# Suspension spectra of matrix algebras, the rank filtration, and rational
noncommutative CW-spectra

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

In a companion paper [ABS1] we introduced the stable $\infty$-category of
noncommutative CW-spectra, which we denoted $\mathtt{NSp}$. Let $\mathcal{M}$
denote the full spectrally enriched subcategory of $\mathtt{NSp}$ whose
objects are the non-commutative suspension spectra of matrix algebras. In
[ABS1] we proved that $\mathtt{NSp}$ is equivalent to the $\infty$-category of
spectral presheaves on $\mathcal{M}$. In this paper we investigate the
structure of $\mathcal{M}$, and derive some consequences regarding the
structure of $\mathtt{NSp}$.

To begin with, we introduce a rank filtration of $\mathcal{M}$. We show that
the mapping spectra of $\mathcal{M}$ map naturally to the connective
$K$-theory spectrum $ku$, and that the rank filtration of $\mathcal{M}$ is a
lift of the classical rank filtration of $ku$. We describe the subquotients of
the rank filtration in terms of complexes of direct-sum decompositions which
also arose in the study of $K$-theory and of Weiss's orthogonal calculus. We
prove that the rank filtration stabilizes rationally after the first stage.
Using this we give an explicit model of the rationalization of $\mathtt{NSp}$
as presheaves of rational spectra on the category of finite-dimensional
Hilbert spaces and unitary transformations up to scaling. Our results also
have consequences for the $p$-localization and the chromatic localization of
$\mathcal{M}$.

###### Contents

  1. 1 Introduction
  2. 2 The functors $G_{k,l}$ and their stabilization
  3. 3 The rank filtration
  4. 4 The restriction of $G_{k,l}$ to finite sets
  5. 5 $G_{k,l}$ is a cofibrant $\Gamma$-space
  6. 6 Connection with topological $K$-theory
  7. 7 Subquotients of the rank filtration
  8. 8 Connection with the complex of direct-sum decompositions
  9. 9 Some calculations of $\mathbb{S}^{k,l}$
  10. 10 On the rationalization and $p$-localization of $\mathcal{M}$

##  1 Introduction

In our previous paper [ABS1] we introduced the $\infty$-category of
noncommutative CW-spectra, which we denoted $\mathtt{NSp}$. This is the
stabilization of the $\infty$-category of noncommutative CW-complexes, denoted
$\mathtt{NCW}$, which in turn is the ind completion of the $\infty$-category
of finite noncommutative CW-complexes, denoted $\mathtt{NCW}^{f}$. The latter
is defined as opposite of the topological nerve of the topological category
whose objects are the $C^{*}$-algebras which are noncommutative CW-complexes
in the sense of [ELP] and whose hom-spaces are given by taking the topology of
pointwise norm convergence on the sets of $*$-homomorphisms.

The main result of [ABS1] says that $\mathtt{NSp}$ is equivalent to the
$\infty$-category of spectral presheaves over a small spectrally enriched
$\infty$-category $\mathcal{M}$. The spectral $\infty$-category $\mathcal{M}$
is defined to be the full spectral subcategory of $\mathtt{NSp}$, whose
objects are noncommutative suspension spectra of matrix algebras. In this
paper we analyze the category $\mathcal{M}$ in considerable detail. We
introduce a rank filtration of $\mathcal{M}$, describe the subquotients of the
rank filtration and use this to give an explicit model for the rationalization
of $\mathtt{NSp}$.

######  Remark 1.1.

In [ABS1] we made extensive use of Hinich's theory of enriched
$\infty$-categories (see [Hin2, Hin3]). In this paper we also use this theory
and terminology on a few occasions. The interested reader is referred also to
[ABS1, Section 3] for a summery of the parts of this theory relevant to us.

Given an integer $k\geq 1$, let $M_{k}$ be the $C^{*}$-algebra of $k\times k$
matrices over $\mathbb{C}$. We have the stabilization functor

$\Sigma^{\infty}_{\mathtt{NC}}:\mathtt{NCW}\to\mathtt{NSp}$

and the set of objects of $\mathcal{M}$ is
$\\{\Sigma^{\infty}_{\mathtt{NC}}M_{k}|\,k\geq 1\\}$. Thus the objects of
$\mathcal{M}$ are in one to one correspondence with the positive integers.
Given two integers $k,l$, we denote the spectral mapping object in
$\mathcal{M}$ by

$\mathbb{S}^{k,l}:=\operatorname{Hom}_{\mathcal{M}}(\Sigma^{\infty}_{\mathtt{NC}}M_{k},\Sigma^{\infty}_{\mathtt{NC}}M_{l})\simeq\operatorname{Hom}_{\mathtt{NSp}}(\Sigma^{\infty}_{\mathtt{NC}}M_{k},\Sigma^{\infty}_{\mathtt{NC}}M_{l})\in\mathtt{Sp},$

where $\mathtt{Sp}$ is the $\infty$-category of (ordinary) spectra.

Our goal in this paper is to analyze the category $\mathcal{M}$. For this, it
will be convenient to use a model category that models the $\infty$-category
of spectra $\mathtt{Sp}$. Analyzing $\mathcal{M}$ means, firstly, that we want
to describe, for each $k$ and $l$, the homotopy type of $\mathbb{S}^{k,l}$.
For this purpose it is adequate to use the simple model for spectra, as a
sequence of pointed spaces with structure maps. But we also want to model the
composition maps,

$\mathbb{S}^{k,l}\wedge\mathbb{S}^{j,k}\to\mathbb{S}^{j,l}.$ (1)

To do this properly, we need to use a more sophisticated model for spectra,
which incorporates a smash product.

To be more explicit, in [ABS1] we defined a ``strict" model of $\mathcal{M}$.
That is, we defined a category $\mathcal{M}_{s}$ strictly enriched in a
certain monoidal model category of spectra $\mathtt{Sp^{M}}$, such that the
enriched $\infty$-localization of $\mathcal{M}_{s}$ is equivalent to
$\mathcal{M}$. The model category $\mathtt{Sp^{M}}$ is the category of pointed
continuous functors from pointed finite CW-complexes to pointed topological
spaces, endowed with the stable model structure. (By a topological space in
this paper we always mean a compactly generated weak Hausdorff space.) Day
convolution turns $\mathtt{Sp^{M}}$ into a symmetric monoidal model category.
See [Lyd2, MMSS, Lyd1] for more details on this model structure.

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 $\mathcal{C}$.

We have a canonical equivalence of symmetric monoidal $\infty$-categories

$\mathtt{Sp^{M}}_{\infty}\simeq\mathtt{Sp}.$

We will identify the two $\infty$-categories above through this equivalence.
Similarly, if $\operatorname{Top}$ is the category of pointed topological
spaces endowed with the Quillen model structure [Qui], then we have a
canonical equivalence

$\operatorname{Top}_{\infty}\simeq\mathcal{S}_{*}$

where $\mathcal{S}_{*}$ is the $\infty$-category of pointed spaces. Again, we
identify the two $\infty$-categories above through this equivalence.

We denote the localization functor from $\mathtt{Sp^{M}}$ to
$\mathtt{Sp^{M}}_{\infty}=\mathtt{Sp}$ by $\partial_{1}$

$\partial_{1}:\mathtt{Sp^{M}}\to\mathtt{Sp}.$

If $G\in\mathtt{Sp^{M}}$ is a pointed continuous functors from pointed finite
CW-complexes to pointed topological spaces, then $\partial_{1}G$ is the
spectrum corresponding to the sequence of spaces
$\\{G(S^{0}),G(S^{1}),\ldots$} (where we identify a pointed topological space
with its image in $\operatorname{Top}_{\infty}=\mathcal{S}_{*}$). In other
words we can write

$\partial_{1}G\simeq{\mathop{\operatorname{hocolim}}}_{n}\Sigma^{-n}\Sigma^{\infty}G(S^{n}),$

where by $\mathop{\operatorname{hocolim}}$ here we mean $\infty$-colimit in
$\mathtt{Sp}$. This is known as the stabilization, or the first derivative of
the functor $G$.

The way we define $\mathcal{M}_{s}$ in [ABS1] is by letting for every $k,l$

$\operatorname{Hom}_{\mathcal{M}_{s}}(M_{k},M_{l}):=G_{k,l}\in\mathtt{Sp^{M}},$

where

$G_{k,l}(X):=\mathtt{SC^{*}}(\mathrm{C}_{0}(X,M_{l}),M_{k})$

(see also Definition 2.2). Here $\mathrm{C}_{0}(X,M_{l})$ is the space of
pointed map from $X$ to $M_{l}$, considered as a $C^{*}$-algebra, and
$\mathtt{SC^{*}}(-,-)$ denotes the space of $C^{*}$-algebra maps with the
topology of pointwise norm convergence. Since $\mathcal{M}_{s}$ is a model for
$\mathcal{M}$ we have that $G_{k,l}$ is a model for $\mathbb{S}^{k,l}$, or in
other words, the stabilization of $G_{k,l}$ is $\mathbb{S}^{k,l}$:

$\mathbb{S}^{k,l}\simeq\partial_{1}G_{k,l}\simeq{\mathop{\operatorname{hocolim}}}_{n}\Sigma^{-n}\Sigma^{\infty}G_{k,l}(S^{n}).$

######  Remark 1.2.

In this paper we use both $\mathcal{M}$ and $\mathcal{M}_{s}$. For convenience
of notation we denote both categories by $\mathcal{M}$ trusting that it is
clear from the context which is meant.

We proceed to investigate the homotopy type of $\mathbb{S}^{k,l}$. It is not
hard to show that if $l>k$ then $\mathbb{S}^{k,l}$ is contractible (see
corollary 5.8), so we generally assume that $k\geq l$. To analyze
$\mathbb{S}^{k,l}$ further, we introduce a natural filtration of $\mathcal{M}$
in Section 3, which we call the rank filtration. More precisely, for each $k$
and $l$, we define a sequence of subfunctors $G_{k,l,i}\in\mathtt{Sp^{M}}$

$G_{k,l,1}\subset G_{k,l,2}\subset\cdots\subset
G_{k,l,\lfloor\frac{k}{l}\rfloor}=G_{k,l,\lfloor\frac{k}{l}\rfloor+1}=\cdots=G_{k,l}.$

Upon stabilization, we obtain a sequence of spectra

$\mathbb{S}^{k,l,1}\hookrightarrow\mathbb{S}^{k,l,2}\hookrightarrow\cdots\hookrightarrow\mathbb{S}^{k,l,\lfloor\frac{k}{l}\rfloor}=\mathbb{S}^{k,l}.$

We think of this sequence as defining a filtration of $\mathbb{S}^{k,l}$. This
filtration is multiplicatively compatible with composition. This means that
there are maps, compatible with (1)

$\mathbb{S}^{k,l,m}\wedge\mathbb{S}^{j,k,n}\to\mathbb{S}^{j,l,mn}.$

One can say that the category $\mathcal{M}$ is filtered by the multiplicative
monoid of positive integers.

The functors $G_{k,l,m}$ are, by definition, functors from the category of
pointed finite CW-complexes to pointed topological spaces. But it turns out
that they are determined by their restriction to the category of pointed
finite sets. Specifically, we show in Theorem 5.7 that $G_{k,l,m}$ is
equivalent to both the strict and the derived left Kan extension of its
restriction to pointed finite sets. In other words, $G_{k,l,m}$ are
$\Gamma$-spaces.

In Section 4 we describe explicitly the restriction of $G_{k,l}$ to pointed
finite sets (Proposition 4.5). In Section 5 we show that the $\Gamma$-spaces
$G_{k,l,m}$ are in fact cofibrant in a generalized Reedy model structure
considered by Bousfield-Friedlander [BF], Lydakis [Lyd1] and Berger-Moerdijk
[BM].

Proposition 4.5 indicates that $G_{k,l}$ is similar to the $\Gamma$-space that
models the K-theory spectrum $ku$. This is not a coincidence. The inclusion of
matrix algebras $M_{k}\to M_{k+1}$ that sends a matrix $a$ to $a\oplus 0$
indues a natural transformation $G_{k,l}\to G_{k+1,l}$, which in turn induces
a map of spectra $\mathbb{S}^{k,l}\to\mathbb{S}^{k+1,l}$. We prove in Section
6 that for each fixed $l$,
$\mathop{\operatorname{hocolim}}_{k\to\infty}\mathbb{S}^{k,l}\simeq ku$ (thus
all the mapping spectra $\mathbb{S}^{k,l}$ are naturally spectra over $ku$).
The case $l=1$ of this observation goes back to Segal [Se2] and was exploited
extensively by Dadarlat with various collaborators (for example see [DM]).
Furthermore, it turns out that the functor
$l\mapsto\mathop{\operatorname{hocolim}}_{k\to\infty}\mathbb{S}^{k,l}$ takes
values in module spectra over $ku$. This allows for a natural way do define
bivariant connective $K$-theory on $\mathtt{NSp}$, which generalizes the
definition of Dadarlat and McClure [DM] to the noncommutative setting. We also
observe in this section that the rank filtration of $\mathcal{M}$ is a lift of
the classical rank filtration of $ku$. Later in the paper we prove that the
map $\mathbb{S}^{k,l}\to ku$ induces an isomorphism on $\pi_{0}$ (Lemma 9.4).

The results of Sections 4 and 5 are useful for the homotopic analysis of
$\mathbb{S}^{k,l}$ and of the rank filtration. One of our main results is an
explicit description of the associated graded filtration. Thus we describe,
for each $k,l,m$, the homotopy cofiber spectrum
$\mathbb{S}^{k,l}_{m}:=\mathbb{S}^{k,l,m}/\mathbb{S}^{k,l,m-1}$, as well as
the induced maps

$\mathbb{S}^{k,l}_{m}\wedge\mathbb{S}^{j,k}_{n}\to\mathbb{S}^{j,l}_{mn}.$

Our description features certain $U(m)$-complexes
$\mathcal{L}_{m}^{\diamond}$. Roughly speaking $\mathcal{L}_{m}^{\diamond}$ is
the space of direct-sum decompositions of $\mathbb{C}^{m}$ (see Definition
8.6). These complexes have some remarkable homotopical properties, which we
will review below. But first let us state the result describing the rank
filtration in terms of $\mathcal{L}_{m}^{\diamond}$. When $U$ and $V$ are
unitary vector spaces, let $\operatorname{Inj}(U,V)$ denote the space of
(necessarily injective) linear transformations of $U$ into $V$ that preserve
the unitary product. Note for future reference that there is a homeomorphism
$\operatorname{Inj}(\mathbb{C}^{m},\mathbb{C}^{n})\cong U(n)/U(n-m)$. The
following is Theorem 8.8 in the text:

######  Theorem 1.3.

There is an equivalence of spectra

$\mathbb{S}^{k,l}_{m}\simeq\Sigma^{\infty}\mathcal{L}_{m}^{\diamond}\wedge_{U(m)}\operatorname{Inj}(\mathbb{C}^{lm},\mathbb{C}^{k})_{+},$

where $U(m)$ acts through the identification
$\mathbb{C}^{lm}\cong\mathbb{C}^{m}\otimes\mathbb{C}^{l}$. The composition map
$\mathbb{S}^{k,l}_{n}\wedge\mathbb{S}^{j,k}_{m}\to\mathbb{S}^{j,l}_{mn}$ is
determined by the map
$\mathcal{L}_{n}^{\diamond}\wedge\mathcal{L}_{m}^{\diamond}\to\mathcal{L}_{mn}^{\diamond}$
defined by tensor product of decompositions, and the obvious composition map

$\operatorname{Inj}(\mathbb{C}^{ln},\mathbb{C}^{k})\times\operatorname{Inj}(\mathbb{C}^{km},\mathbb{C}^{j})\to\operatorname{Inj}(\mathbb{C}^{lmn},\mathbb{C}^{km})\times\operatorname{Inj}(\mathbb{C}^{km},\mathbb{C}^{j})\to\operatorname{Inj}(\mathbb{C}^{lmn},\mathbb{C}^{j}).$

The complexes $\mathcal{L}^{\diamond}_{m}$ were first introduced in [Ar1], and
were studied in detail in [BJL+] and [AL3]. They play a role in describing the
subquotients of the rank filtration of $K$-theory [AL1, AL2]. Therefore it is
perhaps not surprising that they play a similar role in the rank filtration of
$\mathcal{M}$, given the connection between $\mathcal{M}$ and $\mathtt{ku}$.

Next proposition lists some relevant facts about the complexes
$\mathcal{L}_{m}^{\diamond}$ (Proposition 9.1 in the paper).

######  Proposition 1.4.

  1. 1.

$\mathcal{L}^{\diamond}_{1}=S^{0}$.

  2. 2.

The complex $\mathcal{L}^{\diamond}_{m}$ is rationally contractible for all
$m>1$.

  3. 3.

The complex $\mathcal{L}^{\diamond}_{m}$ is contractible unless $m$ is a prime
power.

  4. 4.

If $m=p^{k}$ where $p$ is a prime and $k>0$ then
$\mathcal{L}^{\diamond}_{p^{k}}$ is $p$-local.

  5. 5.

The complex $\mathcal{L}^{\diamond}_{p^{k}}$ has chromatic type $k$.

Here are some consequences of Theorem 1.3 and Proposition 1.4. To begin with,
we have a simple description of the endomorphisms in $\mathcal{M}$. The
endomorphism spectrum of $\Sigma^{\infty}_{\mathtt{NC}}M_{k}$ is the group
ring spectrum of the projective unitary group $PU(k)$.

