and show that

${\mathop{\operatorname{colim}}}_{i\in I}U(J(d_{i}))(c)\to U(J(d))(c)$

in an equivalence. We have an equivalence natural in $e\in\mathcal{D}$

$U(J(e))(c)=J(e)(c)=\operatorname{Hom}_{P_{\mathtt{Sp}}(\mathcal{C})}(Y(c),J(e))\simeq$
$\operatorname{Hom}_{\mathcal{D}}(I(Y(c)),e)\simeq\operatorname{Hom}_{\mathcal{D}}(i(c),e)\simeq\operatorname{Hom}_{\mathcal{D}}(c,e),$

so it is enough to show that

${\mathop{\operatorname{colim}}}_{i\in
I}\operatorname{Hom}_{\mathcal{D}}(c,d_{i})\to\operatorname{Hom}_{\mathcal{D}}(c,d)$

is an equivalence, which is true by the compactness of $c$ in $\mathcal{D}$.

Since both range and domain of $J$ are stable, and in a stable
$\infty$-category every pullback square is a pushout square and vice versa, it
follows that $J$ sends pushout squares to pushout squares. Thus $J$ commutes
with all small colimits. ∎

Using the results in [Hin3] one can prove an extension to Theorem 4.1:

######  Theorem 4.3.

In the situation of Theorem 4.1, suppose $\mathcal{D}$ is symmetric monoidal
and the set $C$ is closed under the monoidal product in $\mathcal{D}$ and
contains the unit of $\mathcal{D}$. Then $\mathcal{C}$ acquires a canonical
symmetric monoidal $\mathtt{Sp}$-enriched structure, the category of
presheaves $P_{\mathtt{Sp}}(\mathcal{C})$ acquires a canonical symmetric
monoidal left $\mathtt{Sp}$-tensored structure and the equivalence
$P_{\mathtt{Sp}}(\mathcal{C})\xrightarrow{\sim}\mathcal{D}$ acquires a
canonical symmetric monoidal left $\mathtt{Sp}$-tensored structure.

##  5 Stabilization of categories

In this section we review the notion of stabilization of an $\infty$-category.
We will use the framework established by Lurie in [Lur2, Section 1.4]. We
present in subsection 5.0.1 a similar procedure that can be applied to an
ordinary topologically enriched category. We will compare the strict and the
$\infty$-categorical versions of stabilization.

Let $\mathcal{C}$ be a pointed $\infty$-category. To ensure that $\mathcal{C}$
has all the good properties we may want, we will assume that
$\mathcal{C}\simeq\operatorname{Ind}(\mathcal{C}_{0})$ where $\mathcal{C}_{0}$
is a small pointed $\infty$-category that is closed under finite colimits.
This includes $\mathtt{NCW}\simeq\operatorname{Ind}(\mathtt{NCW}^{f})$. By
[Lur1, Theorem 5.5.1.1] $\mathcal{C}$ is presentable, and therefore has small
limits and colimits [op. cit., Corollary 5.5.2.4]. Furthermore, filtered
colimits commute with finite limits in $\mathcal{C}$ by the remark immediately
following [Lur1, Definition 5.5.7.1]. It follows, in particular, that
$\mathcal{C}$ is differentiable in the sense of [Lur2, Definition 6.1.1.6] and
therefore the results of [op. cit., Chapter 6] apply to $\mathcal{C}$.

Recall that $\mathtt{CW}^{f}_{*}$ is the $\infty$-category of pointed finite
CW-complexes. Let $F\colon\mathtt{CW}^{f}_{*}\to\mathcal{C}$ be a functor.
Recall that $F$ is called reduced if $F(*)$ is a final object of
$\mathcal{C}$, and $F$ is called $1$-excisive if $F$ takes pushout squares to
pullback squares. Let linear functors be functors that are both reduced and
$1$-excisive. Linear functors provide a good framework for defining spectra in
the context of general $\infty$-categories.

######  Definition 5.1 ([Lur2], Definition 1.4.2.8).

A spectrum object in $\mathcal{C}$ is a linear functor
$\mathtt{CW}^{f}_{*}\to\mathcal{C}$. Let $\mathtt{Sp}(\mathcal{C})$ be the
$\infty$-category of linear functors $\mathtt{CW}^{f}_{*}\to\mathcal{C}$.

$\mathtt{Sp}(\mathcal{C})$ is called the category of spectra of $\mathcal{C}$,
or the stabilization of $\mathcal{C}$. By results in [Lur2],
$\mathtt{Sp}(\mathcal{C})$ is a stable and presentable $\infty$-category
(Corollary 1.4.2.17 and Proposition 1.4.4.4 respectively).

Let $\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\mathcal{C})$ be the category
of all pointed functors from $\mathtt{CW}^{f}_{*}$ to $\mathcal{C}$. Then
$\mathtt{Sp}(\mathcal{C})$ is by definition a full subcategory of
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\mathcal{C})$. The fully faithful
functor
$\mathtt{Sp}(\mathcal{C})\hookrightarrow\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\mathcal{C})$
has a left adjoint

$\mathcal{L}\colon\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\mathcal{C})\to\mathtt{Sp}(\mathcal{C})$

called linearization. Explicitly, if
$F\colon\mathtt{CW}^{f}_{*}\to\mathcal{C}$ is a pointed functor, then the
linearization of $F$ is given by the following formula

$\mathcal{L}F(X)=\mathop{\operatorname{colim}}_{n\to\infty}\Omega^{n}_{\mathcal{C}}F(\Sigma^{n}X).$

See [Lur2, Example 6.1.1.28] for a discussion of this formula in the context
of $\infty$-categories (of course this formula is older than [Lur2] and goes
back at least to [Goo]). The stabilization $\mathtt{Sp}(\mathcal{C})$ is the
left Bousfield localization of
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\mathcal{C})$ at the stable
equivalences and $\mathcal{L}$ is the localization functor (a map between
functors is a stable equivalence if it induces an equivalence between
linearizations).

There is an adjoint pair of functors

$\Sigma_{\mathcal{C}}^{\infty}\colon\mathcal{C}\leftrightarrows\mathtt{Sp}(\mathcal{C})\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}\Omega^{\infty}_{\mathcal{C}},$

where
$\Sigma^{\infty}_{\mathcal{C}}x(K)=\mathop{\operatorname{colim}}_{n\to\infty}\Omega^{n}_{\mathcal{C}}\Sigma^{n}_{\mathcal{C}}(K\wedge
x)$ and $\Omega^{\infty}_{\mathcal{C}}G=G(S^{0})$. This formula for
$\Omega^{\infty}_{\mathcal{C}}$ agrees with the one in [Lur2, Notation
1.4.2.20], and therefore our $\Sigma^{\infty}_{\mathcal{C}}$, being left
adjoint to $\Omega^{\infty}_{\mathcal{C}}$ is also equivalent to Lurie's. The
functor $\Sigma^{\infty}_{\mathcal{C}}$ satisfies the following universal
property: For every stable presentable $\infty$-category $\mathcal{D}$, pre-
composition with $\Sigma^{\infty}_{\mathcal{C}}$ induces an equivalence of
$\infty$-categories

$\mathtt{Fun^{L}}(\mathtt{Sp}(\mathcal{C}),\mathcal{D})\xrightarrow{\simeq}\mathtt{Fun^{L}}(\mathcal{C},\mathcal{D}),$

where $\mathtt{Fun^{L}}$ denotes left functors (that is, colimit preserving
functors).

An important special case is when $\mathcal{C}=\mathcal{S}_{*}$ is the
$\infty$-category of pointed spaces. In this case
$\mathtt{Sp}:=\mathtt{Sp}(\mathcal{S}_{*})$ is the classical $\infty$-category
of spectra, presented as the category of linear functors from
$\mathtt{CW}^{f}_{*}$ to $\mathcal{S}_{*}$. Whenever
$\mathcal{C}=\mathcal{S}_{*}$ we write $\mathtt{Sp}$, $\Sigma^{\infty}$ or
$\Omega^{\infty}$, omitting the subscript $\mathcal{C}$.

There is another useful way to construct $\mathtt{Sp}(\mathcal{C})$ when
$\mathcal{C}=\operatorname{Ind}(\mathcal{C}_{0})$, with $\mathcal{C}_{0}$
closed under finite colimits. We will now describe it.

######  Definition 5.2.

Let $\mathcal{C}_{0}$ be an $\infty$-category closed under finite colimits.
Define the Spanier-Whitehead category of $\mathcal{C}_{0}$, which we denote by
$\mathtt{SW}(\mathcal{C}_{0})$, to be the colimit of the sequence

$\mathcal{C}_{0}\xrightarrow{\Sigma_{\mathcal{C}_{0}}}\mathcal{C}_{0}\xrightarrow{\Sigma_{\mathcal{C}_{0}}}\cdots$

in $\mathtt{Cat}$.

Thus, the objects of $\mathtt{SW}(\mathcal{C}_{0})$ are pairs $(X,n)$ where
$X\in\mathcal{C}_{0}$ and $n\in\mathbb{N}$. The pair $(X,n)$ represents the
$n$-fold desuspension of $X$. The mapping spaces in
$\mathtt{SW}(\mathcal{C}_{0})$ are given by

$\operatorname{Map}_{\mathtt{SW}(\mathcal{C}_{0})}((X,n),(Y,m))={\mathop{\operatorname{colim}}}_{k\in\mathbb{N}}\operatorname{Map}_{\mathcal{C}_{0}}(\Sigma_{\mathcal{C}}^{k-n}X,\Sigma_{\mathcal{C}}^{k-m}Y),$

where the colimit is taken in the $\infty$-category of spaces. Clearly,
$\mathtt{SW}(\mathcal{C}_{0})$ is a stable $\infty$-category. It is closed
under finite colimits, but not under arbitrary colimits. It plays the role of
the category of finite spectra over $\mathcal{C}$. There is a finite
suspension spectrum functor

${\Sigma^{\infty}_{\mathcal{C}_{0}}}^{f}:\mathcal{C}_{0}\to\mathtt{SW}(\mathcal{C}_{0})$

given by $X\mapsto(X,0)$, which satisfies the following universal property:
For every stable $\infty$-category $\mathcal{D}$, pre-composition with
${\Sigma^{\infty}_{\mathcal{C}_{0}}}^{f}$ induces an equivalence of
$\infty$-categories

$\mathtt{Fun_{*}^{fc}}(\mathtt{SW}(\mathcal{C}_{0}),\mathcal{D})\xrightarrow{\simeq}\mathtt{Fun_{*}^{fc}}(\mathcal{C}_{0},\mathcal{D}),$

where $\mathtt{Fun_{*}^{fc}}$ denotes pointed finite colimit preserving
functors.

One has the following description of
$\mathtt{Sp}(\operatorname{Ind}(\mathcal{C}_{0}))$:

######  Proposition 5.3.

Let $\mathcal{C}_{0}$ be a small pointed finitely cocomplete
${\infty}$-category, and let
$\mathcal{C}:=\operatorname{Ind}(\mathcal{C}_{0})$. Then there is a natural
equivalence

$\mathtt{Sp}(\mathcal{C})\simeq\operatorname{Ind}(\mathtt{SW}(\mathcal{C}_{0})).$

and under this equivalence,

$\Sigma^{\infty}_{\mathcal{C}}:\operatorname{Ind}(\mathcal{C}_{0})\simeq\mathcal{C}\to\mathtt{Sp}(\mathcal{C})\simeq\operatorname{Ind}(\mathtt{SW}(\mathcal{C}_{0})),$

is just the prolongation of

${\Sigma^{\infty}_{\mathcal{C}_{0}}}^{f}:{\mathcal{C}_{0}}\to\mathtt{SW}({\mathcal{C}_{0}}).$

###### Proof.

This is proved in [Lur2, Chapter 1.4] in the case when
$\mathcal{C}_{0}=\mathtt{CW}^{f}_{*}$ and
$\mathcal{C}=\operatorname{Ind}(\mathtt{CW}^{f}_{*})\simeq\mathcal{S}_{*}$.
The proof in the general case is similar. In brief, one can check that
$\operatorname{Ind}(\mathtt{SW}(\mathcal{C}_{0}))$ satisfies the same
universal property as $\mathtt{Sp}(\mathcal{C})$. ∎

This proposition has the following rather important corollary.

######  Corollary 5.4.

Let $\mathcal{C}_{0},\mathcal{C}$ be as before. Suppose $A$ is a set of
objects of $\mathcal{C}_{0}$ that generates $\mathcal{C}_{0}$ under finite
colimits, in the sense that $\mathcal{C}_{0}$ is the only subcategory of
$\mathcal{C}_{0}$ that contains $A$ and is closed under finite colimits and
equivalences. Then the set of suspension spectra
$\\{\Sigma^{\infty}_{\mathcal{C}}x\mid x\in A\\}$ generates
$\mathtt{SW}(\mathcal{C}_{0})$ under finite colimits and desuspensions, and
generates $\mathtt{Sp}(\mathcal{C})$ under arbitrary colimits and
desuspensions.

The following notation is going to be used quite a lot.

######  Definition 5.5.

Suppose $\mathcal{C}$ is a pointed $\infty$-category with finite colimits. Let
$x$ and $y$ be object of $\mathcal{C}$. Then the functor
$G^{\mathcal{C}}_{x,y}\colon\mathtt{CW}^{f}_{*}\to\mathcal{S}_{*}$ is defined
by $G^{\mathcal{C}}_{x,y}(K)=\operatorname{Map}_{\mathcal{C}}(x,K\wedge y)$.
Sometimes we will omit the superscript $\mathcal{C}$ and write simply
$G_{x,y}$.