######  Corollary 1.5.

$\mathbb{S}^{k,k}\simeq\Sigma^{\infty}PU(k)_{+}.$

But our main application of Theorem 1.3 and Proposition 1.4(2) is to give a
simplified description of the rational homotopy type of $\mathbb{S}^{k,l}$,
and consequently of the rationalization of $\mathtt{NSp}$. Recall that the
composition maps $\mathbb{S}^{j,k}\wedge\mathbb{S}^{k,l}\to\mathbb{S}^{j,l}$
restrict to maps of the form
$\mathbb{S}^{j,k,1}\wedge\mathbb{S}^{k,l,1}\to\mathbb{S}^{j,l,1}$. It follows
that the spectra $\mathbb{S}^{k,l,1}$ assemble to a spectral category, that we
denote $\mathcal{M}^{1}$, which has the same objects as $\mathcal{M}$, and
equipped with a functor $\mathcal{M}^{1}\to\mathcal{M}$ that is the identity
on objects. Informally speaking, $\mathcal{M}^{1}$ is the first stage of the
rank filtration of $\mathcal{M}$.

It follows from Theorem 1.3 that there is an equivalence

$\mathbb{S}^{k,l,1}=\mathbb{S}^{k,l}_{1}\simeq\Sigma^{\infty}{\operatorname{Inj}(\mathbb{C}^{l},\mathbb{C}^{k})/_{U(1)}}_{+}.$

It is worth noting that $\mathbb{S}^{k,l,1}$ is a suspension spectrum. Let
$\mathbb{P}\operatorname{Inj}$ be the topologically enriched symmetric
monoidal category of finite positive dimensional Hilbert spaces and embeddings
up to scalar. That is, up to isomorphism the objects of
$\mathbb{P}\operatorname{Inj}$ are given by $\mathbb{C}^{k}$ for $k\geq 1$ and

$\mathbb{P}\operatorname{Inj}(\mathbb{C}^{l},\mathbb{C}^{k})=\operatorname{Inj}(\mathbb{C}^{l},\mathbb{C}^{k})/_{U(1)}=U(k)/(U(1)\times
U(k-l)).$

$\mathbb{P}\operatorname{Inj}$ is a symmetric monoidal category, with the
monoidal structure given by the tensor product. Since the functor
$\Sigma^{\infty}_{+},$ from topological spaces to our model of spectra
$\mathtt{Sp^{M}}$, is symmetric monoidal, we can define a category enriched in
$\mathtt{Sp^{M}}$, which we denote
$\mathbb{P}\operatorname{Inj}^{\mathtt{Sp}}$, by applying
$\Sigma^{\infty}_{+}$ to the mapping spaces of $\mathbb{P}\operatorname{Inj}$.
We show in Section 5.1 that

$\mathcal{M}^{1}\simeq(\mathbb{P}\operatorname{Inj}^{\mathtt{Sp}})^{\operatorname{op}}.$

The following is an easy consequence of Proposition 1.4(2)

######  Corollary 1.6.

The natural map

$\Sigma^{\infty}{\operatorname{Inj}(\mathbb{C}^{l},\mathbb{C}^{k})/_{U(1)}}_{+}\simeq\mathbb{S}^{k,l,1}\xrightarrow{\simeq_{\mathbb{Q}}}\mathbb{S}^{k,l}$

is a rational homotopy equivalence.

######  Remark 1.7.

If one lets $k$ go to $\infty$ in corollary 1.6, one obtains the classical
fact that the canonical map $\Sigma^{\infty}\mathbb{C}P^{\infty}_{+}\to ku$ is
a rational equivalence. So corollary 1.6 can be thought of as a lift of this
fact.

Corollary 1.6 says that the functor $\mathcal{M}^{1}\to\mathcal{M}$ is a
rational equivalence of spectral categories. Using this, we can give a rather
explicit description of the rationalization of $\mathtt{NSp}$. We discuss the
general construction of rational localization and $p$-localization of a
stable, monoidal, $\infty$-category in Section 10. Let
$\mathtt{NSp}_{\mathbb{Q}}$ denote the rational localization of the
$\infty$-category of noncommutative spectra $\mathtt{NSp}$ and let
$\mathtt{Sp}_{\mathbb{Q}}$ denote the rational localization of the usual
$\infty$-category of spectra $\mathtt{Sp}$ . It is well known that
$\mathtt{Sp}_{\mathbb{Q}}$ is a symmetric monoidal presentable
$\infty$-category, and the rationalization functor

$L_{\mathbb{Q}}\colon\mathtt{Sp}\to\mathtt{Sp}_{\mathbb{Q}}$

is symmetric monoidal.

Let $\mathbb{P}\operatorname{Inj}_{\infty}$ denote the topological nerve of
the topological category $\mathbb{P}\operatorname{Inj}$ defined above.
Applying the symmetric monoidal functor

$L_{\mathbb{Q}}\circ\Sigma^{\infty}_{+}:\mathcal{S}\to\mathtt{Sp}_{\mathbb{Q}}$

to the mapping spaces of $\mathbb{P}\operatorname{Inj}_{\infty}$ we obtain an
$\infty$-category enriched in $\mathtt{Sp}_{\mathbb{Q}}$ which we denote by
$\mathbb{P}\operatorname{Inj}_{\infty}^{\mathtt{Sp}_{\mathbb{Q}}}$. Let
$P_{\mathtt{Sp}_{\mathbb{Q}}}((\mathbb{P}\operatorname{Inj}_{\infty}^{\mathtt{Sp}_{\mathbb{Q}}})^{\operatorname{op}})$
denote the $\infty$-category of $\mathtt{Sp}_{\mathbb{Q}}$-enriched functors
from $\mathbb{P}\operatorname{Inj}_{\infty}^{\mathtt{Sp}_{\mathbb{Q}}}$ to
$\mathtt{Sp}_{\mathbb{Q}}$. The following theorem summarizes the results of
Section 10 about $\mathtt{Sp}_{\mathbb{Q}}$:

######  Theorem 1.8 (Theorem 10.5).

There are equivalences of symmetric monoidal $\infty$-categories

$\mathtt{NSp}_{\mathbb{Q}}\simeq
P_{\mathtt{Sp}_{\mathbb{Q}}}((\mathbb{P}\operatorname{Inj}_{\infty}^{\mathtt{Sp}_{\mathbb{Q}}})^{\operatorname{op}})\simeq\mathrm{Fun}(\mathbb{P}\operatorname{Inj}_{\infty},\mathtt{Sp}_{\mathbb{\mathbb{Q}}}).$

######  Remark 1.9.

Note that the expression on the right in Theorem 1.8 does not use enriched
$\infty$-categories. This is the usual $\infty$-category of functors from
$\mathbb{P}\operatorname{Inj}_{\infty}$ to
$\mathtt{Sp}_{\mathbb{\mathbb{Q}}}$. Note also that the mapping spaces in
$\mathbb{P}\operatorname{Inj}$ are all finite connected CW-complexes
(manifolds, even). It is natural to wonder if one can given a more direct
algebraic model of $\mathtt{NSp}_{\mathbb{Q}}$ as a dg-category.

####  $p$-local and chromatic picture

Now instead of rationalizing, suppose we fix a prime $p$ and localize
everything at $p$. One can obtain further information about the
$p$-localization of $\mathcal{M}$. It follows from Proposition 1.4 parts (3)
and (4) that the filtration is $p$-locally constant except at powers of $p$.
Therefore it is natural to regrade the filtration of $\mathbb{S}^{k,l}$ as
follows

$\mathbb{S}^{k,l,1}\hookrightarrow\mathbb{S}^{k,l,p}\hookrightarrow\mathbb{S}^{k,l,p^{2}}\hookrightarrow\cdots\hookrightarrow\mathbb{S}^{k,l,p^{i}}\cdots$

With this grading, (the $p$-localization) of $\mathcal{M}$ is a filtered
category in the usual sense, that composition adds degrees. Furthermore, we
have

######  Corollary 1.10.

Fix a prime $p$ and localize everything at $p$. The map

$\mathbb{S}^{k,l,p^{n}}\to\mathbb{S}^{k,l}$

induces an isomorphism on Morava $K(i)$-theory for $i\leq n$.

The last corollary may have consequences for ``noncommutative chromatic
homotopy theory'', but we will not pursue it here.

#### Section by section outline of the paper

In Section 2 we set the stage by recalling some relevant definitions from
[ABS1]. We introduce functors $G_{k,l}$, where $k,l$ are positive integers.
The functors $G_{k,l}$ encode all the information about morphisms in
$\mathcal{M}$. More precisely, the stabilization of $G_{k,l}$ is the spectral
mapping object from $k$ to $l$ in $\mathcal{M}$. In Section 3 we introduce a
natural filtration of the functors $G_{k,l}$, which we call the rank
filtration.

In Section 4 we give an explicit description of the restriction of $G_{k,l}$
to pointed finite sets. In Section 5 we establish various properties of the
restriction of $G_{k,l}$ to finite sets. Most importantly, we show that the
functor $G_{k,l}$ is a $\Gamma$-space, in the sense that it is determined by
its values on finite sets, and we also observe that the restriction of
$G_{k,l}$ to finite sets is cofibrant in the Reedy model structure on
$\Gamma$-spaces.

In Section 6 we show that all the mapping spectra in $\mathcal{M}$ are
equipped with a natural map to the connective $K$-theory spectrum $ku$. We
observe that the rank filtration of $\mathcal{M}$ is a lift of the classical
rank filtration of $ku$. We also discuss how one can use our models to
represent the $K$-theory functor on non-commutative complexes. We make
connection with some work of Dadarlat and McClure [DM].

In Sections 7 and 8 we describe the subquotients of the rank filtration in
terms of complexes of direct-sum decompositions that arose earlier in the
study of the rank filtration of $ku$. Since complexes of direct-sum
decompositions are well-studied, we obtain interesting consequences about
$\mathcal{M}$ and $\mathtt{NSp}$. In Section 9 we use the results of preceding
sections to calculate the mapping spectra in $\mathcal{M}$ in some cases. In
Section 10 we use those results to give an explicit model for the
rationalization of $\mathcal{M}$ and $\mathtt{NSp}$. Rationally,
$\mathtt{NSp}$ is equivalent to the $\infty$-category of presheaves of
rational spectra on the $\infty$-category whose objects are finite-dimensional
Hilbert spaces, and whose hom-spaces are linear embeddings modulo scalars. We
also point out some consequences that our models have for the $p$-localization
and potentially chromatic localization of $\mathcal{M}$ and $\mathtt{NSp}$.

#### Acknowledgements

We would like to thank Jeffrey Carlson for fruitful correspondences during the
early stages of our work. We are grateful to Vladimir Hinich for explaining to
us his theory of enriched infinity categories and its relevance to our work.

##  2 The functors $G_{k,l}$ and their stabilization

In this section we recall the construction of $\mathcal{M}$ as a category
strictly enriched in $\mathtt{Sp^{M}}$ (see Remark 1.2). We will introduce
certain functors $G_{k,l}\in\mathtt{Sp^{M}}$, which will represent the mapping
spectra in $\mathcal{M}$.

To begin with, let $\mathtt{SC^{*}}$ denote the category of (non-unital)
separable $C^{*}$-algebras and $*$-homomorphisms. Note that the matrix
algebras $\\{M_{k}\mid k=1,2,\ldots\\}$ are objects of $\mathtt{SC^{*}}$.
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. It is
well-known that $\mathtt{SC^{*}}$ is cotensored over the category of pointed
finite CW-complexes [AG]. For a finite pointed CW-complex $X$ and a
$C^{*}$-algebra $A$ we denote the contensoring by $\mathrm{C}_{0}(X,A)$.

Next, we want to use $\mathtt{SC^{*}}$ to define $\mathcal{M}$ as a category
enriched in $\mathtt{Sp^{M}}$. Recall that the underlying category of
$\mathtt{Sp^{M}}$ is the category of pointed continuous functors from pointed
finite CW-complexes to pointed topological spaces.

######  Definition 2.1.

Let $G_{k,l}\colon\mathtt{CW}^{f}_{*}\to\operatorname{Top}$ be the functor
defined as follows

$G_{k,l}(X)=\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(X,M_{l}),M_{k}).$

We will consider $G_{k,l}$ to be an object of $\mathtt{Sp^{M}}$. Notice that
for all $k,l,m$, there is a natural map $G_{k,l}(X)\wedge G_{l,m}(Y)\to
G_{k,m}(X\wedge Y)$, defined as a composition of the following maps.

$\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(X,M_{l}),M_{k})\wedge\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(Y,M_{m}),M_{l})\to\\\
\to\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(X,M_{l}),M_{k})\wedge\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(X\wedge
Y,M_{m}),\mathrm{C}_{0}(X,M_{l}))\to\\\
\to\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(X\wedge
Y,M_{m}),M_{k}).$

Here the second map is composition, and the first map is induced by the
cotensoring

$\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(Y,M_{m}),M_{l})\to\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(X\wedge
Y,M_{m}),\mathrm{C}_{0}(X,M_{l})).$

This map induces natural maps

$G_{k,l}\wedge G_{l,m}\to G_{k,m}$ (2)

where $\wedge$ denotes internal smash product (aka Day convolution).

######  Definition 2.2.

Let $\mathcal{M}$ be the following $\mathtt{Sp^{M}}$-enriched category. The
objects of $\mathcal{M}$ are positive integers. Given two integers $k,l$, the
mapping spectrum from $k$ to $l$ is given by $G_{k,l}$. The composition law in
$\mathcal{M}$ is defined by the structure maps like in (2), for all $k,l,m$.

######  Remark 2.3.

Recall that in the infinity categorical picture, $\mathcal{M}$ is the full
spectral subcategory of $\mathtt{NSp}$ whose objects are
$\\{\Sigma^{\infty}_{\mathtt{NC}}M_{k}\mid k=1,2\ldots\\}$. We show in [ABS1]
that the enriched coherent nerve of $\mathcal{M}$ from Definition 2.2 is
equivalent to $\mathcal{M}$ defined above. The objects of $\mathcal{M}$
provide a set of compact generators of $\mathtt{NSp}$. The main result of
[ABS1] says that the $\infty$-category $\mathtt{NSp}$ is equivalent to the
$\infty$-category $P_{\mathtt{Sp}}(\mathcal{M})$ of spectral presheaves on
$\mathcal{M}$. In this paper we investigate the category $\mathcal{M}$, with
the eventual goal in mind of understanding $\mathtt{NSp}$. In view of the
results of [ABS1] in this paper we identify $\mathtt{NSp}$ with
$P_{\mathtt{Sp}}(\mathcal{M})$. In the remaining part of the paper we will
study the category $\mathcal{M}$ mainly using the explicit model provided in
Definition 2.2, but will state the results also in the infinity categorical
picture.

Recall from the introduction that we identify
$\mathtt{Sp^{M}}_{\infty}=\mathtt{Sp}$ and we denote by
$\partial_{1}:\mathtt{Sp^{M}}\to\mathtt{Sp}$, the localization functor. If
$G\in\mathtt{Sp^{M}}$ is a pointed continuous functors from pointed finite CW-
complexes to pointed topological spaces, then $\partial_{1}G$ is the spectrum
corresponding to the sequence of spaces $\\{G(S^{0}),G(S^{1}),\ldots$}, or in
other words

$\partial_{1}G\simeq{\mathop{\operatorname{hocolim}}}_{n}\Sigma^{-n}\Sigma^{\infty}G(S^{n}).$

This is known as the stabilization, or the first derivative of the functor
$G$. We have shown in [ABS1] that for all natural numbers $k,l$, we have

$\mathbb{S}^{k,l}:=\operatorname{Hom}_{\mathtt{NSp}}(\Sigma^{\infty}_{\mathtt{NC}}M_{k},\Sigma^{\infty}_{\mathtt{NC}}M_{l})\simeq\partial_{1}G_{k,l}\simeq{\mathop{\operatorname{hocolim}}}_{n}\Sigma^{-n}\Sigma^{\infty}G_{k,l}(S^{n}).$

######  Example 1.

Let us consider the case $k=l=1$. The functor $G_{1,1}$ is given as follows

$G_{1,1}(X)=\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(X,M_{1}),M_{1})=\operatorname{Map}_{\mathtt{SC^{*}}}(\mathrm{C}_{0}(X,\mathbb{C}),\mathbb{C}).$

By the Gelfand-Naimark theorem, it follows that $G_{1,1}(X)\cong X$, and
therefore $\mathbb{S}^{1,1}=\Sigma^{\infty}S^{0}$ is the ordinary sphere
spectrum.

######  Remark 2.4.

Let us interpret $\mathbb{S}^{1,1}=\Sigma^{\infty}S^{0}$ as the endomorphism
spectrum
$\operatorname{End}_{\mathtt{NSp}}(\Sigma^{\infty}_{\mathtt{NC}}M_{1})$. We
have identified $\mathtt{NSp}$ with the $\infty$-category of spectral
presheaves on $\mathcal{M}$. In this picture, the $\infty$-category of
spectral presheaves on the full $\mathtt{Sp}$-enriched subcategory of
$\mathcal{M}$ consisting of the object $\Sigma^{\infty}_{\mathtt{NC}}M_{1}$
can be identified with the $\infty$-category $\mathtt{Sp}$ of ``commutative''
or ``ordinary'' spectra. There is an ``inclusion'' functor of
$\mathtt{Sp}$-tensored categories $\mathtt{Sp}\to\mathtt{NSp}$ which in terms
of presheaves is defined by an $\mathtt{Sp}$-enriched left Kan extension
(weighted colimit) from $\\{\Sigma^{\infty}_{\mathtt{NC}}M_{1}\\}$ to
$\mathcal{M}$. The inclusion functor has a right adjoint
$\mathtt{NSp}\to\mathtt{Sp}$, a kind of ``abelianization'' functor, defined by
restriction of presheaves. We will say a little more about it in Section 6.