######  Lemma 5.6.

If $\mathcal{C}$ is a stable $\infty$-category then $G_{x,y}$ is linear for
any two objects $x,y$.

###### Proof.

We need to prove that $G_{x,y}(*)\simeq*$ and that $G_{x,y}$ is $1$-excisive.
The first condition holds because $*\wedge y$ is equivalent to a final object
of $\mathcal{C}$. Now let us prove that $G_{x,y}$ is $1$-excisive. We have
equivalences

$\operatorname{Map}_{\mathcal{C}}(x,K\wedge
y)\xrightarrow{\simeq}\operatorname{Map}_{\mathcal{C}}(S^{1}\wedge
x,S^{1}\wedge K\wedge
y)\xrightarrow{\simeq}\operatorname{Map}_{\mathcal{C}}(x,\Omega(S^{1}\wedge
K\wedge y)).$

Here the first map is an equivalence because $\mathcal{C}$ is stable, and the
second equivalence is a standard adjunction. The composite equivalence can be
reinterpreted as saying that the canonical map $G_{x,y}\to\Omega
G_{x,y}\Sigma$ is an equivalence. It follows that the map
$G_{x,y}\to\mathcal{L}G_{x,y}$ is an equivalence, so $G_{x,y}$ is linear. ∎

If $\mathcal{C}$ is a stable $\infty$-category then $G^{\mathcal{C}}_{x,y}$ is
in fact equivalent to the canonical enrichment of $\mathcal{C}$ over spectra.
The next lemma and remark say that more generally the linearization of
$G^{\mathcal{C}}$ gives, in favorable circumstances, the spectral enrichment
of the stabilization of $\mathcal{C}$.

######  Lemma 5.7.

Let $\mathcal{C}_{0}$ be a small finitely cocomplete $\infty$-category. To
simplify notation, let $S\colon\mathcal{C}_{0}\to\mathtt{SW}(\mathcal{C}_{0})$
be the finite suspension spectrum functor. Then the natural map

$G^{\mathcal{C}_{0}}_{x,y}\to G^{\mathtt{SW}(\mathcal{C}_{0})}_{S(x),S(y)}$

induced by the finite suspension spectrum functor is equivalent to the
linearization map

$G^{\mathcal{C}_{0}}_{x,y}\to\mathcal{L}G^{\mathcal{C}_{0}}_{x,y}.$

###### Proof.

By definition, we have equivalences

$G^{\mathtt{SW}(\mathcal{C}_{0})}_{S(x),S(y)}(K)=\operatorname{Map}_{\mathtt{SW}(\mathcal{C}_{0})}(S(x),K\wedge
S(y))=\mathop{\operatorname{colim}}_{n\to\infty}\operatorname{Map}_{\mathcal{C}_{0}}(S^{n}\wedge
x,S^{n}\wedge K\wedge y)=\\\
=\mathop{\operatorname{colim}}_{n\to\infty}\Omega^{n}\operatorname{Map}_{\mathcal{C}_{0}}(x,S^{n}\wedge
K\wedge
y)=\mathop{\operatorname{colim}}_{n\to\infty}\Omega^{n}G^{\mathcal{C}_{0}}_{x,y}(S^{n}\wedge
K)=\mathcal{L}G^{\mathcal{C}_{0}}_{x,y}(K)$

and the map from $G^{\mathcal{C}_{0}}_{x,y}$ is precisely the linearization
map. ∎

######  Remark 5.8.

If $\mathcal{C}=\operatorname{Ind}\mathcal{C}_{0}$, with $\mathcal{C}_{0}$ as
in the previous lemma, and $x,y$ are objects of $\mathcal{C}$, it is not true
that
$G^{\mathtt{Sp}(\mathcal{C})}_{\Sigma^{\infty}_{\mathcal{C}}(x),\Sigma^{\infty}_{\mathcal{C}}(y)}$
is equivalent to the linearization of $G^{\mathcal{C}}_{x,y}$. Rather, there
is an equivalence

$G^{\mathtt{Sp}(\mathcal{C})}_{\Sigma^{\infty}_{\mathcal{C}}(x),\Sigma^{\infty}_{\mathcal{C}}(y)}(K)=\operatorname{Map}_{\mathcal{C}}(x,\mathop{\operatorname{colim}}_{n\to\infty}\Omega^{n}(S^{n}\wedge
y)).$

This is equivalent to $\mathcal{L}G^{\mathcal{C}}_{x,y}(K)$ if $x$ is a
compact object, but not in general. In particular, it is true when
$x\in\mathcal{C}_{0}$, which is the case considered in the previous lemma.

###  5.0.1 Spectral enrichment of pointed topological categories

Suppose $\mathcal{C}$ is an $\infty$-category and $x,y$ are objects of
$\mathcal{C}$. We saw that functors of the form $G^{\mathcal{C}}_{x,y}$ can be
used to define the spectral enrichment of the stabilization of $\mathcal{C}$.
If $\mathcal{C}$ is an ordinary topological category, one can use similar
functors $G^{\mathcal{C}}_{x,y}$ to define a spectral enrichment of
$\mathcal{C}$, using the more traditional view of spectra as modeled by the
Quillen model category of continuous functors. In this subsection we define a
strict spectral enrichment of pointed topological categories and compare it
with the $\infty$-categorical enrichment.

Let $\operatorname{Top}$ denote the category of pointed compactly generated
weak Hausdorff spaces with the standard model structure of Quillen [Qui].
Every object in $\operatorname{Top}$ is fibrant, and every CW-complex is
cofibrant. The model category $\operatorname{Top}$ is a model for the
$\infty$-category of pointed spaces $\mathcal{S}_{*}$. This means that the
$\infty$-localization of $\operatorname{Top}$ with respect to the weak
equivalences (see Remark 2.1) is canonically equivalent to $\mathcal{S}_{*}$.
Note that any pointed topological category is naturally enriched in
$\operatorname{Top}$.

######  Definition 5.9.

A pointed topological category $\mathcal{C}$ is called tensored closed over
$\mathtt{CW}^{f}_{*}$ if we are given a bi-continuous left action
$\wedge:\mathtt{CW}^{f}_{*}\times\mathcal{C}\to\mathcal{C}$ such that the
following hold:

  1. 1.

The $\infty$-category $\mathcal{C}_{\infty}$ is finitely cocomplete, where
$\mathcal{C}_{\infty}$ is the topological nerve of $\mathcal{C}$.

  2. 2.

After application of the topological nerve the functor

$\wedge_{\infty}:\mathtt{CW}^{f}_{*}\times\mathcal{C}_{\infty}\to\mathcal{C}_{\infty}$

commutes with finite colimits in each variable.