##  3 The rank filtration

In this section we introduce the rank filtration of $G_{k,l}$, which induces a
rank filtration of the spectral category $\mathcal{M}$. In later sections we
will see that the rank filtration of $\mathcal{M}$ is a lift of the classical
rank filtration of the connective $K$-theory spectrum $ku$.

Let $l,k\geq 1$, let $X$ be a finite pointed CW-complex and let

$f\in G_{k,l}(X)=\mathtt{SC^{*}}(\mathrm{C}_{0}(X,M_{l}),M_{k}),$

be a map. Let $A_{f}\subseteq M_{k}$ be the image of $f$. $A_{f}$ acts as non-
unital $C^{*}$-Algebra on the Hilbert space $\mathbb{C}^{k}$ and thus we get
an orthogonal decomposition $\mathbb{C}^{k}=\mathrm{Ker}A_{f}\oplus
A_{f}\cdot\mathbb{C}^{k}$. Denote
$V_{f}:=A_{f}\cdot\mathbb{C}^{k}\subseteq\mathbb{C}^{k}$. We shall filter the
space $G_{k,l}(X)$ according to the dimension of $V_{f}$. The following
theorem is useful in that analysis.

######  Theorem 3.1.

Let $X$ be a pointed compact metrizable space and let $l\geq 1$. There is a
bijection between the closed subsets of $X\setminus\\{*\\}$ and closed two-
sided ideals of $\mathrm{C}_{0}(X,M_{l})$, defined by the following
correspondence

$F\mapsto I_{F}:=\\{f\in\mathrm{C}_{0}(X,M_{l})\mid\forall x\in
F.\,f(x)=0\\}.$

###### Proof.

The case $l=1$ is well-known. We will show that it implies the rest. Two
$C^{*}$-algebras $A$ and $B$ are called strongly Morita equivalent if they are
related by a $B$-$A$-imprimitivity bimodule in the sense of [Rie]. Let
$\mathbb{K}$ be the algebra of compact operators on an infinite dimensional
separable Hilbert space. It is shown in [BGR] that if $A$ and $B$ are
separable, then $A$ and $B$ are strongly Morita equivalent iff

$A\otimes\mathbb{K}\cong B\otimes\mathbb{K}.$

It is not hard to see that
$\mathrm{C}_{0}(X,M_{l})\cong\mathrm{C}_{0}(X)\otimes M_{l}$ and
$M_{l}\otimes\mathbb{K}\cong\mathbb{K}$, so we have

$\mathrm{C}_{0}(X,M_{l})\otimes\mathbb{K}\cong(\mathrm{C}_{0}(X)\otimes
M_{l})\otimes\mathbb{K}\cong\mathrm{C}_{0}(X)\otimes(M_{l}\otimes\mathbb{K})\cong\mathrm{C}_{0}(X)\otimes\mathbb{K}.$

Thus, $\mathrm{C}_{0}(X,M_{l})$ and $\mathrm{C}_{0}(X)$ are strongly Morita
equivalent. By [Zet], we have an isomorphism between the sets of closed two-
sided ideals of $\mathrm{C}_{0}(X,M_{l})$ and of $\mathrm{C}_{0}(X)$. ∎

######  Lemma 3.2.

Let $X\in\mathtt{CW}^{f}_{*}$ be pointed finite CW-complex and let

$f\in G_{k,l}(X)={\mathtt{SC^{*}}}(\mathrm{C}_{0}(X,M_{l}),M_{k}).$

Then $f$ admits a unique factorization of the following form

$\mathrm{C}_{0}(X,M_{l})\twoheadrightarrow\mathrm{C}_{0}(F_{f}\cup\\{*\\},M_{l})\stackrel{{\scriptstyle
f^{\prime}}}{{\rightarrowtail}}M_{k}$ (3)

where the first map is the surjective restriction to a finite subset
$F_{f}\subset X\setminus\\{\ast\\}$ and the second map
$f^{\prime}\colon\mathrm{C}_{0}(F_{f}\cup\\{*\\},M_{l})\rightarrowtail M_{k}$
is a monomorphism. In particular we have $V_{f}=V_{f^{\prime}}$.

###### Proof.

First, note that $\ker(f)$ is a closed two sided $*$-ideal of
$\mathrm{C}_{0}(X,M_{l})$. By Theorem 3.1, there exists a closed subset
$F_{f}$ of $X\setminus\\{*\\}$ such that

$\ker(f)=I_{F_{f}}=\\{g\in\mathrm{C}_{0}(X,M_{l})\mid g|_{F_{f}}=0\\}.$

Notice that $\ker(f)=I_{F_{f}}$ is also the kernel of the restriction
homomorphism

$\mathrm{C}_{0}(X,M_{l})\to\mathrm{C}_{0}(F\cup\\{*\\},M_{l}).$

We claim that the restriction homomorphism is surjective. This amounts to
showing that any map from a closed subset of $X$ to $M_{l}$ can be extended to
a map from $X$ to $M_{l}$. This in turn follows immediately from the Tietze
extension theorem.

It follows that $f$ admits a unique factorization of the following form

$\mathrm{C}_{0}(X,M_{l})\twoheadrightarrow\mathrm{C}_{0}(F_{f}\cup\\{*\\},M_{l})\stackrel{{\scriptstyle
f^{\prime}}}{{\rightarrowtail}}M_{k}$ (4)

where the first map is restriction to a subset $F_{f}$ and the second map
$f^{\prime}\colon\mathrm{C}_{0}(F_{f}\cup\\{*\\},M_{l})\to M_{k}$ is a
monomorphism. Moreover $F_{f}$ is the minimal subset of $X\setminus\\{*\\}$
for which the map $f$ factors through
$\mathrm{C}_{0}(F_{f}\cup\\{*\\},M_{l})$. Notice that
$\textrm{Im\,}(f^{\prime})$ is finite-dimensional as a vector space over
$\mathbb{C}$. This implies that $F_{f}$ is finite. ∎

######  Lemma 3.3.

let

$f\in G_{k,l}(X)=\mathtt{SC^{*}}(\mathrm{C}_{0}(X,M_{l}),M_{k}),$

we have

$l|\dim V_{f}$

and

$l\cdot|F_{f}|\leq\dim V_{f}\leq k$

###### Proof.

By lemma 3.2 the function $f$ can be factored as a surjection followed by an
injection

$\mathrm{C}_{0}(X,M_{l})\to M_{l}^{F_{f}}\stackrel{{\scriptstyle
f^{\prime}}}{{\to}}M_{k}$

Thus we have an isomorphism $M_{l}^{F_{f}}\cong A_{f}$. $1\in M_{l}^{F_{f}}$
now acts on $\mathbb{C}^{k}$ as a projection onto $V_{f}$ and thus we get a
unital action of $M_{l}^{F_{f}}$ on $V_{f}$. Now for $x\in F_{f}$ denote by
$W_{x}$ the unital $M_{l}^{F_{f}}$ module obtained by the canonical action on
$\mathbb{C}^{l}$ via map $M_{l}^{F_{f}}\to M_{l}^{\\{x\\}}=M_{l}$. Every
finite dimensional unital $M_{l}^{F_{f}}$ module is a direct sum of finitely
many copies of the $W_{x}$'s. We thus get that

$V_{f}=\bigoplus_{x\in F_{f}}=W_{x}^{e_{x}}.$

The injectivity of the map $M_{l}^{F_{f}}\stackrel{{\scriptstyle
f^{\prime}}}{{\to}}M_{k}$ implies that $e_{x}\geq 1$ for every $x\in F_{f}$.
Since $\dim V_{f}=l\sum e_{x}$ and $V_{f}\subseteq\mathbb{C}^{k}$ we get the
claim. ∎

######  Definition 3.4.

Let $l,k\geq 1$, let $X$ be a finite pointed CW-complex and let

$f\in G_{k,l}(X)=\mathtt{SC^{*}}(\mathrm{C}_{0}(X,M_{l}),M_{k}).$

We define the _rank of $f$_ to be the non-negative integer

$\mathrm{rank}(f):=\frac{\dim V_{f}}{l}\in\mathbb{Z}_{\geq 0}.$

Suppose we have a map $\alpha\colon X\to Y$ in $\mathtt{CW}^{f}_{*}$. By
functoriality, it induces a map $G_{k,l}(X)\to G_{k,l}(Y)$. Suppose $f\in
G_{k,l}(X)$. By definition, $f$ is a $*$-homomorphism
$f\colon\mathrm{C}_{0}(X,M_{l})\to M_{k}$, and the image of $f$ in
$G_{k,l}(Y)$ is the composite homomorphism

$\mathrm{C}_{0}(Y,M_{l})\xrightarrow{\alpha^{*}}\mathrm{C}_{0}(X,M_{l})\xrightarrow{f}M_{k}.$

Therefore the rank of the image of $f$ in $G_{k,l}(Y)$ is at most the rank of
$f$. Because of this, the following definition really does describe a functor.

######  Definition 3.5.

Let $k,l\geq 1$, $m\geq 0$. Define the functors
$G_{k,l,m}\colon\mathtt{CW}^{f}_{*}\to\operatorname{Top}$ as follows

$G_{k,l,m}(X)=\left\\{f\in G_{k,l}(X)|\quad\mathrm{rank}(f)\leq
m\right\\}\subseteq G_{k,l}(X).$

Similarly, define $\mathbb{S}^{k,l,m}$ to be the stabilization of $G_{k,l,m}$.
Explicitly, $\mathbb{S}^{k,l,m}$ is the spectrum
$\\{G_{k,l,m}(S^{0}),G_{k,l,m}(S^{1}),\ldots\\}$.

######  Remark 3.6.

Note that for all $X\in\mathtt{CW}^{f}_{*}$ and $k,l\geq 1$ we have
$G_{k,l,0}(X)=\ast$. Additionally by lemma 3.3 for
$m\geq\lfloor\frac{k}{l}\rfloor$ we get $G_{k,l,m}=G_{k,l}$.

We have defined a filtration of $G_{k,l}$ by sequence of subfunctors

$*=G_{k,l,0}\subset G_{k,l,1}\subset\cdots\subset
G_{k,l,\lfloor\frac{k}{l}\rfloor}=G_{k,l}.$

We call this filtration _the rank filtration_. Now recall that the functors
$G_{k,l}$ represent mapping spectra in $\mathcal{M}$ and that composition in
$\mathcal{M}$ is determined by maps of the following form

$G_{k,l}(X)\wedge G_{l,m}(Y)\to G_{k,m}(X\wedge Y).$

The following proposition tells how the rank filtration interacts with
composition.

######  Proposition 3.7.

For all $r$ and $s$, the composition map above restricts to a natural map

$G_{k,l,r}(X)\wedge G_{l,m,s}(Y)\to G_{k,m,rs}(X\wedge Y)$

###### Proof.

Recall that $G_{k,l}(X)=\mathtt{SC^{*}}(\mathrm{C}_{0}(X,M_{l}),M_{k})$.
Written in these terms, the composition map has the following form

$\mathtt{SC^{*}}(\mathrm{C}_{0}(X,M_{l}),M_{k})\wedge\mathtt{SC^{*}}(\mathrm{C}_{0}(Y,M_{m}),M_{l})\to\\\
\to\mathtt{SC^{*}}(\mathrm{C}_{0}(X,M_{l}),M_{k})\wedge\mathtt{SC^{*}}(\mathrm{C}_{0}(X\wedge
Y,M_{m}),\mathrm{C}_{0}(X,M_{l}))\to\\\
\to\mathtt{SC^{*}}(\mathrm{C}_{0}(X\wedge Y,M_{m}),M_{k}).$

Suppose that $f\in\mathtt{SC^{*}}(\mathrm{C}_{0}(X,M_{l}),M_{k})$ and
$g\in\mathtt{SC^{*}}(\mathrm{C}_{0}(Y,M_{m}),M_{l})$ have ranks $r$ and $s$
respectively. Let $f\odot g$ denote the image of $f\wedge g$ in
$\mathtt{SC^{*}}(\mathrm{C}_{0}(X\wedge Y,M_{m}),M_{k})$. Our goal is to show
that $f\odot g$ has rank $rs$.

By Lemma 3.2, there exist finite subsets $F_{f}\subset
X\smallsetminus\\{\ast\\}$, $F_{g}\subset Y\smallsetminus\\{\ast\\}$ such that
$f$ factors as
$\mathrm{C}_{0}(X,M_{l})\twoheadrightarrow\mathrm{C}_{0}(F_{f}\cup\\{*\\},M_{l})\stackrel{{\scriptstyle
f^{\prime}}}{{\rightarrowtail}}M_{k}$, and there is a similar factorization of
$g$. It follows that $f\odot g$ factors as follows

$\mathrm{C}_{0}(X\wedge Y,M_{m})\twoheadrightarrow\mathrm{C}_{0}(F_{f}\times
F_{g}\cup\\{*\\},M_{m})\stackrel{{\scriptstyle f^{\prime}\odot
g^{\prime}}}{{\rightarrowtail}}M_{k}$

where the second map is itself the following composite

$(M_{m}^{F_{g}})^{F_{f}}\xrightarrow{g^{\prime\times
F_{f}}}M_{l}^{F_{f}}\xrightarrow{f^{\prime}}M_{k}.$

Here the first map is the cartesian product of $|F_{f}|$ copies of the
$g^{\prime}$ with itself. This map determines an action of $M_{m}^{F_{g}\times
F_{f}}$ on $\mathbb{C}^{k}$. Our goal is to show that $M_{m}^{F_{g}\times
F_{f}}\cdot\mathbb{C}^{k}$ has dimension $rsm$.

Recall that $A_{f}=A_{f^{\prime}}$ is the image of $f^{\prime}$. Since
$\mathrm{rank}(f)=r$, $A_{f}\cdot\mathbb{C}^{k}$ has dimension $rl$. If
$B\subset M_{l}$ is a $C^{*}$-subalgebra such that $B\cdot\mathbb{C}^{l}$ has
dimension $d$, and we let $B^{F_{f}}$ act on $\mathbb{C}^{k}$ via the map
$f^{\prime}$, then $B^{F_{f}}\cdot\mathbb{C}^{k}$ has dimension $rd$. Now take
$B$ to be the image of $g^{\prime}$. Since $\mathrm{rank}(g)=s$,
$B\cdot\mathbb{C}^{l}$ has dimension $sm$, so finally we conclude that
$M_{m}^{F_{g}\times F_{f}}\cdot\mathbb{C}^{k}$ has dimension $rsm$. This means
that $f\odot g$ has rank $rs$. ∎

######  Remark 3.8.

Proposition 3.7 can be intepreted as follows: the rank filtration is a
filtration of the category $\mathcal{M}$ by the multiplicative monoid of
natural numbers.

##  4 The restriction of $G_{k,l}$ to finite sets

It will turn out that the functor $G_{k,l}$, whose domain is the category of
pointed finite CW-complexes, is determined by its restriction to the category
of pointed finite sets. In this section we give an explicit description of the
restriction of $G_{k,l}$ to finite sets.

Let us begin with a definition, which also serves to establish some notation.

######  Definition 4.1.

For a natural number $i$, let $[i]=\\{0,1,\ldots,i\\}$, considered as a
pointed set with basepoint $0$. Let $\mathtt{Fin}_{*}$ be the category whose
objects are $\\{[0],[1],\dots,[k],\ldots\\}$ and whose morphisms are
basepoint-preserving functions. For $k\geq 0$, let $\mathtt{Fin^{\leq k}_{*}}$
denote the full subcategory of $\mathtt{Fin}_{*}$ spanned by the objects
$\\{[0],[1],\dots,[k]\\}$. We will also use the notation $\underline{i}$ for
the unpointed set $\\{1,\ldots,i\\}$.

######  Remark 4.2.

The category $\mathtt{Fin}_{*}$ is denoted $\Gamma$ in some sources, and
$\Gamma^{\operatorname{op}}$ in some other sources. We find the notation
$\mathtt{Fin}_{*}$ to be more descriptive. But following the tradition
established by Segal [Se1], we call pointed functors
$\mathtt{Fin}_{*}\to\operatorname{Top}$ $\Gamma$-spaces.

We will now examine the restriction of $G_{k,l}$ to $\mathtt{Fin}_{*}$. For a
finite pointed set $[t]$, $G_{k,l}([t])$ is the space of non-unital
$C^{*}$-algebra homomorphisms from $M_{l}^{t}$ to $M_{k}$. Spaces of such
homomorphisms are well-understood. We want to describe them in a way that
makes the functoriality in $[t]$ explicit. We need a few definitions.

######  Definition 4.3.

The category of pointed multisets is defined as follows. The objects are
ordered $t$-tuples $(m_{1},\ldots,m_{t})$ of natural numbers. The possibility
$t=0$ is included, in which case the tuple is empty. A morphism
$(m_{1},\ldots,m_{t})\to(n_{1},\ldots,n_{s})$ consists of a pointed function
$\alpha\colon[t]\to[s]$ such that $n_{j}=\Sigma_{i\in\alpha^{-1}(j)}m_{i}$ for
all $1\leq j\leq s$. In particular, if $j$ is not in the image of $\alpha$
then $n_{j}=0$. Note that there are no restrictions on $m_{i}$ for
$i\in\alpha^{-1}(0)$.