######  Definition 5.10.

Let $\mathcal{C}$ be a pointed topological category, tensored closed over
$\mathtt{CW}^{f}_{*}$. Let $x$ and $y$ be objects of $\mathcal{C}$. In keeping
with notation we introduced in Definition 5.5, we define the pointed
topological functor
$G^{\mathcal{C}}_{x,y}\colon\mathtt{CW}^{f}_{*}\to\operatorname{Top}$ by
$G^{\mathcal{C}}_{x,y}(K)=\operatorname{Map}_{\mathcal{C}}(x,K\wedge y)$.
Again, we may omit the superscript $\mathcal{C}$ and write simply $G_{x,y}$.

######  Remark 5.11.

We saw earlier that pointed $\infty$-functors from $\mathtt{CW}^{f}_{*}$ to
$\mathcal{S}_{*}$ provide a way of defining the $\infty$-category of spectra.
This is known also in the more traditional approach to spectra via model
categories. There is a Quillen model structure on the category
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})$ of continuous
pointed functors, called the stable model structure, and it provides one of
the models for the category of spectra. We refer the reader to [Lyd, MMSS] for
more details about this model structure.

We denote the category
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})$ with the
stable model structure by $\mathtt{Sp^{M}}$. We denote the category
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})$ with the
projective model structure simply by
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})$. The model
category $\mathtt{Sp^{M}}$ is a left Bousfield localization of
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})$, so we have a
Quillen pair

$\operatorname{Id}:\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})\rightleftarrows\mathtt{Sp^{M}}:\operatorname{Id}.$

Applying $\infty$-localization, this Quillen pair becomes the localization
adjunction

$\mathcal{L}:\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\mathcal{S}_{*})\rightleftarrows\mathtt{Sp}:\i.$

Let $\mathcal{C}$ be a pointed topological category, tensored closed over
$\mathtt{CW}^{f}_{*}$ and let $x,y\in\mathcal{C}$. By the previous definition
we have a functor
$G^{\mathcal{C}}_{x,y}\colon\mathtt{CW}^{f}_{*}\to\operatorname{Top}$. Under
the identification
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})_{\infty}\simeq\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\mathcal{S}_{*})$,
$G^{\mathcal{C}}_{x,y}$ can be thought of as an object in
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\mathcal{S}_{*})$. We will now
compare this with the functor
$G^{\mathcal{C}_{\infty}}_{x,y}\colon\mathtt{CW}^{f}_{*}\to\mathcal{S}_{*}$
from Definition 5.5.

Clearly, the bi-functor

$\wedge_{\infty}:\mathtt{CW}^{f}_{*}\times\mathcal{C}_{\infty}\to\mathcal{C}_{\infty}$

is a left action of the monoidal $\infty$-category $\mathtt{CW}^{f}_{*}$ on
$\mathcal{C}_{\infty}$. It follows that we have an induced action on the ind-
categories

$\wedge_{\infty}:\mathcal{S}_{*}\times\operatorname{Ind}(\mathcal{C}_{\infty})\to\operatorname{Ind}(\mathcal{C}_{\infty})$

and this action commutes with small colimits in each variable. Since
$\mathcal{S}_{*}$ is a mode in the sense of [CSY, Section 5], this action
coincides with the canonical action of $\mathcal{S}_{*}$ on
$\operatorname{Ind}(\mathcal{C}_{\infty})$ as a presentable pointed
$\infty$-category. In particular we get that the restriction

$\wedge_{\infty}:\mathtt{CW}^{f}_{*}\times\mathcal{C}_{\infty}\to\mathcal{C}_{\infty}$

coincides with the canonical action of $\mathtt{CW}^{f}_{*}$ on
$\mathcal{C}_{\infty}$ as an $\infty$-category with finite colimits.

It follows that under the identification
$\operatorname{Top}_{\infty}\simeq\mathcal{S}_{*}$, for any
$K\in\mathtt{CW}^{f}_{*}$ and $z\in\mathcal{C}$ we have natural equivalences

$\operatorname{Map}_{\mathcal{C}}(x,z)\simeq\operatorname{Map}_{\mathcal{C}_{\infty}}(x,z)$
$K\wedge y\simeq K\wedge_{\infty}y.$

Thus we have

$G^{\mathcal{C}_{\infty}}_{x,y}(K)=\operatorname{Map}_{\mathcal{C}_{\infty}}(x,K\wedge_{\infty}y)\simeq\operatorname{Map}_{\mathcal{C}}(x,K\wedge
y)=G^{\mathcal{C}}_{x,y}(K)$ (1)

or $G^{\mathcal{C}_{\infty}}_{x,y}\simeq G^{\mathcal{C}}_{x,y}.$

######  Definition 5.12.

Let $\mathcal{C}$ be a pointed topological category, tensored closed over
$\mathtt{CW}^{f}_{*}$. We define a strict enrichment of $\mathcal{C}$ over
$\mathtt{Sp^{M}}$ as follows: If $x$ and $y$ are objects of $\mathcal{C}$ we
define

$\operatorname{Hom}_{\mathcal{C}}(x,y):=G^{\mathcal{C}}_{x,y}\in\mathtt{Sp^{M}}.$

Let $x,y,z$ be objects of $\mathcal{C}$ and let $K$ and $L$ be finite CW-
complexes. Note that there is a natural map $G_{x,y}(K)\wedge G_{y,z}(L)\to
G_{x,z}(K\wedge L)$, defined as the composite:

$\operatorname{Map}_{\mathcal{C}}(x,K\wedge
y)\wedge\operatorname{Map}_{\mathcal{C}}(y,L\wedge z)\to\\\
\to\operatorname{Map}_{\mathcal{C}}(x,K\wedge
y)\wedge\operatorname{Map}_{\mathcal{C}}(K\wedge y,K\wedge L\wedge z)\to\\\
\to\operatorname{Map}_{\mathcal{C}}(x,K\wedge L\wedge z)$

where the first map is induced by the topological functor
$K\wedge(-):\mathcal{C}\to\mathcal{C}$ and the second map is given by
composition. This map induces a natural map $G_{x,y}\otimes G_{y,z}\to
G_{x,z}$, where $\otimes$ denotes Day convolution which is the tensor product
in $\mathtt{Sp^{M}}$. Thus we have defined composition and it can be checked
that the above indeed defines a strict enrichment of $\mathcal{C}$ over
$\mathtt{Sp^{M}}$.

######  Theorem 5.13.

Let $\mathcal{C}$ be a small pointed topological category, tensored closed
over $\mathtt{CW}^{f}_{*}$. Then under the identification
$\mathtt{Sp^{M}}_{\infty}\simeq\mathtt{Sp}$, for any two objects $x$ and $y$
of $\mathcal{C}$ we have a natural equivalence

$\operatorname{Hom}_{\mathcal{C}}(x,y)\simeq\operatorname{Hom}_{\mathtt{Sp}(\operatorname{Ind}(\mathcal{C}_{\infty}))}(\Sigma^{\infty}(x),\Sigma^{\infty}(y)).$

###### Proof.

Let $x$ and $y$ be objects in $\mathcal{C}$. Recall that
$\operatorname{Hom}_{\mathcal{C}}(x,y)=G^{\mathcal{C}}_{x,y}$ and consider
$G^{\mathcal{C}}_{x,y}$ as an object in
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})$. By (1), we
have $G^{\mathcal{C}}_{x,y}\simeq G^{\mathcal{C}_{\infty}}_{x,y}.$

We have a commutative square

$\textstyle{\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{L}\operatorname{Id}}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\mathcal{S}_{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}}$$\textstyle{\mathtt{Sp^{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathtt{Sp}}$

Let $(G^{\mathcal{C}}_{x,y})^{f}$ be a fibrant replacement to
$G^{\mathcal{C}}_{x,y}$ in $\mathtt{Sp^{M}}$. Then the map
$G^{\mathcal{C}}_{x,y}\to(G^{\mathcal{C}}_{x,y})^{f}$, considered in
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})$, translates
to
$G^{\mathcal{C}_{\infty}}_{x,y}\to\mathcal{L}G^{\mathcal{C}_{\infty}}_{x,y}$
under the top horizontal map. By Lemma 5.7 the last map is equivalent to
$G^{\mathcal{C}_{\infty}}_{x,y}\to
G^{\mathtt{SW}(\mathcal{C}_{\infty})}_{S(x),S(y)}.$ After applying
$\mathcal{L}$, this map becomes an equivalence, so we have a natural
equivalence in $\mathtt{Sp}$

$G^{\mathcal{C}}_{x,y}\simeq
G^{\mathtt{SW}(\mathcal{C}_{\infty})}_{S(x),S(y)}\simeq\operatorname{Hom}_{\mathtt{SW}(\mathcal{C}_{\infty})}(S(x),S(y)).$

By Proposition 5.3 we are done. ∎

##  6 The $\infty$-category of noncommutative CW-spectra

In this section we define the $\infty$-category of noncommutative CW-spectra
$\mathtt{NSp}$. The suspension spectra of matrix algebras form a set of
compact generators of $\mathtt{NSp}$. We denote by $\mathcal{M}$ the full
spectral subcategory of $\mathtt{NSp}$ spanned by this set of generators (see
Proposition 3.5). Thus $\mathcal{M}$ is an $\mathtt{Sp}$-enriched
$\infty$-category. We prove our main theorem: there is an equivalence of
$\infty$-categories between $\mathtt{NSp}$ and the category of spectral
presheaves on $\mathcal{M}$. We give two versions and two independent proofs
of this result. One version is formulated fully in the language of enriched
$\infty$-categories, using Hinich's theory. In the second approach, we first
define a strict version of $\mathcal{M}$, denoted $\mathcal{M}_{s}$, which is
a category strictly enriched in $\mathtt{Sp^{M}}$. We then prove that
$\mathtt{NSp}$ is modelled by a Quillen model category of
$\mathtt{Sp^{M}}$-valued presheaves on $\mathcal{M}_{s}$. Finally we prove
that our two models of $\mathcal{M}$ are equivalent, in the sense that
$\mathcal{M}$ is equivalent to the enriched $\infty$-localization of
$\mathcal{M}_{s}$ (see Definition 3.6).

Let us proceed with the definition of $\mathtt{NSp}$. Recall that in Section 2
we defined the $\infty$-category of finite noncommutative CW-complexes and
denoted it by $\mathtt{NCW}^{f}$. We then defined the $\infty$-category of all
noncommutative CW-complexes by the formula
$\mathtt{NCW}:=\operatorname{Ind}(\mathtt{NCW}^{f}).$ We now define the
$\infty$-category of noncommutative CW-spectra to be

$\mathtt{NSp}:=\mathtt{Sp}(\mathtt{NCW}).$

By the results in Section 5 we know that $\mathtt{NSp}$ is a presentable
stable $\infty$-category. In particular, $\mathtt{NSp}$ is naturally left-
tensored over spectra. By [Lur2, Corollary 4.8.2.19] the monoidal structure on
$\mathtt{NCW}$ induces a closed symmetric monoidal structure on
$\mathtt{NSp}$, such that
$\Sigma^{\infty}_{\mathtt{NC}}:\mathtt{NCW}\longrightarrow\mathtt{NSp}$ is
symmetric monoidal.

Recall that $M_{n}$ is the algebra of $n\times n$ matrices over $\mathbb{C}$.
Since the set of objects $\\{M_{i}\mid i\in\mathbb{N}\\}$ generates
$\mathtt{NCW}^{f}$ under finite colimits, it follows by Corollary 5.4 that
$M:=\\{\Sigma^{\infty}_{\mathtt{NC}}M_{i}\mid i\in\mathbb{N}\\}$ generates
$\mathtt{NSp}$ under small colimits and desuspensions. The following is one of
the main definitions of the paper:

######  Definition 6.1.

Let $\mathcal{M}$ be the full $\mathtt{Sp}$-enriched subcategory of
$\mathtt{NSp}$ spanned by the spectra
$\\{\Sigma^{\infty}_{\mathtt{NC}}M_{i}\mid i\in\mathbb{N}\\}$.

Since $\mathcal{M}$ is closed under the monoidal product in $\mathtt{NSp}$,
the following theorem is a special case of Theorem 4.3:

######  Theorem 6.2.

The $\mathtt{Sp}$-enriched category $\mathcal{M}$ acquires a canonical
symmetric monoidal structure, the category of presheaves
$P_{\mathtt{Sp}}(\mathcal{M})$ acquires a canonical symmetric monoidal left
$\mathtt{Sp}$-tensored structure and we have a natural symmetric monoidal left
$\mathtt{Sp}$-tensored functor

$P_{\mathtt{Sp}}(\mathcal{M})\xrightarrow{\sim}\mathtt{NSp},$

which is an equivalence of the underlying $\infty$-categories and sends each
representable presheaf $Y(\Sigma^{\infty}_{\mathtt{NC}}M_{n})\in
P_{\mathtt{Sp}}(\mathcal{C})$ to $\Sigma^{\infty}_{\mathtt{NC}}M_{n}$.

###  6.0.1 Strictification of $\mathcal{M}$

In this subsection we give a strict model for the category $\mathcal{M}$ as a
monodial spectrally enriched category, as well as a strict version of Theorem
6.2.

In the context of Section 5.0.1, let us consider the example
$\mathcal{C}=\mathtt{NCW}^{f}$, considered as a topological category. Then
$\mathtt{NCW}^{f}$ is a pointed topological category, tensored closed over
$\mathtt{CW}^{f}_{*}$ (see Definition 5.9). As explained in Definition 5.12,
we have a strict enrichment of $\mathtt{NCW}^{f}$ over the model category of
spectra $\mathtt{Sp^{M}}$ using the functors $G^{\mathtt{NCW}^{f}}_{x,y}$.