Given a pointed multiset $(m_{1},\ldots,m_{t})$ and a pointed function of sets
$\alpha\colon[t]\to[s]$ we define $\alpha_{*}(m_{1},\ldots,m_{t})$ to be the
multiset $(n_{1},\ldots,n_{s})$ with $n_{j}=\Sigma_{i\in\alpha^{-1}(j)}m_{i}$
for all $1\leq j\leq s$.

######  Example 2.

Let $\alpha\colon[3]\to[2]$ be the function defined by
$\alpha(0)=\alpha(1)=0$, $\alpha(2)=\alpha(3)=1$. Then
$\alpha_{*}(4,2,3)=(5,0)$.

Suppose we have a multiset $(m_{1},\ldots,m_{t})$ and a natural number $l$. We
will make much use of the unitary vector space
$\mathbb{C}^{(m_{1}+\cdots+m_{t})l}$. We identify this vector space with

$\mathbb{C}^{m_{1}+\cdots+m_{t}}\otimes\mathbb{C}^{l}\cong\mathbb{C}^{m_{1}}\otimes\mathbb{C}^{l}\oplus\cdots\oplus\mathbb{C}^{m_{t}}\otimes\mathbb{C}^{l}.$

Notice that there are commuting actions of $U(m_{1})\times\cdots\times
U(m_{t})$ and $U(l)$ on $\mathbb{C}^{(m_{1}+\cdots+m_{t})l}$. It follows that
these groups act on any space obtained by applying a continuous functor to
this vector space.

Now suppose we have a morphism of pointed multisets
$\alpha\colon(m_{1},\ldots,m_{t})\to(n_{1},\ldots,n_{s})$, so
$(n_{1},\ldots,n_{s})=\alpha_{*}(m_{1},\ldots,m_{t})$. Choose unitary
isomorphisms
$\mathbb{C}^{n_{j}}\stackrel{{\scriptstyle\cong}}{{\to}}\mathbb{C}^{\Sigma_{i\in\alpha^{-1}(j)}m_{i}}$
for all $1\leq j\leq s$. The function $\alpha$ together with these
isomorphisms determine an inner-product-preserving inclusion
$\mathbb{C}^{(n_{1}+\cdots+n_{s})l}\to\mathbb{C}^{(m_{1}+\cdots+m_{t})l}$.
This inclusion in turn defines a map of spaces (where $k$ is another natural
number)

$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})\to\operatorname{Inj}(\mathbb{C}^{(n_{1}+\cdots+n_{s})l},\mathbb{C}^{k})$

A different choice of isomorphisms
$\mathbb{C}^{n_{j}}\stackrel{{\scriptstyle\cong}}{{\to}}\mathbb{C}^{\Sigma_{i\in\alpha^{-1}(j)}m_{i}}$
will change the map by precomposition with a unitary automorphism of
$\mathbb{C}^{(n_{1}+\cdots+n_{s})l}$ that is induced by automorphisms of
$\mathbb{C}^{n_{1}},\ldots,\mathbb{C}^{n_{s}}$. Therefore we get a well-
defined (i.e., independent of choices of isomorphisms) map

$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})\to\operatorname{Inj}(\mathbb{C}^{(n_{1}+\cdots+n_{s})l},\mathbb{C}^{k})/_{\prod_{j=1}^{s}U(n_{j})}$

Moreover, it is easy to see that the map passes to a well-defined map between
quotients

$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}\to\operatorname{Inj}(\mathbb{C}^{(n_{1}+\cdots+n_{s})l},\mathbb{C}^{k})/_{\prod_{j=1}^{s}U(n_{j})}$
(5)

The upshot is that we have defined a functor from the category of pointed
multi-sets to spaces that sends $(m_{1},\ldots,m_{t})$ to
$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}$.

######  Remark 4.4.

Here is a slightly different way to think of the map (5). Let
$m=m_{1}+\cdots+m_{t}$. There are homeomorphisms

$\operatorname{Inj}(\mathbb{C}^{ml},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}\cong
U(k)/\prod_{i=1}^{t}U(m_{i})\times U(k-ml)$

and similarly

$\operatorname{Inj}(\mathbb{C}^{nl},\mathbb{C}^{k})/_{\prod_{j=1}^{s}U(n_{j})}\cong
U(k)/\prod_{j=1}^{s}U(n_{j})\times U(k-nl).$

A morphism of multisets
$\alpha\colon(m_{1},\ldots,m_{t})\to(n_{1},\ldots,n_{s})$ gives a canonical
way to conjugate $\prod_{i=1}^{t}U(m_{i})\times U(k-ml)$ into a subgroup of
$\prod_{j=1}^{s}U(n_{j})\times U(k-nl)$, and therefore gives rise to a
$U(k)$-equivariant map

$U(k)/\prod_{i=1}^{t}U(m_{i})\times U(k-ml)\to
U(k)/\prod_{j=1}^{s}U(n_{j})\times U(k-nl).$

Now we can describe the functor $G_{k,l}$ on finite sets. The following
proposition is essentially due to Bratelli [Br].

######  Proposition 4.5.

There is a homeomorphism

$G_{k,l}([t])\cong\bigvee_{(m_{1},\ldots,m_{t})}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}$
(6)

The wedge sum on the right is indexed on non-zero ordered $t$-tuples
$(m_{1},\ldots,m_{t})$ of non-negative integers (the zero tuple corresponds to
the basepoint). The functoriality on the right hand side is defined as
follows. A pointed map $\alpha\colon[t]\to[s]$, induces a map

$\bigvee_{(m_{1},\ldots,m_{t})}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}\to\\\
\to\bigvee_{(n_{1},\ldots,n_{s})}{\operatorname{Inj}(\mathbb{C}^{(n_{1}+\cdots+n_{s})l},\mathbb{C}^{k})/_{\prod_{j=1}^{s}U(n_{j})}}_{+}$

that sends the wedge summand corresponding to $(m_{1},\ldots,m_{t})$ to the
wedge summand corresponding to $\alpha_{*}(m_{1},\ldots,m_{t})$ by the map
(5), assuming $\alpha_{*}(m_{1},\ldots,m_{t})$ is not a tuple of zeros. If
$\alpha_{*}(m_{1},\ldots,m_{t})$ consists just of zeros, then $\alpha$ sends
the corresponding wedge summand to the basepoint.

###### Proof.

By definiton 2.1, there is a homeomorphism

$G_{k,l}([t])\cong{\mathtt{SC^{*}}}(M_{l}^{t},M_{k}).$

For every multi-set $(m_{1},\ldots,m_{t})$, we define a map

$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}\to{\mathtt{SC^{*}}}(M_{l}^{t},M_{k})$
(7)

as follows. Suppose we have a unitary isometric inclusion
$\mathbb{C}^{(m_{1}+\cdots+m_{t})l}\hookrightarrow\mathbb{C}^{k}$. From this,
we get a unitary isomorphism (determined up to an automorphism of
$\mathbb{C}^{k-ml}$)

$\mathbb{C}^{m_{1}}\otimes\mathbb{C}^{l}\oplus\cdots\oplus\mathbb{C}^{m_{t}}\otimes\mathbb{C}^{l}\oplus\mathbb{C}^{k-ml}\stackrel{{\scriptstyle\cong}}{{\to}}\mathbb{C}^{k}$
(8)

Having fixed such an isomorphism, we associate with it a $C^{*}$-algebra
homomorphism $M_{l}^{t}\to M_{k}$ as follows: the $i$-th factor $M_{l}$ of
$M_{l}^{t}$ acts on $\mathbb{C}^{m_{i}}\otimes\mathbb{C}^{l}$ by identity on
$\mathbb{C}^{m_{i}}$ and by the standard action on $\mathbb{C}^{l}$. Note that
the action of $M_{l}^{t}$ on $\mathbb{C}^{k-ml}$ is multiplication by zero.

Automorphisms of $\mathbb{C}^{k-ml}$ commute with the action of $M_{l}^{t}$ on
$\mathbb{C}^{m_{1}}\otimes\mathbb{C}^{l}\oplus\cdots\oplus\mathbb{C}^{m_{t}}\otimes\mathbb{C}^{l}\oplus\mathbb{C}^{k-ml}$.
It follows that changing isomorphism 8 by an automorphism of
$\mathbb{C}^{k-ml}$ does not change the resulting algebra homomorphism from
$M_{l}^{t}$ to $M_{k}$. It follows in turn that we have a well-defined map

$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})\to{\mathtt{SC^{*}}}(M_{l}^{t},M_{k}).$

It follows from elementary representation theory (Schur Lemma) that two
elements of
$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})$ induce
the same algebra homomorphism if and only if they differ by an action of
$U(m_{1})\times\cdots\times U(m_{t})$. Therefore we get a well-defined
injective map in (7).

Taking union over multi-sets of the form $(m_{1},\ldots,m_{t})$ with fixed
$t$, we obtain a map

$\bigvee_{(m_{1},\ldots,m_{t})}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}\stackrel{{\scriptstyle\cong}}{{\to}}{\mathtt{SC^{*}}}(M_{l}^{t},M_{k})$
(9)

which we claim is a homeomorphism. Indeed, we already know that it is
injective. Next we need to show that the map (9) is surjective. Let $f\colon
M_{l}^{t}\to M^{k}$ be a $C^{*}$-algebra homomorphism. Let
$I_{1},\ldots,I_{t}$ be the identity elements of the $t$ factors $M_{l}$ of
$M_{l}^{t}$. Then $f(I_{1}),\ldots,f(I_{t})$ are pairwise commuting hermitian
idempotents in $M_{k}$. It follows that for $i=1,\ldots,t$, $f(I_{i})$ is
hermitian projection onto $U_{i}$, where $U_{1},\ldots,U_{t}$ are pairwise
orthogonal subspaces of $\mathbb{C}^{k}$. Now suppose that $1\leq i\leq t$ and
$A_{i}$ is an element of the $i$th factor $M_{l}$ of $M_{l}^{t}$. Then
$f(A_{i})=f(A_{i})f(I_{i})=f(I_{i})f(A_{i})$. Thus $f(A_{i})$ commutes with
the hermitian idempotent $f(I_{i})$. It follows that $f(A_{i})$ leaves
invariant $U_{i}$ and the orthogonal complement of $U_{i}$. Moreover, since
$f(A_{i})=f(A_{i})f(I_{i})$ it follows that $f(A_{i})$ is the composition of
projection onto $U_{i}$ and a linear transformation of $U_{i}$. It follows
that the restriction of $f$ to the $i$th factor of $M_{l}^{t}$ defines a
unital representation of the algebra $M_{l}$ on $U_{i}$. Since $M_{l}$ is
Morita equivalent to $\mathbb{C}$, $U_{i}$ is isomorphic to a sum of copies of
the standard representation of $M_{l}$. This means that we can write
$U_{i}\cong\mathbb{C}^{m_{i}}\otimes\mathbb{C}^{l}$, where
$m_{1},\ldots,m_{t}$ are some non-negative integers. With this identification
the $i$-th $M_{l}$ acts on $U_{i}$ via standard action, and it follows that
$f$ is in the image of the map (9)

We have shown that the map (9) is a bijection. To show that it is a
homeomorphism, observe that $U(k)$ acts continuously on the source and the
target. Moreover, both the source and the target are topologized as the
disjoint union of $U(k)$-orbits. This is true by definition for the source. To
see this for the target, notice that the map that associates to an algebra
morphism $f\colon M_{l}^{t}\to M_{k}$ the integers $(m_{1},\ldots,m_{t})$ is
continuous and therefore locally constant, and $U(k)$ acts transitively on the
preimage of any $t$-tuple of integers. Thus the map (9) is a
$U(k)$-equivariant bijection between disjoint unions of orbits of a continuous
action $U(k)$. It follows that it is a homeomorphism.

The statement about functoriality follows by straightforward diagram-chasing.
∎

In the previous section we defined the rank filtration of $G_{k,l}$. Unwinding
the definitions, we find that if $f\in G_{k,l}([t])$ belongs to the wedge
summand corresponding to $(m_{1},\ldots,m_{t})$ in Proposition 4.5, then

$\mathrm{rank}(f)=m_{1}+\cdots+m_{t}.$

Thus, on finite sets the rank filtration is given by the following formula:

$G_{k,l,m}([t])=\bigvee_{\underset{m_{1}+\cdots+m_{t}\leq
m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}.$
(10)

##  5 $G_{k,l}$ is a cofibrant $\Gamma$-space

In this section we observe that for all $k,l,m$ the restriction of the functor
$G_{k,l,m}$ from finite complexes to finite sets is cofibrant, in a certain
well-known model structure on $\Gamma$ spaces. This implies that the strict
smash product between these functors is equivalent to the derived smash
product. We also show that the value of the functor $G_{k,l,m}$ on pointed
finite CW-complexes is equivalent to both the strict and the derived left Kan
extension of the restriction of $G_{k,l,m}$ to the category of pointed finite
sets. Furthermore the $\Gamma$-space $G_{k,l,m}$ is
$\min(\lfloor\frac{k}{l}\rfloor,m)$-skeletal. This implies that $G_{k,l,m}$ is
determined by its restriction to the category of sets of cardinality at most
$\min(\lfloor\frac{k}{l}\rfloor,m)$.

For any fixed $[t]$, there is a canonical map

$\mathop{\operatorname{colim}}_{U}G_{k,l,m}(U_{+})\to G_{k,l,m}([t])$ (11)

where $U$ ranges over the poset of proper subsets of
$\underline{t}=\\{1,\ldots,t\\}$ and $U_{+}=U\cup\\{0\\}$. Recall once again
that there is an isomorphism

$G_{k,l,m}([t])=\bigvee_{\underset{m_{1}+\cdots+m_{t}\leq
m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}.$
(12)

With this isomorphism in mind, the following lemma is proved by routine
manipulations of colimits

######  Lemma 5.1.

There is an isomorphism

$\mathop{\operatorname{colim}}_{U\subsetneq\underline{t}}G_{k,l,m}(U_{+})\cong\bigvee_{\begin{subarray}{c}\\{(m_{1},\ldots,m_{t})\mid
m_{i}=0\mbox{ \small for some }1\leq i\leq t\\\ \mbox{ and
}m_{1}+\cdots+m_{t}\leq
m\\}\end{subarray}}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}$

The map

$\mathop{\operatorname{colim}}_{U\subsetneq\underline{t}}G_{k,l}(U_{+})\to
G_{k,l}([t])$

corresponds, under this isomorphism, to inclusion of the wedge sum of all
summands indexed by tuples $(m_{1},\ldots,m_{t})$ where $m_{i}=0$ for at least
one $i$.

It follows that the map (11) is an inclusion of a union of path components.
Furthermore, the action of $\Sigma_{t}$ on the quotient space of this
inclusion is free. This means that (11) is a $\Sigma_{t}$-equivariant
cofibration, and this in turn means that as a functor on $\mathtt{Fin}_{*}$,
$G_{k,l,m}$ is cofibrant in the model structure defined for $\Gamma$-spaces in
[BF, Lyd1] (technically, there references work with $\Gamma$-simplicial sets,
but an analogous structure exists for $\Gamma$-spaces). This model structure
is also discussed in [BM] as an example of a generalized Reedy model
structure. We will refer to this model structure simply as the Reedy model
structure.

Let us note that the quotient space of (11) is given by the wedge sum

$\bigvee_{\begin{subarray}{c}\\{(m_{1},\ldots,m_{t})\mid
m_{1},\ldots,m_{t}\geq 1\\\ \mbox{ and }m_{1}+\cdots+m_{t}\leq
m\\}\end{subarray}}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}.$

Notice that the quotient is trivial for $t>m$ and for
$t>\lfloor\frac{k}{l}\rfloor$. In the terminology of [BF, Lyd1], this means
that the $\Gamma$-space $G_{k,l,m}$ is
$\min(m,\lfloor\frac{k}{l}\rfloor)$-skeletal. This implies that $G_{k,l,m}$ is
determined on $\mathtt{Fin}_{*}$, via left Kan extension, by its restriction
to the subcategory of sets of cardinality at most
$\min(m,\lfloor\frac{k}{l}\rfloor)$.

We want to verify that $G_{k,l}$ is equivalent, as a functor on
$\mathtt{CW}^{f}_{*}$, to both the strict and the derived left Kan extension
of the restriction of $G_{k,l,m}$ to $\mathtt{Fin}_{*}$, and even to
$\mathtt{Fin}_{*}^{\leq\min(m,\lfloor\frac{k}{l}\rfloor)}$, where
$\mathtt{Fin}_{*}^{\leq j}\subset\mathtt{Fin}_{*}$ is the full subcategory
consisting of $[0],\ldots,[j]$. Left Kan extension can be described as a
coend. Let us introduce some notation.

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. Suppose $\mathcal{C}$ is a small category, and we have a pair of
functors $F\colon\mathcal{C}^{\operatorname{op}}\to\operatorname{Top}$ and
$G\colon\mathcal{C}\to\operatorname{Top}$. We denote the coend of $F$ and $G$
by $\int_{s}^{\mathcal{C}}F\wedge G$, or $\int_{s}^{x\in\mathcal{C}}F(x)\wedge
G(x)$. The subscript $s$ is there to indicate that this is a strict coend, as
opposed to the homotopy coend. Let us recall the definition and construction
of the latter.