The topological category $\mathtt{NCW}^{f}$ has a continuous symmetric
monoidal structure induced by tensor product in $\mathtt{SC^{*}}$. The
spectral enrichment respects the monoidal structure, in the sense that given
objects $x,x_{1},y,y_{1}$, there is a natural transformation

$G_{x,y}^{\mathtt{NCW}^{f}}(K)\wedge G_{x_{1},y_{1}}^{\mathtt{NCW}^{f}}(L)\to
G_{x\otimes x_{1},y\otimes y_{1}}^{\mathtt{NCW}^{f}}(K\wedge L).$

Thus the spectral enrichment of $\mathtt{NCW}^{f}$ is symmetric monoidal.

######  Definition 6.3.

Let $\mathcal{M}_{s}$ be the full (strict) $\mathtt{Sp^{M}}$-enriched
subcategory of $\mathtt{NCW}^{f}$ spanned by $\\{M_{n}\mid n\in\mathbb{N}\\}$.
That is, the objects of $\mathcal{M}_{s}$ are $\\{M_{n}\mid n\in\mathbb{N}\\}$
and for any $m,n\in\mathbb{N}$ we have

$\operatorname{Hom}_{\mathcal{M}_{s}}(M_{m},M_{n})=G^{\mathtt{NCW}^{f}}_{M_{m},M_{n}}\in\mathtt{Sp^{M}}.$

Since $\mathcal{M}_{s}$ is a category enriched over $\mathtt{Sp^{M}}$, we can
define the strict category of spectral presheaves on $\mathcal{M}_{s}$, which
we denote by $P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})$, to be the category of
enriched functors $\mathcal{M}_{s}^{\operatorname{op}}\to\mathtt{Sp^{M}}$. We
endow $P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})$ with the projective model
structure (see, for example, [GM] on the projective model structure in the
enriched setting). Since both $\mathcal{M}_{s}^{\operatorname{op}}$ and
$\mathtt{Sp^{M}}$ have a symmetric monoidal structure, the category
$P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})$ has a symmetric monoidal structure
given by enriched Day convolution turning it into a symmetric monoidal model
category.

Consider $\mathtt{NCW}^{f}$ as a category enriched in $\mathtt{Sp^{M}}$. There
is a canonical strict spectral functor

$RY\colon\mathtt{NCW}^{f}\to P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}),$ (2)

which is the composition of the enriched Yoneda embedding $\mathtt{NCW}^{f}\to
P_{\mathtt{Sp^{M}}}(\mathtt{NCW}^{f})$ followed by restriction
$P_{\mathtt{Sp^{M}}}(\mathtt{NCW}^{f})\to
P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})$. It is well-known, and easy to check
that $RY$ is lax symmetric monoidal.

We call a map $A\to B$ in $\mathtt{NCW}^{f}$ a weak equivalence if it is a
homotopy equivalence in $\mathtt{NCW}^{f}$ considered as a topological
category.

######  Lemma 6.4.

The functor $RY$ sends weak equivalences to weak equivalences.

###### Proof.

Let $A\to B$ be a weak equivalence in $\mathtt{NCW}^{f}$. We need to show that
$RY(A)\to RY(B)$ is a levelwise weak equivalence in
$P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})$. Let $n\geq 1$. We need to show that
the induced map
$\operatorname{Hom}_{\mathtt{NCW}^{f}}(M_{n},A)\to\operatorname{Hom}_{\mathtt{NCW}^{f}}(M_{n},B)$
is a weak equivalence in $\mathtt{Sp^{M}}$. Since $\mathtt{Sp^{M}}$ is a
localization of the projective model structure, is enough to show that
$\operatorname{Hom}_{\mathtt{NCW}^{f}}(M_{n},A)\to\operatorname{Hom}_{\mathtt{NCW}^{f}}(M_{n},B)$
is a levelwise weak equivalence in
$\operatorname{Fun}_{*}(\mathtt{CW}^{f}_{*},\operatorname{Top})$. That is, it
is enough to show that for every finite pointed CW-complex $K$,

$\operatorname{Map}_{\mathtt{NCW}^{f}}(M_{n},K\wedge
A)\to\operatorname{Map}_{\mathtt{NCW}^{f}}(M_{n},K\wedge B)$

is a weak equivalence. Since $\mathtt{NCW}^{f}$ is a topological category, and
a weak equivalence in $\mathtt{NCW}^{f}$ is just a homotopy equivalence, we
are done. ∎

By the lemma above, we can apply $\infty$-localization with respect to weak
equivalences (see Remark 2.1) to $RY$ and obtain a functor of
$\infty$-categories

$RY_{\infty}\colon\mathtt{NCW}^{f}_{\infty}\to
P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}.$

######  Lemma 6.5.

The $\infty$-category $\mathtt{NCW}^{f}_{\infty}$ is naturally equivalent to
$\mathtt{NCW}^{f}$ defined above as the topological nerve of the topological
category $\mathtt{NCW}^{f}$.

###### Proof.

The category $\mathtt{SC^{*}}^{\operatorname{op}}$ (defined in the beginning
of Section 2) has the structure of a category of cofibrant objects with the
weak equivalences given by the homotopy equivalences and the cofibrations by
Schochet cofibrations (see, for instance, in [AG, Uuy]). We say that a map in
$\mathtt{NCW}^{f}$ is a weak equivalence (resp. cofibration) if it is a weak
equivalence (resp. cofibration) when regarded as a map in
$\mathtt{SC^{*}}^{\operatorname{op}}$. Since
$\mathtt{SC^{*}}^{\operatorname{op}}$ is a category of cofibrant objects and
$\mathtt{NCW}^{f}\subseteq\mathtt{SC^{*}}^{\operatorname{op}}$ is a full
subcategory which is closed under weak equivalences and pushouts along
cofibrations it follows that $\mathtt{NCW}^{f}$ inherits a structure of a
category of cofibrant objects. In exactly the same way as in [BHH, Lemma
7.1.1] one can show that the natural map between $\infty$-localizations with
respect to weak equivalences:

$(\mathtt{NCW}^{f})_{\infty}\longrightarrow(\mathtt{SC^{*}}^{\operatorname{op}})_{\infty}$

is fully faithful.

By [BJM, Proposition 3.17] we have that the $\infty$-localization of
$\mathtt{SC^{*}}^{\operatorname{op}}$ is equivalent to the topological nerve
of the topological category structure on $\mathtt{SC^{*}}^{\operatorname{op}}$
described in Section 2 (see Remark 2.1 and the paragraph before). Since
$\mathtt{NCW}^{f}$ is a full topological subcategory of
$\mathtt{SC^{*}}^{\operatorname{op}}$, we are done. ∎

######  Lemma 6.6.

The functor $RY_{\infty}$ preserves finite colimits.