######  Definition 5.2.

For a functor $F$, let $Q_{o}F$ denote an objectwise cofibrant replacement of
$F$ and let $Q_{p}F$ denote a cofibrant replacement in a projective model
structure. If $\mathcal{C}$ is a generalized Reedy category in the sense of
[BM], then let $Q_{r}F$ denote a cofibrant replacement in the Reedy model
structure.

The following lemma is standard

######  Lemma 5.3.

There are natural equivalences

$\int_{s}^{\mathcal{C}}Q_{p}F\wedge
Q_{p}G\simeq\int_{s}^{\mathcal{C}}Q_{o}F\wedge
Q_{p}G\simeq\int_{s}^{\mathcal{C}}Q_{p}F\wedge
Q_{o}G\simeq\int_{s}^{\mathcal{C}}Q_{r}F\wedge Q_{r}G$

###### Proof.

Let us prove, for example, the equivalence $\int_{s}^{\mathcal{C}}Q_{p}F\wedge
Q_{o}G\simeq\int_{s}^{\mathcal{C}}Q_{r}F\wedge Q_{r}G$. The other equivalences
are similar. It is enough to prove that for all spaces $Z$ there is an
equivalence, natural in $Z$

$\operatorname{Map}_{\operatorname{Top}}\left(\int_{s}^{\mathcal{C}}Q_{p}F\wedge
Q_{o}G,Z\right)\simeq\operatorname{Map}_{\operatorname{Top}}\left(\int_{s}^{\mathcal{C}}Q_{r}F\wedge
Q_{r}G,Z\right).$

Note that these are derived mapping spaces as the source is cofibrant and the
target fibrant. For any covariant/contravariant pair of functors $F,G$, there
is a homeomorphism

$\operatorname{Map}_{\operatorname{Top}}\left(\int_{s}^{\mathcal{C}}F\wedge
G,Z\right)\cong\mathrm{Nat}(F,\operatorname{Map}(G,Z)).$

Therefore it is enough to show that there is a natural equivalence

$\mathrm{Nat}\left(Q_{p}F,\operatorname{Map}(Q_{o}G,Z)\right)\simeq\mathrm{Nat}\left(Q_{r}F,\operatorname{Map}(Q_{r}G,Z)\right).$
(13)

The key observation is that if $Q_{r}G(-)$ is Reedy cofibrant then the functor
$\operatorname{Map}(Q_{r}G(-),Z)$ is Reedy fibrant. The two functors
$\operatorname{Map}(Q_{o}G(-),Z)$ and $\operatorname{Map}(Q_{r}G(-),Z)$ are
weakly equivalent functors from $\mathcal{C}^{\operatorname{op}}$ to
$\operatorname{Top}$. They are fibrant in the projective and the Reedy model
structure respectively. On the other hand, the functors $Q_{p}F$ and $Q_{r}F$
are weakly equivalent functors that are cofibrant in the projective and the
Reedy model structure. It follows that the two sides of (13) are the derived
mapping spaces from $F$ to $\operatorname{Map}(Q_{o}(G),Z)$ in the projective
and the Reedy model structure respectively. Since the two model structures
have the same week equivalences, the two derived mapping spaces are
equivalent. ∎

######  Definition 5.4.

The homotopy coend of $F$ and $G$, denoted by $\int_{h}^{\mathcal{C}}F\wedge
G$, is defined to be any one of the equivalent coends in Lemma 5.3.

######  Remark 5.5.

As mentioned in the introduction, we identify the $\infty$-localization of
$\operatorname{Top}$ with the $\infty$-category of pointed spaces
$\operatorname{Top}_{\infty}=\mathcal{S}_{*}.$ Let
$F_{\infty}\colon\mathcal{C}^{\operatorname{op}}\to\mathcal{S}_{*}$ and
$G_{\infty}\colon\mathcal{C}\to\mathcal{S}_{*}$ be the compositions of $F$ and
$G$ with the localization functor
$\operatorname{Top}\to\operatorname{Top}_{\infty}$. Then it is known (see for
example [BHH, Proposition 2.5.6]) that the image of the homotopy coend of $F$
and $G$ under the localization functor
$\operatorname{Top}\to\operatorname{Top}_{\infty}$ is equivalent to the
$\infty$-coend of $F_{\infty}$ and $G_{\infty}$. In the sequel we will
sometimes abuse notation and identify the homotopy coend with the
$\infty$-coend and more generally homotopy colimits with $\infty$-colimits.

The case we are interested in is of the covariant functor
$G_{k,l,m}\colon\mathtt{Fin}_{*}\to\operatorname{Top}$ and the contravariant
functor
$X^{-}\colon\mathtt{Fin}_{*}^{\operatorname{op}}\to\operatorname{Top}$, where
$X$ is a pointed finite CW complex. The strict coend
$\int^{[t]\in\mathtt{Fin}_{*}}_{\mathrm{s}}X^{t}\wedge G_{k,l,m}([t])$ is a
model for the strict (continuous) Kan extension of $G_{k,l,m}$ from
$\mathtt{Fin}_{*}$ to $\operatorname{Top}$. The homotopy coend of same
functors is a model for the homotopy Kan extension. We saw above that
$G_{k,l,m}$ is cofibrant in the Reedy model structure. The functor $X^{-}$ is
also Reedy cofibrant if $X$ is a CW complex. This amounts to saying that for
all $t$, the inclusion of the fat diagonal $\Delta^{t}X$ into $X^{t}$ is a
$\Sigma_{t}$-equivariant cofibration. Since both functors are Reedy cofibrant,
their homotopy coend is in fact equivalent to the strict coend.

######  Lemma 5.6.

Let $X$ be a pointed CW complex and $j$ any integer satisfying $\infty\geq
j\geq\min(m,\lfloor\frac{k}{l}\rfloor)$. All the maps in the following diagram
are equivalences

$\begin{array}[]{ccc}\int^{[t]\in\mathtt{Fin}_{*}^{\leq
j}}_{\mathrm{s}}X^{t}\wedge
G_{k,l,m}([t])&\to&\int^{[t]\in\mathtt{Fin}_{*}}_{\mathrm{s}}X^{t}\wedge
G_{k,l,m}([t])\\\ \downarrow&&\downarrow\\\ \int^{[t]\in\mathtt{Fin}_{*}^{\leq
j}}_{\mathrm{h}}X^{t}\wedge
G_{k,l,m}([t])&\to&\int^{[t]\in\mathtt{Fin}_{*}}_{\mathrm{h}}X^{t}\wedge
G_{k,l,m}([t])\end{array}$

###### Proof.

The vertical maps are equivalences because the functors $X^{-}$ and
$G_{k,l,m}$ are each Reedy cofibrant. The top map is an equivalence because
$G_{k,l,m}$ is $j$-skeletal. ∎

Now comes the main result of this section: $G_{k,l,m}$ is equivalent to both
the strict and the derived left Kan extension of its restriction to the
category of finite sets of size at most $\min(m,\lfloor\frac{k}{l}\rfloor)$.

######  Theorem 5.7.

For all $k,l,m,$ and $\infty\geq j\geq\min(m,\lfloor\frac{k}{l}\rfloor)$, the
functor $G_{k,l,m}$ is equivalent to both the strict and the derived left Kan
extension of $G_{k,l,m}|_{\mathtt{Fin}_{*}^{\leq j}}$ along the inclusion
$\mathtt{Fin}_{*}^{\leq j}\subseteq\mathtt{CW}_{*}$.

###### Proof.

By lemma 5.6, the map from the derived left Kan extension to the strict one is
an equivalence. So it is enough to prove the statement for strict Kan
extension. So throughout this proof, $\int$ stands for the strict coend. There
is a natural assembly map

$\int^{[t]\in\mathtt{Fin}_{*}}X^{t}\wedge G_{k,l,m}([t])\to G_{k,l,m}(X).$
(14)

We will prove that it is a homeomorphism, if $X$ is a finite CW complex. This
is enough for proving the theorem. Thus we need to prove that (14) is
bijective and bi-continuous.

First we prove surjectivity. Let

$f\in G_{k,l,m}(X)\subseteq{\mathtt{SC^{*}}}(\mathrm{C}_{0}(X,M_{l}),M_{k}).$

Factor $f$ using lemma 3.2. Denote $t=|F_{f}|$ and choose a pointed bijection
$[t]\xrightarrow{\cong}F_{f}\cup\\{*\\}$. It follows that $f$ admits a
factorization

$\mathrm{C}_{0}(X,M_{l})\to M_{l}^{t}\stackrel{{\scriptstyle
f^{\prime}}}{{\to}}M_{k}.$

Note that since $V_{f}=V_{f}^{\prime}$ we have $f^{\prime}\in G_{k,l,m}([t])$
This means that $f$ is in the image of the map $X^{t}\wedge G_{k,l,m}([t])\to
G_{k,l,m}(X)$. Thus $f$ is in the image of the assembly map. Since $f$ was an
arbitrary element of $G_{k,l,m}(X)$, we have proved surjectivity of the
assembly map.

Next we show that the assembly map is injective. Suppose that $(\alpha,g)\in
X^{t}\wedge G_{k,l,m}([t])$ and $(\alpha_{1},g_{1})\in X^{t_{1}}\wedge
G_{k,l,m}([t_{1}])$ represent two elements of
$\int^{[t]\in\mathtt{Fin}_{*}}X^{t}\wedge G_{k,l}([t])$ that are mapped to the
same element of $G_{k,l,m}(X)$. We have to show that $(\alpha,g)$ and
$(\alpha_{1},g_{1})$ represent the same element of
$\int^{[t]\in\mathtt{Fin}_{*}}X^{t}\wedge G_{k,l}([t])$. Without loss of
generality we may assume that the functions $\alpha\colon[t]\to X$ and
$\alpha_{1}\colon[t_{1}]\to X$ are injective. Indeed, suppose for example that
$\alpha$ is not injective. Then $\alpha$ can be factored as a surjection
followed by injection, say
$[t]\twoheadrightarrow[t^{\prime}]\stackrel{{\scriptstyle\alpha^{\prime}}}{{\hookrightarrow}}X$.
Let $g^{\prime}$ be the image of $g$ under the map $G_{k,l,m}([t])\to
G_{k,l,m}([t^{\prime}])$. Then $(\alpha^{\prime},g^{\prime})$ represents the
same element as $(\alpha,g)$ in the coend.

Assuming $\alpha$ and $\alpha_{1}$ are injective, let $f$ be the common image
of $(\alpha,g)$ and $(\alpha_{1},g_{1})$ in $G_{k,l,m}(X)$. Then there is a
unique subset $F\subset X\setminus\\{*\\}$ such that $f$ factors as in lemma
3.2. By the minimality of $F$, $F$ is contained in the image of $\alpha$ and
in the image of $\alpha_{1}$. Choose a pointed bijection
$\alpha^{\prime}\colon[t_{0}]\stackrel{{\scriptstyle\cong}}{{\to}}F\cup\\{*\\}$.
This bijection factors through $\alpha$ and $\alpha_{1}$. By slight abuse of
notation, let use $\alpha^{\prime}$ to denote also the element of $X^{t_{0}}$
that is the composed map
$[t_{0}]\stackrel{{\scriptstyle\alpha^{\prime}}}{{\to}}F\cup\\{*\\}\hookrightarrow
X$. Putting it all together we obtain a commutative diagram

It follows that $(\alpha,g)$ and $(\alpha_{1},g_{1})$ are both mapped to the
same element $(\alpha^{\prime},f^{\prime})$ of the coend
$\int^{[t]\in\mathtt{Fin}_{*}}X^{t}\wedge G_{k,l,m}([t])$.

We have shown that the assembly map (14) is a bijection. Its domain is compact
and its codomain is Hausdorff, so it is a homeomorphism. ∎

######  Corollary 5.8.

If $l>k$ then $\mathbb{S}^{k,l}\simeq*$.

###### Proof.

It follows immediately from Proposition 4.5 that if $l>k$ then the restriction
of $G_{k,l}$ to $\mathtt{Fin}_{*}$ is trivial, i.e., $G_{k,l}([t])\cong*$ for
every finite pointed set $[t]$. By Theorem 5.7 $G_{k,l}$ is equivalent to the
homotopy left Kan extension of the restriction of $G_{k,l}$ to
$\mathtt{Fin^{\leq k}_{*}}$, so $G_{k,l}\simeq*$. Since $\mathbb{S}^{k,l}$ is
the stabilization of $G_{k,l}$, it follows that $\mathbb{S}^{k,l}\simeq*$ as
well. ∎

###  5.1 The first stage of the rank filtration

The functor $G_{k,l,1}$ is especially well behaved. The following lemma is an
easy consequence of Theorem 5.7 taken with $j=1$. We also give a direct proof

######  Lemma 5.9.

Let $l,k\geq 1$ and let $X$ be a finite pointed CW-complex. Then the assembly
map

$a_{X}\colon X\wedge G_{k,l,1}(S^{0})\to G_{k,l,1}(X)$

is a homeomorphism.

###### Proof.

Since $X\wedge G_{k,l,1}(S^{0})$ is compact and $G_{k,l,1}(X)$ is Hausdorff it
is enough to show that $a_{X}$ is a bijection. Since $M_{l}$ is simple we have
$G_{k,l}(S^{0})=\mathtt{SC^{*}}^{\mathrm{inj}}(M_{l},M_{k})_{+}$ where
$\mathtt{SC^{*}}^{\mathrm{inj}}(M_{l},M_{k})\subset\mathtt{SC^{*}}(M_{l},M_{k})$
is the space of injective $C^{*}$-algebra maps. similarly we get
$G_{k,l,1}(S^{0})=\mathtt{SC^{*}}^{,1}(M_{l},M_{k})_{+}$ where
$\mathtt{SC^{*}}^{,1}(M_{l},M_{k})\subset\mathtt{SC^{*}}^{\mathrm{inj}}(M_{l},M_{k})$
is the space of maps $f$ with $\mathrm{rank}(f)=1$. We get that $(X\wedge
G_{k,l,1}(S^{0}))\smallsetminus\\{\ast\\}=(X\smallsetminus\\{\ast\\})\times\mathtt{SC^{*}}^{,1}(M_{l},M_{k})$
and the map $a_{X}(x_{0},f)\in G_{k,l,1}(X)\subset G_{k,l}(X)$ is the
composition

$\mathrm{C}_{0}(X,M_{l})\xrightarrow{x_{0}^{*}}M_{l}\xrightarrow{f}M_{k}.$

The injectivity now follows from the uniqueness of the factorisation in lemma
3.2 and the surjectivity from the existence in lemma 3.2 and lemma 3.3. ∎

######  Corollary 5.10.

Let $l,k\geq 1$, the natural map

$\Sigma^{\infty}G_{k,l,1}(S^{0})\to\mathbb{S}^{k,l,1}$

is an equivalence.

Let $\mathbb{P}\operatorname{Inj}$ be the topologically enriched symmetric
monoidal category of finite positive dimensional Hilbert spaces and isometric
embeddings up to scalar. That is, up to isomorphism objects are given by
$\mathbb{C}^{k}$ for $k\in\mathbb{Z}_{\geq 1}$ and

$\mathbb{P}\operatorname{Inj}(\mathbb{C}^{l},\mathbb{C}^{k})=U(k)/(U(1)\times
U(k-l)).$

The symmetric monoidal structure is given by the tensor product. We have a
topologically enriched symmetric monoidal functor

$\textrm{End}\colon\mathbb{P}\operatorname{Inj}\to\mathtt{SC^{*}},$

that sends the Hilbert space $V$ to the $C^{*}$-algebra $\textrm{End}(V)$ of
linear maps $V\to V$ and the embedding $i\colon V\to W$ is sent to the
$*$-homomorphism $\textrm{End}(V)\to\textrm{End}(W)$ sending
$A\in\textrm{End}(V)$ to $i\circ A\circ i^{-1}\circ p\in\textrm{End}(W)$,
where $p\colon W\to\textrm{Im\,}(i)$ is the orthogonal projection. The
monoidal coherence maps of End are given by the natural isomorphisms

$\textrm{End}(V_{1})\otimes\textrm{End}(V_{2})\xrightarrow{\sim}\textrm{End}(V_{1}\otimes
V_{2})$

and

$\mathbb{C}\xrightarrow{\sim}\textrm{End}(\mathbb{C}).$

######  Lemma 5.11.

The map

$\mathbb{P}\operatorname{Inj}(\mathbb{C}^{l},\mathbb{C}^{k})_{+}\to\mathtt{SC^{*}}(M_{l},M_{k})=G_{k,l}(S^{0})$

induced by the functor End is an embedding with image $G_{k,l,1}(S^{0})$.

###### Proof.

We first show that the map is surjective on $G_{k,l,1}(S^{0})$. Let
$i\colon\mathbb{C}^{l}\to\mathbb{C}^{k}$ be an isometric embedding. The map
$f_{i}=\textrm{End}(i)\in\mathtt{SC^{*}}(M_{l},M_{k})$ clearly satisfies
$V_{f}=\mathrm{Im}(i)\subseteq\mathbb{C}^{k}$ and thus
$\mathrm{rank}(f)=\frac{\dim V_{f_{i}}}{l}=\frac{l}{l}=1$. On the other hand
if $f_{i}\in G_{k,l,1}(S^{0})$, then either $f_{i}=0$ and thus is in the image
of the base point or $\dim V_{f_{i}}=1$. In the case $\dim V_{f_{i}}=1$ we get
that the map $f$ factors as an isomorphism followed by an injection.

$M_{l}\xrightarrow{\sim}\textrm{End}(V_{f_{i}})\to M_{k}$

subjectivity now follows from the fact that every automorphism of $M_{l}$ is
inner. We now prove injectivity. First since $V_{f_{i}}=\mathrm{Im}(i)$,
$f_{i}$ determines $\mathrm{Im}(i)$. We are thus reduced to show that if two
embeddings $i,j\colon\mathbb{C}^{l}\to\mathbb{C}^{k}$ have the same image $V$
and the induced maps $M_{l}\to\textrm{End}(V)$ are the same then $i$ and $j$
differ by a scalar. This follows from the fact that the center of $M_{l}$ is
exactly the scalar matrices. We thus get a continuous bijection
$\mathbb{P}\operatorname{Inj}(\mathbb{C}^{l},\mathbb{C}^{k})_{+}\to
G_{k,l,1}(S^{0})$. Since it has compact domain and Hausdorff target, it is a
homeomorphism. ∎

Applying the topological nerve to $\mathbb{P}\operatorname{Inj}$ we get a
symmetric monoidal $\infty$-category $\mathbb{P}\operatorname{Inj}_{\infty}$.
We thus get that End induces a symmetric monoidal functor

$\widetilde{\textrm{End}}\colon\mathbb{P}\operatorname{Inj}_{\infty}^{\mathrm{op}}\to\mathtt{NCW}.$

Composing with the symmetric monoidal functor
$\Sigma^{\infty}\colon\mathtt{NCW}\to\mathtt{NSp}$ we get a symmetric monoidal
functor

$\Sigma^{\infty}\circ\widetilde{\textrm{End}}\colon\mathbb{P}\operatorname{Inj}_{\infty}^{\mathrm{op}}\to\mathtt{NSp}.$

For a closed symmetric monoidal $\infty$-category $\mathcal{M}$ denote by
$\mathtt{Cat}^{\otimes}_{\mathcal{M}}$ the $\infty$-category of symmetric
monoidal $\mathcal{M}$-enriched $\infty$-categories. If $\mathcal{M}$ and
$\mathcal{N}$ are closed symmetric monoidal $\infty$-categories and
$a\colon\mathcal{M}\to\mathcal{N}$ is a symmetric monoidal functor which
admits a right adjoint $b$, then we have an induced adjunction

$a_{!}\colon\mathtt{Cat}^{\otimes}_{\mathcal{M}}\rightleftarrows\mathtt{Cat}^{\otimes}_{\mathcal{N}}\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}b_{!}.$

We shall especially use the case where
$a=\Sigma^{\infty}_{+}\colon\mathcal{S}\to\mathtt{Sp}$. Using the
identification
$\mathtt{Cat}^{\otimes}_{\mathcal{S}}\cong\mathtt{Cat}^{\otimes}$, we obtain
an adjunction

$(\Sigma_{+}^{\infty})_{!}\colon\mathtt{Cat}^{\otimes}\leftrightarrows\mathtt{Cat}^{\otimes}_{\mathtt{Sp}}\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}(\Omega^{\infty})_{!}$

Given a symmetric monoidal $\infty$-category
$\mathcal{C}\in\mathtt{Cat}^{\otimes}$ we denote

$\mathcal{C}^{\mathtt{Sp}}:=(\Sigma^{\infty}_{+})_{!}(\mathcal{C})\in\mathtt{Cat}^{\otimes}_{\mathtt{Sp}}.$

As a stable $\infty$-category $\mathtt{NSp}$ is naturally
$\mathtt{Sp}$-enriched. Thus we have natural isomorphisms

$\operatorname{Map}_{\mathtt{Cat}^{\otimes}_{\mathtt{Sp}}}((\mathbb{P}\operatorname{Inj}^{\mathtt{Sp}}_{\infty})^{\mathrm{op}},\mathtt{NSp})\simeq\operatorname{Map}_{\mathtt{Cat}^{\otimes}_{\mathtt{Sp}}}((\Sigma_{+}^{\infty})_{!}(\mathbb{P}\operatorname{Inj}_{\infty}^{\mathrm{op}}),\mathtt{NSp})\simeq$
$\operatorname{Map}_{\mathtt{Cat}^{\otimes}}(\mathbb{P}\operatorname{Inj}_{\infty}^{\mathrm{op}},(\Omega^{\infty})_{!}\mathtt{NSp})\simeq\operatorname{Map}_{\mathtt{Cat}^{\otimes}}(\mathbb{P}\operatorname{Inj}_{\infty}^{\mathrm{op}},\mathtt{NSp}).$

We denote the mate of
$\Sigma^{\infty}\circ\widetilde{\textrm{End}}\in\operatorname{Map}_{\mathtt{Cat}^{\otimes}}(\mathbb{P}\operatorname{Inj}_{\infty}^{\mathrm{op}},\mathtt{NSp})$
under this adjunction by

$\widetilde{\mathrm{E}}\colon(\mathbb{P}\operatorname{Inj}_{\infty}^{\mathtt{Sp}})^{\mathrm{op}}\to\mathtt{NSp}.$

This is a symmetric monoidal $\mathtt{Sp}$-enriched functor.

######  Lemma 5.12.

We get the following commutative diagram for every $k,l\geq 1$

---
$\textstyle{\operatorname{Map}_{(\mathbb{P}\operatorname{Inj}_{\infty}^{\mathtt{Sp}})^{\mathrm{op}}}(\mathbb{C}^{k},\mathbb{C}^{l})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\tilde{\mathrm{E}}}$$\textstyle{\operatorname{Map}_{\mathtt{NSp}}(\Sigma^{\infty}M^{k},\Sigma^{\infty}M^{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\Sigma^{\infty}_{+}\operatorname{Map}_{\mathbb{P}\operatorname{Inj}_{\infty}}(\mathbb{C}^{l},\mathbb{C}^{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$End$\textstyle{\Sigma^{\infty}\operatorname{Map}_{\mathtt{NCW}}(M^{k},M^{l})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\Sigma^{\infty}G_{k,l,1}(S^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\Sigma^{\infty}G_{k,l}(S^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\partial_{1}(G_{k,l})}$$\textstyle{\partial_{1}(G_{k,l,1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

###### Proof.

The commutation of the lower triangle and the right trapezoid is clear. The
left square is a consequence of lemma 5.11. To see the commutation of the top
trapezoid consider the digram in $\mathtt{Cat}_{\mathtt{Sp}}^{\otimes}$.

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
22.15837pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\mathbb{P}\operatorname{Inj}_{\infty}^{\mathtt{Sp}})^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
64.31952pt\raise-33.59503pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
46.15837pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
46.15837pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{(\Sigma^{\infty})_{!}(\mathtt{NCW})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
76.62509pt\raise-30.17888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
64.31952pt\raise-40.01219pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathtt{NSp}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

∎

##  6 Connection with topological $K$-theory

In this section we show that for every fixed $l$, the spectrum
$\mathbb{S}^{\infty,l}:=\mathop{\operatorname{colim}}_{k\to\infty}\mathbb{S}^{k,l}$,
is equivalent to the connective $K$-theory spectrum $ku$. We will use this
observation to show how the spectra $\mathbb{S}^{\infty,l}$ together represent
$K$-theory of noncommutative CW-complexes.

Recall that for a pointed finite set $[t]$,

$G_{k,l}([t])=\mathtt{SC^{*}}(M_{l}^{t},M_{k})\cong\bigvee_{(m_{1},\ldots,m_{t})}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}.$

The inclusions of algebras $M_{k}\to M_{k+1}$ that send a matrix $a$ to
$a\oplus 0$ induce a sequence of natural transformations, for each fixed $l$,

$\cdots\to G_{k,l}\to G_{k+1,l}\to\cdots.$

######  Definition 6.1.

For each fixed $l$, define the functor
$G_{\infty,l}:=\mathop{\operatorname{colim}}_{k\to\infty}G_{k,l}$. Here by
$\mathop{\operatorname{colim}}$ we mean strict rather than homotopy colimit.

######  Lemma 6.2.

$G_{\infty,l}$ is equivalent to the homotopy colimit
$\mathop{\operatorname{hocolim}}_{k\to\infty}G_{k,l}$. Moreover there is a
homeomorphism (where as usual the wedge sum is indexed on the set of non-zero
$t$-tuples of non-negative integers)

$G_{\infty,l}([t])=\bigvee_{(m_{1},\ldots,m_{t})}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{\infty})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}$

and a homotopy equivalence

$G_{\infty,l}([t])\simeq\bigvee_{(m_{1},\ldots,m_{t})}BU(m_{1})\times\cdots\times
BU(m_{t})_{+}.$

###### Proof.

The map

$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}\to\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k+1})/_{\prod_{i=1}^{t}U(m_{i})}$

is an inclusion of a submanifold, and in particular a cofibration. It follows
that the colimit is equivalent to the homotopy colimit. As $k$ goes to
$\infty$, the colimit of $G_{k,l}$ is, by definition, homeomorphic to

$\bigvee_{(m_{1},\ldots,m_{t})}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{\infty})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}.$

Note that
$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{\infty})$
is a contractible space with a free action of $\prod_{i=1}^{t}U(m_{i})$, and
moreover the quotient map

$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{\infty})\to\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{\infty})/_{\prod_{i=1}^{t}U(m_{i})}$

is a fiber bundle. It follows that the quotient space is equivalent to the
classifying space $BU(m_{1})\times\cdots\times BU(m_{t})$. ∎

Recall $\mathbb{S}^{k,l}$ is the stabilization of $G_{k,l}$. We define
$\mathbb{S}^{\infty,l}$ accordingly.

######  Definition 6.3.

The spectrum $\mathbb{S}^{\infty,l}$ is defined as follows

$\mathbb{S}^{\infty,l}:=\mathop{\operatorname{colim}}_{k\to\infty}\mathbb{S}^{k,l}.$

Since stabilization commutes with homotopy colimits of pointed functors,
$\mathbb{S}^{\infty,l}$ is the stabilization of $G_{\infty,l}$. It is worth
noting that the homotopy type of $\mathbb{S}^{\infty,l}$ is independent of
$l$. Indeed, the following statement is an immediate consequence of Lemma 6.2.

######  Corollary 6.4.

A choice of an inclusion $\mathbb{C}\hookrightarrow\mathbb{C}^{l}$ induces an
equivalence of $\Gamma$-spaces $G_{\infty,l}\to G_{\infty,1}$.

In fact, the spectrum $\mathbb{S}^{\infty,l}$ is homotopy equivalent to the
connective $K$-theory spectrum $ku$ for each $l$. We record this observation
in a lemma.

######  Lemma 6.5.

For each $l$ the spectrum $\mathbb{S}^{\infty,l}$ is homotopy equivalent to
$ku$.

###### Proof.

The $\Gamma$-space $G_{\infty,1}$ is equivalent to the $\Gamma$-space
constructed by Segal in [Se1, Section 2]. It is, in Segal's terminology, a
special $\Gamma$-space. This means that for any two pointed finite sets
$[s],[t]$, the natural map $G_{\infty,1}([s]\vee[t])\to
G_{\infty,1}([s])\times G_{\infty,1}([t])$ is an equivalence. It follows from
Lemma 6.2 that

$G_{\infty,1}([1])\cong\bigvee_{m=1}^{\infty}BU(m)_{+}.$

So $\mathbb{S}^{\infty,1}$ is the spectrum associated with the group-
completion of $\bigvee_{m=1}^{\infty}BU(m)_{+}$, which is well-known to be
$ku$. By corollary 6.4, it follows that $\mathbb{S}^{\infty,l}\simeq ku$ for
all $l$. ∎

It follows that for each $k,l$ there is a natural map
$\mathbb{S}^{k,l}\to\mathbb{S}^{\infty,l}\simeq ku$. We will show later, after
we analyze the subquotients of the rank filtration, that this map induces an
isomorphism on $\pi_{0}$ (Lemma 9.4). In particular, it is not trivial.

####  $K$-theory of noncommutative CW-complexes.

We will now discuss how the spectra $\mathbb{S}^{\infty,l}$ can be used to
represent the $K$ theory functor on the category $\mathtt{NSp}$. In this
subsection we consider $\mathcal{M}$ as an $\infty$-category and use the
$\infty$-categorical picture.

Consider the enriched Yoneda embedding of $\mathcal{M}^{\operatorname{op}}$
(which is an $\mathtt{Sp}^{rev}$-functor):

$Y\colon\mathcal{M}^{\operatorname{op}}\to\operatorname{Fun}_{\mathtt{Sp}}(\mathcal{M},\mathtt{Sp}).$

The sequence of algebras $\cdots\to M_{k}\to M_{k+1}\to\cdots$ gives rise to a
direct sequence in $\mathcal{M}^{\operatorname{op}}$:

$\cdots\to\Sigma^{\infty}_{\mathtt{NC}}M_{k}\to\Sigma^{\infty}_{\mathtt{NC}}M_{k+1}\to\cdots$

Applying the Yoneda embedding $Y$ to this sequence, we obtain a sequence of
spectral functors
$Y(\Sigma^{\infty}_{\mathtt{NC}}M_{k})\in\operatorname{Fun}_{\mathtt{Sp}}(\mathcal{M},\mathtt{Sp})$,
that are characterized by the following property

$Y(\Sigma^{\infty}_{\mathtt{NC}}M_{k})(\Sigma^{\infty}_{\mathtt{NC}}M_{l})\simeq\operatorname{Hom}_{\mathcal{M}}(\Sigma^{\infty}_{\mathtt{NC}}M_{k},\Sigma^{\infty}_{\mathtt{NC}}M_{l})=\mathbb{S}^{k,l}.$

The homotopy colimit of this sequence is a spectral functor that will
represent connective $K$-theory. Let us give this functor a name:

######  Definition 6.6.

We define the spectrally enriched functor
$\mathtt{ku}\in\operatorname{Fun}_{\mathtt{Sp}}(\mathcal{M},\mathtt{Sp})$ to
be the following colimit:

$\mathtt{ku}=\mathop{\operatorname{colim}}_{k\to\infty}Y(\Sigma^{\infty}_{\mathtt{NC}}M_{k}).$

Note that for every $l$ we have the following equivalences, the last of which
follows from Lemma 6.5. The last equivalence justifies the notation
$\mathtt{ku}$ for this functor.

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

We know by the main result of [ABS1] that we can identify $\mathtt{NSp}$ with
$P_{\mathtt{Sp}}(\mathcal{M})$, the category of contravariant spectrally
enriched functors from $\mathcal{M}$ to $\mathtt{Sp}$. This implies that
covariant enriched functors from $\mathcal{M}$ to $\mathtt{Sp}$, such as
$\mathtt{ku}$, can be used to define homology theories on $\mathtt{NSp}$, and
therefore also on $\mathtt{NCW}$.

Indeed suppose $h\colon\mathcal{M}\to\mathtt{Sp}$ is a covariant
$\mathtt{Sp}$-functor. Then $h$ determines an $\mathtt{Sp}$-tensored functor

$h\wedge_{\mathcal{M}}(-):\mathtt{NSp}\to\mathtt{Sp}$

by the universal property of the enriched Yoneda embedding, such that there is
a natural equivalence

$h(k)\wedge_{k\in\mathcal{M}}\mathbb{S}^{k,l}\simeq
h(\Sigma^{\infty}_{\mathtt{NC}}M_{l}).$

The universal property also tells us that this induces an equivalence between
the $\infty$-category of $\mathtt{Sp}$-tensored functors
$\mathtt{NSp}\to\mathtt{Sp}$ and the $\infty$-category of
$\mathtt{Sp}$-functors $\mathcal{M}\to\mathtt{Sp}$. By taking homotopy groups
$\pi_{*}(h\wedge_{\mathcal{M}}(-))$ one obtains a generalized homology theory
on noncommutative CW-spectra (see [BJM, Definition 4.1]).

In particular, let us take $h=\mathtt{ku}$. We interpret the property that
$\mathtt{ku}(\Sigma^{\infty}_{\mathtt{NC}}M_{l})\simeq ku$ for all $l$ as
saying that $\mathtt{ku}$ represents a version of connective $K$-theory. We
have a further enhancement of this fact.

######  Lemma 6.7.

The functor $\mathtt{ku}$ takes values in the category of $ku$-module spectra.