###### Proof.

By [Lur1, Corollary 4.4.2.5], it is enough to prove that the functor preserves
initial objects and pushout squares. Since $\mathtt{NCW}^{f}$ is a pointed
category, the inital object of $\mathtt{NCW}^{f}$ is also the final object,
and the first condition obviously holds.

In both $\mathtt{NCW}^{f}_{\infty}$ and
$P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}$ pushouts can be calculated as
homotopy pushouts in an appropriate structure. Suppose we have a homotopy
pushout diagram in $\mathtt{NCW}^{f}$

$\begin{array}[]{ccc}y_{0}&\to&y_{1}\\\ \downarrow&&\downarrow\\\
y_{2}&\to&y_{12}.\end{array}$ (3)

We want to prove that for any $x\in\mathcal{M}_{s}$ the induced diagram of
functors is a homotopy pushout in the stable model structure

$\begin{array}[]{ccc}\operatorname{Map}_{\mathtt{NCW}^{f}}(x,-\wedge
y_{0})&\to&\operatorname{Map}_{\mathtt{NCW}^{f}}(x,-\wedge y_{1})\\\
\downarrow&&\downarrow\\\ \operatorname{Map}_{\mathtt{NCW}^{f}}(x,-\wedge
y_{2})&\to&\operatorname{Map}_{\mathtt{NCW}^{f}}(x,-\wedge
y_{12}).\end{array}$

Since we are working in the stable model structure, a square is a homotopy
pushout if and only if it is a homotopy pullback. A square of functors is a
homotopy pullback in the stable model structure if the induced square of
linearizations is a homotopy pullback. But the linearization of the functor

$\operatorname{Map}_{\mathtt{NCW}^{f}}(x,-\wedge
y)\colon\mathtt{CW}^{f}_{*}\to\operatorname{Top}$

evaluated at $K$ is the same as the linearization of the functor

$\operatorname{Map}_{\mathtt{NCW}^{f}}(x,K\wedge-)\colon\mathtt{NCW}^{f}\to\operatorname{Top}$

evaluated at $y$. Indeed, the two linearizations are given by the equivalent
formulas

$\mathop{\operatorname{hocolim}}_{n\to\infty}\Omega^{n}\operatorname{Map}_{\mathtt{NCW}^{f}}(x,\Sigma^{n}K\wedge
y)=\mathop{\operatorname{hocolim}}_{n\to\infty}\Omega^{n}\operatorname{Map}_{\mathtt{NCW}^{f}}(x,K\wedge\Sigma^{n}_{\mathtt{NCW}^{f}}y).$

We have been thinking of the functor
$\operatorname{Map}_{\mathtt{NCW}^{f}}(x,K\wedge-)\colon\mathtt{NCW}^{f}\to\operatorname{Top}$
as a strict functor, but now let us think of it as a functor between
$\infty$-categories by applying the $\infty$-localization. The
$\infty$-category $\mathtt{NCW}^{f}$ has finite colimits and a final object.
The conditions of [Lur1, Lemma 6.1.1.33] are satisfied, and therefore the
linearization of this functor really is linear, i.e., takes homotopy pushout
squares to homotopy pullback squares. Therefore applying the linearization to
the square (3) yields a homotopy pullback square, which is what we wanted to
prove. ∎

Since $P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}$ is a stable
$\infty$-category, we have, by the lemma above, that $RY_{\infty}$ extends
canonically to a finite-colimit preserving functor

$RY_{\infty}:\mathtt{SW}(\mathtt{NCW}^{f})\to
P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}.$

This functor extends, in turn, to an all-small-colimit-preserving functor

$RY_{\infty}\colon\mathtt{NSp}=\operatorname{Ind}(\mathtt{SW}(\mathtt{NCW}^{f}))\to
P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}.$ (4)

This functor takes an object $\Sigma^{\infty}_{\mathtt{NC}}M_{n}$ to the
presheaf represented by $M_{n}$.

######  Theorem 6.7.

The functor $RY_{\infty}\colon\mathtt{NSp}\to
P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}$ is an equivalence of
$\infty$-categories.

###### Proof.

First let us prove that $RY_{\infty}$ is fully faithful, that is, that for all
objects $x,y$ of $\mathtt{NSp}$ the map of spectral mapping functors

$G^{\mathtt{NSp}}_{x,y}\to
G^{P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}}_{RY_{\infty}(x),RY_{\infty}(y)}$
(5)

is an equivalence. First, consider the case
$x,y\in\Sigma^{\infty}\mathcal{M}_{s}$, i.e.,
$x=\Sigma^{\infty}_{\mathtt{NC}}M_{k},y=\Sigma^{\infty}_{\mathtt{NC}}M_{l}$
for some $k,l$. In this case $x,y$ are in the image of the finite suspension
functor $\mathtt{NCW}^{f}\to\mathtt{SW}(\mathtt{NCW}^{f})$. The functor
$\mathtt{SW}(\mathtt{NCW}^{f})\to\mathtt{NSp}$ is fully faithful, so it
induces an equivalence

$G^{\mathtt{SW}(\mathtt{NCW}^{f})}_{x,y}\xrightarrow{\simeq}G^{\mathtt{NSp}}_{x,y}.$

By Lemma 5.7, the map $G^{\mathtt{NCW}^{f}}_{M_{k},M_{l}}\to
G^{\mathtt{SW}(\mathtt{NCW}^{f})}_{x,y}$ is stabilization.

The functor $RY:\mathtt{NCW}^{f}\to P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})$ when
restricted to $\mathcal{M}_{s}$ is just the enriched Yoneda embedding of
$\mathcal{M}_{s}$

$Y:\mathcal{M}_{s}\to P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}).$

Since the unit in $\mathtt{Sp^{M}}$ is cofibrant,
$RY_{\infty}(\Sigma^{\infty}M_{k})=Y(M_{k})$ is cofibrant in the projective
model structure on $P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})$ (see [GM, Theorem
4.32]). The fibrant replacement in $P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})$ is
levelwise, so using the (strict) enriched Yoneda lemma we get

$G^{P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}}_{RY_{\infty}(x),RY_{\infty}(y)}=\operatorname{Hom}_{P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}}(Y(M_{k}),Y(M_{l}))\simeq$
$\operatorname{Hom}_{P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})}(Y(M_{k}),Y(M_{l})^{f})\cong(Y(M_{l})^{f})(M_{k})\simeq
Y(M_{l})(M_{k})^{f}=(G^{\mathtt{NCW}^{f}}_{M_{k},M_{l}})^{f}.$

But the map
$G^{\mathtt{NCW}^{f}}_{M_{k},M_{l}}\to(G^{\mathtt{NCW}^{f}}_{M_{k},M_{l}})^{f}$
translates to the stabilization
$G^{\mathtt{NCW}^{f}}_{M_{k},M_{l}}\to\mathcal{L}G^{\mathtt{NCW}^{f}}_{M_{k},M_{l}}$
under $\infty$-localization (see the proof of Theorem 5.13). Thus the map

$G^{\mathtt{NCW}^{f}}_{M_{k},M_{l}}\to
G^{P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}}_{RY_{\infty}(x),RY_{\infty}(y)}$

is also the stabilization. By the uniqueness of the stabilization map, we get
that the map (5) is an equivalence in the case when $x,y$ are suspension
spectra of matrix algebras.