###### Proof.

The monoidal structure on $\mathcal{M}$ gives rise to maps
$\mathbb{S}^{k^{\prime},l^{\prime}}\wedge\mathbb{S}^{k,l}\to\mathbb{S}^{kk^{\prime},ll^{\prime}}$.
Fixing $l^{\prime}=1$ we obtain maps
$\mathbb{S}^{k^{\prime},1}\wedge\mathbb{S}^{k,l}\to\mathbb{S}^{kk^{\prime},l}$,
natural in $l$. Taking limits as $k,k^{\prime}\to\infty$ we obtain maps, still
natural in $\Sigma^{\infty}_{\mathtt{NC}}M_{l}$

$\mathtt{ku}(\Sigma^{\infty}_{\mathtt{NC}}M_{1})\wedge\mathtt{ku}(\Sigma^{\infty}_{\mathtt{NC}}M_{l})\to\mathtt{ku}(\Sigma^{\infty}_{\mathtt{NC}}M_{l}).$

Identifying $\mathtt{ku}(\Sigma^{\infty}_{\mathtt{NC}}M_{1})$ with $ku$, this
map endows $\mathtt{ku}(\Sigma^{\infty}_{\mathtt{NC}}M_{l})$ with the
structure of a $ku$-module, functorial in
$\Sigma^{\infty}_{\mathtt{NC}}M_{l}$. ∎

The $\infty$-category of $ku$-module spectra $ku\mathrm{-mod}$ is stable and
thus $\mathtt{Sp}$-tensored. In light of Lemma 6.7, we may now view
$\mathtt{ku}$ as an $\mathtt{Sp}$-functor $\mathtt{ku}\colon\mathcal{M}\to
ku\mathrm{-mod}$. Thus, by the universal property of the enriched Yoneda
embedding, $\mathtt{ku}$ determines an $\mathtt{Sp}$-tensored functor

$\mathtt{ku}\wedge_{\mathcal{M}}(-):\mathtt{NSp}\to ku\mathrm{-mod}.$

This allows for the following definition:

######  Definition 6.8.

For $X,Y\in\mathtt{NSp}$ we define

$kk(X,Y):=[\mathtt{ku}\wedge_{\mathcal{M}}X,\mathtt{ku}\wedge_{\mathcal{M}}Y]_{ku\mathrm{-mod}},$
$kk_{*}(X,Y)=\pi_{*}\operatorname{Map}_{ku\mathrm{-mod}}(\mathtt{ku}\wedge_{\mathcal{M}}X,\mathtt{ku}\wedge_{\mathcal{M}}Y)=kk(\Sigma^{*}X,Y).$

It follows from Lemma 6.7 that for all $k,l\geq 1$

$kk_{*}(\Sigma^{\infty}_{\mathtt{NC}}M_{k},\Sigma^{\infty}_{\mathtt{NC}}M_{l})\simeq\pi_{*}\operatorname{Map}_{ku\mathrm{-mod}}(ku,ku)\simeq\pi_{*}ku.$

Recall from Remark 2.4 that the category of commutative spectra is identified
with the category of spectral presheaves on the full $\mathtt{Sp}$-enriched
subcategory of $\mathcal{M}$ containing the object
$\Sigma^{\infty}_{\mathtt{NC}}M_{1}$. A commutative spectrum is made
noncommutative by means of an $\mathtt{Sp}$-enriched left Kan extension.
Suppose that $X$ is a commutative spectrum, which we also may consider as a
noncommutative spectrum by Kan extension. Standard adjunction implies that
there is a natural equivalence

$\mathtt{ku}\wedge_{\mathcal{M}}X\simeq ku\wedge X,$

where the symbol $\wedge$ on the right hand side denotes the usual smash
product of spectra. It follows that if $X$ and $Y$ are commutative spectra
then there is an equivalence

$kk_{*}(X,Y)\simeq\pi_{*}\operatorname{Map}_{\mathtt{Sp}}(X,ku\wedge Y).$

In the case when $X$ and $Y$ are suspension spectra of pointed finite CW-
complexes, this is essentially [DM, Proposition 3.1]. Thus $kk$ defined above
is a natural extension for the connective bivariant $K$-theory of Dadarlat and
McClure. This also suggests that some results from [loc. cit.] may be
generalized from finite CW-complexes to finite noncommutative CW-complexes.
This will be addressed in future papers.

We conclude by remarking that since the functor $\mathtt{ku}$ takes values in
modules over $ku$, one can invert the Bott element and get a functor
representing non-connective $K$-theory.

####  6.0.1 Connection to the rank filtration of $ku$

It follows immediately from equation (10), that the filtration of $G_{k,l}$ by
$G_{k,l,m}$ interacts well with the stabilization map $G_{k,l}\to G_{k+1,l}$
that we considered in the previous section. To be more specific, there is a
commutative diagram

$\begin{array}[]{ccccccccc}\cdots&\to&G_{k,l,m}&\to&G_{k,l,m+1}&\to&\cdots&\to&G_{k,l}\\\
&&\downarrow&&\downarrow&&\cdots&&\downarrow\\\
\cdots&\to&G_{k+1,l,m}&\to&G_{k+1,l,m+1}&\to&\cdots&\to&G_{k+1,l}\\\
&&\downarrow&&\downarrow&&\cdots&&\downarrow\\\
&&\vdots&&\vdots&&\cdots&&\vdots\\\
\cdots&\to&G_{\infty,l,m}&\to&G_{\infty,l,m+1}&\to&\cdots&\to&G_{\infty,l}.\end{array}$

On finite sets, $G_{\infty,l,m}$ is given, at least up to homotopy, by the
following formula

$G_{\infty,l,m}([t])\simeq\bigvee_{\underset{m_{1}+\cdots+m_{t}\leq
m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}BU(m_{1})\times\cdots\times BU(m_{t})_{+}$

We conclude that the rank filtration of $G_{k,l}$ induces a compatible rank
filtration of $G_{\infty,l}$. Upon passing to stabilization, the rank
filtration of $G_{\infty,l}$ induces a filtration of $ku$ by a sequence of
spectra:

$\cdots\to\mathbb{S}^{\infty,l,m}\to\mathbb{S}^{\infty,l,m+1}\to\cdots\to\mathbb{S}^{\infty,l}\simeq
ku.$

The latter filtration is the familiar rank filtration of the $K$-theory
spectrum $ku$ studied, for example, in [AL2] (we remark that $ku$ is denoted
$bu$ in [loc. cit]). Thus the rank filtration of $\mathbb{S}^{k,l}$ is a lift
of the classical rank filtration of $ku$.

##  7 Subquotients of the rank filtration

In this section we investigate the subquotients of the rank filtration. We
show that on the level of the functors $G_{k,l}$ the subquotients of the rank
filtration have a presentation as a homotopy coend over the category
$\mathtt{Epi}$ of finite sets and surjections, as opposed to the category
$\mathtt{Fin}_{*}$ of pointed sets and all pointed functions.

######  Definition 7.1.

Let $G_{k,l}^{m}\colon\mathtt{CW}_{*}\to\operatorname{Top}$ be the quotient
functor

$G_{k,l}^{m}:=G_{k,l,m}/G_{k,l,m-1}.$

Similarly, let $\mathbb{S}^{k,l}_{m}$ be the homotopy cofiber of the map
$\mathbb{S}^{k,l,m-1}\to\mathbb{S}^{k,l,m}$.

It is not hard to check that for any $X\in\mathtt{CW}_{*}$ the map
$G_{k,l,m-1}(X)\to G_{k,l,m}(X)$ is a cofibration in $\operatorname{Top}$.
Thus, $G_{k,l}^{m}$ the levelwise homotopy cofiber of the map $G_{k,l,m-1}\to
G_{k,l,m}$ in the category of pointed continuous functors
$\operatorname{Fun}_{*}(\mathtt{CW}_{*},\operatorname{Top})$. It follows from
[ADL, Lemma 2.16] that $G_{k,l}^{m}$ is also the homotopy cofiber of the map
$G^{k,l,m-1}\to G^{k,l,m}$ in the projective model structure on
$\operatorname{Fun}_{*}(\mathtt{CW}_{*},\operatorname{Top})$, and thus also in
the stable model structure $\mathtt{Sp^{M}}$.

Since strict (homotopy) left Kan extension commutes with strict (homotopy)
cofiber, we get from Theorem 5.7 that the functor $G_{k,l}^{m}$ is equivalent
to both the strict and the derived left Kan extension of its restriction to
${\mathtt{Fin}_{*}^{\leq j}}$ for any $\infty\geq
j\geq\min(m,\lfloor\frac{k}{l}\rfloor)$. Thus, for any finite pointed CW-
complex $X$, we have the folowing formula:

$G_{k,l}^{m}(X)\cong\int^{[t]\in\mathtt{Fin}_{*}^{\leq
j}}_{\mathrm{s}}X^{t}\wedge
G_{k,l}^{m}([t])\simeq\int^{[t]\in\mathtt{Fin}_{*}^{\leq
j}}_{\mathrm{h}}X^{t}\wedge G_{k,l}^{m}([t]).$ (15)

Since stabilization commutes with homotopy cofibers, $\mathbb{S}^{k,l}_{m}$ is
equivalent to the stabilization of $G_{k,l}^{m}$.

It follows immediately from equation (10) that on objects $G_{k,l}^{m}$ is
given as follows

$G_{k,l}^{m}([t])=\bigvee_{\underset{m_{1}+\cdots+m_{t}=m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}{\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}$
(16)

Which also can be written as

$G_{k,l}^{m}([t])=\bigvee_{\underset{m_{1}+\cdots+m_{t}=m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}{\operatorname{Inj}(\mathbb{C}^{ml},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}$

To understand the effect of $G_{k,l}^{m}$ on morphisms, let
$\alpha\colon[t]\to[s]$ be a pointed function. If
$|\alpha_{*}(m_{1},\ldots,m_{t})|<m$, then $G_{k,l}^{m}(\alpha)$ takes the
summand of $G_{k,l}^{m}([t])$ corresponding to $(m_{1},\ldots,m_{t})$ to the
basepoint. If $|\alpha_{*}(m_{1},\ldots,m_{t})|=m$, then $G_{k,l}^{m}(\alpha)$
takes the corresponding summand of $G_{k,l}^{m}([t])$ to the summand of
$G_{k,l}^{m}([s])$ indexed by $\alpha_{*}(m_{1},\ldots,m_{t})$ by the map (5).

One attractive property of $G_{k,l}^{m}$ is that it has a more compact coend
formula than the one given in equation 15 (see Proposition 7.5). In this
formula the category $\mathtt{Fin}_{*}$ is replaced with the smaller category
$\mathtt{Epi}$ of non-empty unpointed sets and surjections. For $k\leq\infty$
let $\mathtt{Epi}^{\leq k}$ be the category of non-empty finite sets of
cardinality at most $k$ and epimorphisms between them. In the case $k=\infty$,
this is the category of all non-empty finite sets and surjections, and we
denote it simply $\mathtt{Epi}$.

If $\alpha\colon\underline{t}\twoheadrightarrow\underline{s}$ is a morphism in
$\mathtt{Epi}$, and $(m_{1},\ldots,m_{t})$ is a multiset, then we may define
$\alpha_{*}(m_{1},\ldots,m_{t})=(n_{1},\ldots,n_{s})$ in the usual way, by
saying that $n_{j}=\sum_{i\in\alpha^{-1}(j)}m_{i}$. Note that in this case
there is an equality $m_{1}+\cdots+m_{t}=n_{1}+\cdots+n_{s}$. Note also that
if $m_{i}>0$ for all $i$ then $n_{j}>0$ for all $j$.

######  Definition 7.2.

Let $\operatorname{Top_{u}}$ be the category of unpointed topological spaces.
Let $\mathfrak{G}_{k,l}^{m}\colon\mathtt{Epi}\to\operatorname{Top_{u}}$ be the
following functor. On objects, it is defined by the following formula

$\mathfrak{G}_{k,l}^{m}(\underline{t})=\coprod_{\underset{m_{i}>0,m_{1}+\cdots+m_{t}=m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}{\operatorname{Inj}(\mathbb{C}^{ml},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}$

On morphisms, $\mathfrak{G}_{k,l}^{m}$ is defined similarly to $G_{k,l}$ and
$G_{k,l}^{m}$. Given a surjection
$\alpha\colon\underline{t}\twoheadrightarrow\underline{s}$, the summand
indexed by $(m_{1},\ldots,m_{t})$ is mapped to the summand indexed by
$\alpha_{*}(m_{1},\ldots,m_{t})$ by the same map as in (5).

We will make much use of the functor
${\mathfrak{G}_{k,l}^{m}}_{+}\colon\mathtt{Epi}\to\operatorname{Top}$, which
is obtained by adding a disjoint basepoint to $\mathfrak{G}_{k,l}^{m}$. We
introduced an unpointed version of the functor because it will be convenient
to have it at some point.

######  Remark 7.3.

For a multi-set $(m_{1},\ldots,m_{t})$ define its support to be the set
$A=\\{i\mid m_{i}>0\\}\subseteq\underline{t}$. Notice that for any subset
$A\subseteq\underline{t}$ there is a natural way to identify
${\mathfrak{G}_{k,l}^{m}(A)}_{+}$ with a wedge summand of $G_{k,l}^{m}([t])$.
Namely, ${\mathfrak{G}_{k,l}^{m}(A)}_{+}$ is identified, on the right hand
side of (16) as the wedge sum corresponding to indices $(m_{1},\ldots,m_{t})$
whose support is exactly $A$. With this identification, there is a
homeomorphism

$G_{k,l}^{m}([t])\cong\bigvee_{A\subseteq\underline{t}}{\mathfrak{G}_{k,l}^{m}(A)}_{+}$

Moreover, the functoriality in $[t]$ is defined on the right hand side as
follows. Suppose $\alpha\colon[t]\to[s]$ is a pointed function. Suppose
$A\subseteq\underline{t}$. If $\alpha$ sends some element of $A$ to the
basepoint of $[s]$, then the corresponding summand $\mathfrak{G}_{k,l}^{m}(A)$
is sent to the basepoint. Otherwise, this summand is sent to
$\mathfrak{G}_{k,l}^{m}(\alpha(A))$ using the surjection
$A\twoheadrightarrow\alpha(A)$ defined by $\alpha$.

Consider the functor

$\begin{array}[]{ccc}\mathtt{Fin}_{*}\times\mathtt{Epi}^{\operatorname{op}}&\to&\operatorname{Top}\\\
([t],u)&\to&[t]^{\wedge u}\end{array}$

One can think of this functor as a $\mathtt{Fin}_{*}-\mathtt{Epi}$-bimodule.
It is often used to establish connections between categories of
$\mathtt{Fin}_{*}$-modules and $\mathtt{Epi}$-modules. We have the following
proposition:

######  Proposition 7.4.

There is a homeomorphism and an equivalence, natural in $[t]$ ranging over
$\mathtt{Fin}_{*}$:

$G_{k,l}^{m}([t])\cong\int^{u\in\mathtt{Epi}}_{\mathrm{s}}[t]^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}\simeq\int^{u\in\mathtt{Epi}}_{\mathrm{h}}[t]^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}$

###### Proof.

First of all, let us construct a natural map. Fix a surjective function
$u_{1}\twoheadrightarrow u_{2}$. One has a map

$[t]^{\wedge u_{2}}\wedge{\mathfrak{G}_{k,l}^{m}(u_{1})}_{+}\to
G_{k,l}^{m}([t]).$

The map is defined as follows. First, the surjection $u_{1}\twoheadrightarrow
u_{2}$ induces a map
$\mathfrak{G}_{k,l}^{m}(u_{1})\twoheadrightarrow\mathfrak{G}_{k,l}^{m}(u_{2})$,
and therefore $[t]^{\wedge
u_{2}}\wedge{\mathfrak{G}_{k,l}^{m}(u_{1})}_{+}\to[t]^{\wedge
u_{2}}\wedge{\mathfrak{G}_{k,l}^{m}(u_{2})}_{+}$. Second, there is a bijection
of sets $[t]^{\wedge u_{2}}\cong{\underline{t}^{u_{2}}}_{+}$, so a non
basepoint of this set is a map $f\colon u_{2}\to\underline{t}$. This defines a
map $\mathfrak{G}_{k,l}^{m}(u_{2})\to\mathfrak{G}_{k,l}^{m}(f(u_{2}))$.
Finally, there is an inclusion of ${\mathfrak{G}_{k,l}^{m}(f(u_{2}))}_{+}$ as
a wedge summand of $G_{k,l}^{m}([t])$, as in Remark 7.3.

The map is natural in the variable $[t]$ (exercise for the reader).

The following diagram commutes because $\mathfrak{G}_{k,l}^{m}$ is functorial
with respect to surjections.

$\begin{array}[]{ccc}[t]^{\wedge
u_{2}}\wedge{\mathfrak{G}_{k,l}^{m}(u_{1})}_{+}&\to&[t]^{\wedge
u_{2}}\wedge{\mathfrak{G}_{k,l}^{m}(u_{2})}_{+}\\\ \downarrow&&\downarrow\\\
[t]^{\wedge
u_{1}}\wedge{\mathfrak{G}_{k,l}^{m}(u_{1})}_{+}&\to&G_{k,l}^{m}([t])\end{array}$

It follows that there is a natural transformation of functors of $[t]$ (recall
that $\int_{\mathrm{s}}$ denotes strict coend)

$\int^{u\in\mathtt{Epi}}_{\mathrm{s}}[t]^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}\to G_{k,l}^{m}([t]).$ (17)

We claim that this map is in fact an isomorphism. To see this notice that for
each fixed $t$ there is an isomorphism of functors of $u$

$[t]^{\wedge
u}={\underline{t}^{u}}_{+}\cong\bigvee_{A\subseteq\underline{t}}\mathtt{Epi}(u,A)_{+}$
(18)

For each $A$, the functor $u\mapsto\mathtt{Epi}(u,A)$ is a representable
functor $\mathtt{Epi}^{\operatorname{op}}\to\operatorname{Top}$. By coYoneda
lemma, there is an isomorphism

$\int^{u\in\mathtt{Epi}}_{\mathrm{s}}[t]^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}\cong\bigvee_{A\subseteq\underline{t}}{\mathfrak{G}_{k,l}^{m}(A)}_{+}$

and the right hand is identified with $G_{k,l}^{m}([t])$, again as in Remark
7.3.