Next, let $x$ be a fixed suspension spectrum of a matrix algebra, but let $y$
vary. We may consider the functor $y\mapsto G^{\mathtt{NSp}}_{x,y}$ as a
functor $\mathtt{NSp}\to\mathtt{Sp}$. This functor preserves all small
colimits, because $x$ is compact in $\mathtt{NSp}$ and both $\mathtt{NSp}$ and
$\mathtt{Sp}$ are stable. Similarly, the functor $y\mapsto
G^{P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}}_{RY_{\infty}(x),RY_{\infty}(y)}$
is also a functor $\mathtt{NSp}\to\mathtt{Sp}$ that preserves small colimits.
It follows that the category of objects $y$ for which the map (5) is an
equivalence is closed under colimits and also desuspensions. Since this
category contains $\mathcal{M}_{s}$, it is all of $\mathtt{NSp}$.

Now fix $y$, and consider the functors $x\mapsto
G^{\mathtt{NSp}}_{x,y},G^{P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}}_{RY_{\infty}(x),RY_{\infty}(y)}$
as contravariant functors from $\mathtt{NSp}$ to spectra. Since both functors
take small colimits to limits, a similar argument shows that this map is an
equivalence for all $x\in\mathtt{NSp}$.

We have shown that the functor $RY_{\infty}\colon\mathtt{NSp}\to
P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}$ is fully faithful and also
preserves all small colimits, so it is a left adjoint. It follows that the
image of $RY_{\infty}$ is closed under small colimits. Since the image
contains the representable presheaves, $RY_{\infty}$ is essentially
surjective. It follows that $RY_{\infty}$ is an equivalence of categories. ∎

Thus $RY_{\infty}$ is an explicit monoidal model for the inverse of the
equivalence given by Theorem 6.2. This also allows us to show that
$\mathcal{M}_{s}$ is indeed a strictification of $\mathcal{M}$ from Definition
6.1.

######  Theorem 6.8.

We have a natural equivalence $(\mathcal{M}_{s})_{\infty}\simeq\mathcal{M}$
between the enriched $\infty$-localization of $\mathcal{M}_{s}$ and
$\mathcal{M}$.

###### Proof.

By definition, $\mathcal{M}_{s}$ is an $\mathtt{Sp^{M}}$-enriched category,
whose set of objects is the set of natural numbers $\mathbb{N}$. Let
$\mathtt{Sp^{M}}\mathbb{N}$-$\mathtt{Cat}$ be the category of all
$\mathtt{Sp^{M}}$-enriched categories, whose set of objects is $\mathbb{N}$,
and whose morphisms are functors that are the identity on objects. This
category has a Quillen model structure, where fibrations and weak equivalences
are defined levelwise, and where cofibrant objects are levelwise cofibrant
[ScSh2, Proposition 6.3]. Thus, $\mathcal{M}_{s}$ is an object in
$\mathtt{Sp^{M}}\mathbb{N}$-$\mathtt{Cat}$. Let
$\mathcal{M}_{s}\to\mathcal{M}_{s}^{f}$ be a fibrant replacement of
$\mathcal{M}_{s}$ in $\mathtt{Sp^{M}}\mathbb{N}$-$\mathtt{Cat}$. We get a
Quillen adjunction

$\operatorname{LKan}_{i}:P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})\rightleftarrows
P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}^{f}):i^{*}.$

It follows from [GM, Proposition 2.4] that this adjunction is a Quillen
equivalence. To give a little more detail, it follows from the general result
of Guillou and May that it is enough to show that for every cofibrant object
$M$ of $\mathtt{Sp^{M}}$ and every two objects $x,y$ of $\mathcal{M}_{s}$, the
following induced map is an equivalence

$M\wedge\operatorname{Hom}_{\mathcal{M}_{s}}(x,y)\to
M\wedge\operatorname{Hom}_{\mathcal{M}_{s}^{f}}(x,y),$

where $\operatorname{Hom}(-,-)$ denotes the spectral mapping object. The map

$\operatorname{Hom}_{\mathcal{M}_{s}}(x,y)\to\operatorname{Map}_{\mathcal{M}_{s}^{f}}(x,y)$

is an equivalence by definition of $M_{s}^{f}$. It follows by [MMSS,
Proposition 12.3] that the induced map is an equivalence for all cofibrant
$M$.

Applying $\infty$-localization we obtain an equivalence

$P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}\xrightarrow{\sim}P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}^{f})_{\infty}.$

Now, $P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}^{f})$ is an $\mathtt{Sp^{M}}$-model
category, and it is Quillen equivalent to a combinatorial model category. The
enriched Yoneda embedding

$Y:\mathcal{M}_{s}^{f}\to P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}^{f})$

is a fully faithful $\mathtt{Sp^{M}}$-enriched functor and it clearly lands in
the fibrant cofibrant objects. Thus, by Corollary 3.8, the
$\mathtt{Sp}$-functor

$Y_{\infty}:(\mathcal{M}_{s}^{f})_{\infty}\to
P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}^{f})_{\infty}$

is $\mathtt{Sp}$-fully faithful. Note that this claim is made with respect to
the action of $\mathtt{Sp}$ on
$P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}^{f})_{\infty}$ induced by the structure
of $P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}^{f})$ as an $\mathtt{Sp^{M}}$-model
category. Since $\mathtt{Sp}$ is a mode in the sense of [CSY, Section 5], this
action coincides with the canonical action of $\mathtt{Sp}$ on
$P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}^{f})_{\infty}$ as a presentable stable
$\infty$-category.

Now, using Theorem 6.7, we have the following composition

$(\mathcal{M}_{s}^{f})_{\infty}\xrightarrow{Y_{\infty}}P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s}^{f})_{\infty}\xrightarrow{\sim}P_{\mathtt{Sp^{M}}}(\mathcal{M}_{s})_{\infty}\xrightarrow{\sim}\mathtt{NSp}.$

We get a fully faithful $\mathtt{Sp}$-enriched functor
$(\mathcal{M}_{s}^{f})_{\infty}\to\mathtt{NSp}$, with essential image
$\mathcal{M}$, so that $(\mathcal{M}_{s}^{f})_{\infty}\simeq\mathcal{M}$. ∎

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