It follows that the map (17) is in fact an isomorphism. On the other hand,
Equation (18) shows that the functor $u\mapsto[t]^{\wedge u}$ is cofibrant in
the projective model structure on the functor category
$[\mathtt{Epi}^{\operatorname{op}},\operatorname{Top}]$. It follows that the
natural map from the homotopy coend to the strict coend is an equivalence:

$\int^{u\in\mathtt{Epi}}_{\mathrm{h}}[t]^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}\stackrel{{\scriptstyle\simeq}}{{\to}}\int^{u\in\mathtt{Epi}}_{\mathrm{s}}[t]^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}.$

∎

As a consequence, we have a simplified coend formula for $G_{k,l}^{m}(X)$
where $X$ is a CW complex.

######  Proposition 7.5.

Let $X$ be a finite pointed CW-complex. There is a homeomorphism and an
equivalence, natural in $X$

$G_{k,l}^{m}(X)\cong\int^{u\in\mathtt{Epi}}_{\mathrm{s}}X^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}\simeq\int^{u\in\mathtt{Epi}}_{\mathrm{h}}X^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}.$

The statement remains true if $\mathtt{Epi}$ is replaced with
$\mathtt{Epi}^{\leq k}$.

###### Proof.

We prove the part for the homotopy coend, and the proof of the strict part is
identical. By the standard coend formula (see equation 15), there is an
equivalence

$G_{k,l}^{m}(X)\simeq\int^{[t]\in\mathtt{Fin}_{*}}_{\mathrm{h}}X^{[t]}\wedge
G_{k,l}^{m}([t]).$

By Proposition 7.4, there is an equivalence

$G_{k,l}^{m}([t])\simeq\int^{u\in\mathtt{Epi}}_{\mathrm{h}}[t]^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}.$

It follows that there is an equivalence

$G_{k,l}^{m}(X)\simeq\int^{[t]\in\mathtt{Fin}_{*}}_{\mathrm{h}}X^{[t]}\wedge\left(\int^{u\in\mathtt{Epi}}_{\mathrm{h}}[t]^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}\right).$

By associativity of coend (``Fubini theorem''), the right hand side is
equivalent to

$\int^{u\in\mathtt{Epi}}_{\mathrm{h}}\left(\int^{[t]\in\mathtt{Fin}_{*}}_{\mathrm{h}}X^{[t]}\wedge[t]^{\wedge
u}\right)\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}.$

It remains to show that there is a natural equivalence

$\int^{[t]\in\mathtt{Fin}_{*}}_{\mathrm{h}}X^{[t]}\wedge[t]^{\wedge u}\simeq
X^{\wedge u}.$

This is elementary. The argument goes as follows. The set $[t]^{\wedge u}$ is
equivalent to the total homotopy cofiber of the cubical diagram
$A\mapsto\mathtt{Fin}_{*}(A_{+},[t])$, where $A$ ranges over subsets of $u$,
and the maps are induced by collapsing the complement of a subset to the
basepoint. By the coYoneda lemma,
$\int^{[t]\in\mathtt{Fin}_{*}}_{\mathrm{h}}X^{[t]}\wedge\mathtt{Fin}_{*}(A_{+},[t])\simeq
X^{A}$. It follows that
$\int^{[t]\in\mathtt{Fin}_{*}}_{\mathrm{h}}X^{[t]}\wedge[t]^{\wedge u}$ is
equivalent to the total homotopy cofiber of the cubical diagram $A\mapsto
X^{A}$, where $A$ ranges over subsets of $u$. The total cofiber is equivalent
to $X^{\wedge u}$. ∎

Our next step is to use Proposition 7.5 to describe the subquotient spectra
$\mathbb{S}^{k,l}_{m}$. Let
$\mathtt{I}\colon\mathtt{Epi}^{\operatorname{op}}\to\operatorname{Top}$ be the
(unique) functor defined by

$\mathtt{I}(\underline{t})=\left\\{\begin{array}[]{cc}S^{0}&t=1\\\ &t\neq
1\end{array}\right.$

######  Lemma 7.6.

There are equivalences, where $\mathtt{Epi}$ can be replaced with
$\mathtt{Epi}^{\leq k}$

$\mathbb{S}^{k,l}_{m}\simeq\Sigma^{\infty}\int^{\underline{t}\in\mathtt{Epi}}_{\mathrm{h}}\mathtt{I}\wedge{\mathfrak{G}_{k,l}^{m}}_{+}\simeq\int^{\underline{t}\in\mathtt{Epi}}_{\mathrm{h}}\Sigma^{\infty}\mathtt{I}\wedge{\mathfrak{G}_{k,l}^{m}}_{+}.$

###### Proof.

By definition, $\mathbb{S}^{k,l}_{m}$ is the stabilization of the functor

$X\mapsto\int^{u\in\mathtt{Epi}}_{\mathrm{h}}X^{\wedge
u}\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}$

The right hand side is a weighted homotopy colimit of reduced functors from
$\mathtt{CW}^{f}_{*}\to\operatorname{Top}$. Since stabilization commutes with
such homotopy colimits, it follows that there is an equivalence

$\mathbb{S}^{k,l}_{m}\simeq\int^{u\in\mathtt{Epi}}_{\mathrm{h}}\partial_{1}(X^{\wedge
u})\wedge{\mathfrak{G}_{k,l}^{m}(u)}_{+}$

where $\partial_{1}(X^{\wedge u})$ denotes the stabilization of the functor
$X\mapsto X^{\wedge u}$. Observe that $\partial_{1}(X^{\wedge u})$ is
equivalent to $\Sigma^{\infty}S^{0}$ if $|u|=1$, and is equivalent to $*$ if
$|u|>1$. It follows that the functor $u\mapsto\partial_{1}(X^{\wedge u})$ is
equivalent to $\Sigma^{\infty}\mathtt{I}$ as a functor
$\mathtt{Epi}^{\operatorname{op}}\to\mathtt{Sp}$. The lemma follows. ∎

Recall once again that ${\mathfrak{G}_{k,l}^{m}}_{+}$ is defined by the
following formula,

${\mathfrak{G}_{k,l}^{m}(\underline{t})}_{+}=\bigvee_{\underset{m_{i}>0,\Sigma
m_{i}=m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}{\operatorname{Inj}(\mathbb{C}^{ml},\mathbb{C}^{k})/_{\prod_{i=1}^{t}U(m_{i})}}_{+}\cong\\\
\cong\bigvee_{\underset{m_{i}>0,\Sigma
m_{i}=m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}U(k)/\left(\prod_{i=1}^{t}U(m_{i})\times
U(k-lm)\right)_{+}.$

In the special case $l=1,k=m$ we get that

${\mathfrak{G}_{m,1}^{m}(\underline{t})}_{+}=\bigvee_{\underset{m_{i}>0,\Sigma
m_{i}=m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}U(m)/\prod_{i=1}^{t}U(m_{i})_{+}.$
(19)

And in general, there are equivalences

${\mathfrak{G}_{k,l}^{m}(\underline{t})}_{+}\simeq\bigvee_{\underset{m_{i}>0,\Sigma
m_{i}=m\\}}{\\{(m_{1},\ldots,m_{t})\mid}}U(k)/U(k-lm)_{+}\wedge_{U(m)}U(m)/\prod_{i=1}^{t}U(m_{i})_{+}\simeq\\\
\simeq
U(k)/U(k-lm)_{+}\wedge_{U(m)}{\mathfrak{G}_{m,1}^{m}(\underline{t})}_{+}.$
(20)

It is easily checked that the last equivalence is functorial in
$\underline{t}$ and therefore we have an equivalence of functors
$\mathtt{Epi}\to\operatorname{Top}$

${\mathfrak{G}_{k,l}^{m}}_{+}\simeq
U(k)/U(k-lm)_{+}\wedge_{U(m)}{\mathfrak{G}_{m,1}^{m}}_{+}.$

And upon applying lemma 7.6 we obtain an equivalence

$\mathbb{S}^{k,l}_{m}\simeq
U(k)/U(k-lm)_{+}\wedge_{U(m)}\mathbb{S}^{m,1}_{m}=\operatorname{Inj}(\mathbb{C}^{lm},\mathbb{C}^{k})_{+}\wedge_{U(m)}\mathbb{S}^{m,1}_{m}.$
(21)

We remind the reader that $U(m)$ is considered a subgroup of $U(k)$ via the
diagonal map $U(m)\hookrightarrow U(lm)$ followed by the inclusions
$U(lm)\hookrightarrow U(lm)\times U(k-lm)\hookrightarrow U(k)$. Alternatively,
$U(m)$ acts on $\operatorname{Inj}(\mathbb{C}^{lm},\mathbb{C}^{k})$ through
its obvious action on $\mathbb{C}^{lm}=\mathbb{C}^{l}\otimes\mathbb{C}^{m}$.

##  8 Connection with the complex of direct-sum decompositions

Equation (21) reduces the problem of describing $\mathbb{S}^{k,l}_{m}$ for
general $k,l,m$ to describing $\mathbb{S}^{m,1}_{m}$ for all $m$. In this
section we use Lemma 7.6 to show that $\mathbb{S}^{m,1}_{m}$ is equivalent to
the suspension spectrum of the complex of direct-sum decompositions of
$\mathbb{C}^{m}$, which we denote $\mathcal{L}^{\diamond}_{m}$. This leads to
a complete description of $\mathbb{S}^{k,l}_{m}$ in terms of the complexes
$\mathcal{L}^{\diamond}_{m}$ (Theorem 8.8).

The complexes $\mathcal{L}^{\diamond}_{m}$ were first introduced in [Ar1], and
were studied in detail in [BJL+] and [AL3]. They play a role in orthogonal
calculus, and also in describing the subquotients of the rank filtration of
$K$-theory [AL1, AL2]. They have some remarkable homotopical properties, that
we will recall in the next section (Proposition 9.1).

Our proof that $\mathbb{S}^{m,1}_{m}$ is equivalent to the suspension spectrum
of $\mathcal{L}^{\diamond}$ goes though an intermediate complex, which we call
the complex of ordered direct-sum decompositions. Let us give the formal
definition.

######  Definition 8.1.

Let $\mathcal{D}^{\mathrm{o}}_{m}$ be the following category objects in
topological spaces. Its objects are ordered tuples $(E_{1},\ldots,E_{t})$ of
pairwise orthogonal proper, non-trivial vector subspaces of $\mathbb{C}^{m}$,
whose direct sum is $\mathbb{C}^{m}$. A morphism
$(E_{1},\ldots,E_{t})\to(F_{1},\ldots,F_{s})$ consists of a surjective
function $\alpha\colon\\{1,\ldots,t\\}\twoheadrightarrow\\{1,\ldots,s\\}$ such
that for each $1\leq i\leq t$, $E_{i}\subseteq F_{\alpha(i)}$.

We call the category $\mathcal{D}^{\mathrm{o}}_{m}$ the category of proper,
ordered direct-sum decompositions of $\mathbb{C}^{m}$. The set of objects and
the set of morphisms both have a topology. There is a natural action of $U(m)$
on $\mathcal{D}^{\mathrm{o}}_{m}$, and object and morphism sets of are
topologized as unions of $U(m)$-orbits.

There is a convenient presentation of $\mathcal{D}^{\mathrm{o}}_{m}$ as the
Grothendieck construction applied to the functor $\mathfrak{G}_{m,1}^{m}$ of
Definition 7.2. Let us recall the definition of (a version of) the
Grothendieck construction.

######  Definition 8.2.

Suppose $\mathcal{C}$ is a small category and
$F\colon\mathcal{C}\to\operatorname{Top_{u}}$ is a functor. The Grothendieck
construction on $F$ (a.k.a the wreath product of $\mathcal{C}$ and $F$) is the
following pointed topological category, denoted $\mathcal{C}\wr F$. The
objects of $\mathcal{C}\wr F$ are pairs $(c,x)$ where $c$ is an object of
$\mathcal{C}$, and $x\in F(c)$. A morphism $(c,x)\to(d,y)$ in $\mathcal{C}\wr
F$ is a morphism $\alpha\colon c\to d$ in $\mathcal{C}$ such that
$F(\alpha)(x)=y$. The space of objects of $\mathcal{C}\wr F$ is topologized as
the disjoint union $\coprod_{c}F(c)$ indexed by the objects of $\mathcal{C}$,
and the space of morphisms is topologized as the disjoint union $\coprod_{c\to
d}F(c)$, indexed by morphisms of $\mathcal{C}$.

The following well-known lemma can be thought of as a topological analogue of
Thomason's homotopy colimit theorem.

######  Lemma 8.3.

Suppose $\mathcal{C}$ is a small category and
$F\colon\mathcal{C}\to\operatorname{Top_{u}}$ is a functor. There is a natural
equivalence

$\mathop{\operatorname{hocolim}}_{\mathcal{C}}F\simeq|\mathcal{C}\wr F|.$

###### Proof.

It is easy to see that the simplicial nerve of $\mathcal{C}\wr F$ is
isomorphic, as a simplicial space, to Bousfield and Kan's simplicial model for
$\mathop{\operatorname{hocolim}}_{\mathcal{C}}F$. In fact, both simplicial
spaces are given in simplicial degree $k$ by the space

$\coprod_{c_{0}\to\cdots\to c_{k}}F(c_{0}).$

The $i$-th face map $d_{i}$ is defined by dropping $c_{i}$ and, if $i=0$,
using the functoriality of $F$ to map $F(c_{0})$ to $F(c_{1})$. The degeneracy
map $s_{i}$ is defined by duplicating $c_{i}$. ∎

Now recall that we have a functor
$\mathfrak{G}_{m,1}^{m}\colon\mathtt{Epi}\to\operatorname{Top_{u}}$
(Definition 7.2). Let $\mathtt{Epi}^{>1}$ be the full subcategory of
$\mathtt{Epi}$ consisting of sets of cardinality greater than $1$. By slight
abuse of notation we denote the restriction of $\mathfrak{G}_{m,1}^{m}$ to
$\mathtt{Epi}^{>1}$ also by $\mathfrak{G}_{m,1}^{m}$. The following lemma is
straightforward from the definitions:

######  Lemma 8.4.

There is an isomorphism of topological categories

$\mathtt{Epi}^{>1}\wr\mathfrak{G}_{m,1}^{m}\cong{\mathcal{D}^{\mathrm{o}}_{m}}.$

Given a space $X$, let $X^{\diamond}$ denote the unreduced suspension of $X$.
We have the following connection between $\mathbb{S}^{m,1}_{m}$ and
$\mathcal{D}^{\mathrm{o}}_{m}$.

######  Proposition 8.5.

There is a natural equivalence

$\mathbb{S}^{m,1}_{m}\simeq\Sigma^{\infty}|\mathcal{D}^{\mathrm{o}}_{m}|^{\diamond}.$

###### Proof.

We saw in lemma 7.6 that

$\mathbb{S}^{m,1}_{m}\simeq\Sigma^{\infty}\int^{\mathtt{Epi}}_{\mathrm{h}}\mathtt{I}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}$

where $\mathtt{I}\colon\mathtt{Epi}^{\operatorname{op}}\to\operatorname{Top}$
is the functor that sends $\underline{1}$ to $S^{0}$ and sends all other
objects to $*$. So we need to show that there is an equivalence of pointed
spaces

$\int^{\mathtt{Epi}}_{\mathrm{h}}\mathtt{I}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}\simeq|\mathcal{D}^{\mathrm{o}}_{m}|^{\diamond}.$

Let $\mathtt{S}\colon\mathtt{Epi}^{\operatorname{op}}\to\operatorname{Top}$ be
the constant functor $\mathtt{S}(\underline{t})\equiv S^{0}$. Let
$\mathtt{S}^{>1}\colon\mathtt{Epi}^{\operatorname{op}}\to\operatorname{Top}$
be the functor $\mathtt{S}^{>1}(\underline{1})=*$ and
$\mathtt{S}^{>1}(\underline{t})\equiv S^{0}$ for $t>1$. Then there is a
homotopy cofibration sequence of functors
$\mathtt{S}^{>1}\to\mathtt{S}\to\mathtt{I}$. It follows that there is a
homotopy cofibration sequence of coends

$\int^{\mathtt{Epi}}_{\mathrm{h}}\mathtt{S}^{>1}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}\to\int^{\mathtt{Epi}}_{\mathrm{h}}\mathtt{S}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}\to\int^{\mathtt{Epi}}_{\mathrm{h}}\mathtt{I}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}$

It is a standard fact that

$\int^{\mathtt{Epi}}_{\mathrm{h}}\mathtt{S}\wedge{\mathfrak{G}_{m,1}^{m}}\simeq{\mathop{\operatorname{hocolim}}_{\mathtt{Epi}}}^{*}({\mathfrak{G}_{m,1}^{m}}_{+})\cong(\mathop{\operatorname{hocolim}}_{\mathtt{Epi}}{\mathfrak{G}_{m,1}^{m}})_{+}$

(here $\mathop{\operatorname{hocolim}}^{*}$ denotes pointed homotopy colimit,
while $\mathop{\operatorname{hocolim}}$ denotes unpointed homotopy colimit).
Since $\mathtt{Epi}$ has a final object $\underline{1}$, it follows that

${\mathop{\operatorname{hocolim}}_{\mathtt{Epi}}}^{*}{\mathfrak{G}_{m,1}^{m}}_{+}\simeq{\mathfrak{G}_{m,1}^{m}}_{+}(\underline{1})=S^{0}.$

On the other hand, since $\underline{1}$ is the initial object of
$\mathtt{Epi}^{\operatorname{op}}$, and $\mathtt{S}^{>1}(\underline{1})=*$ it
follows easily that $\mathtt{S}^{>1}$ is equivalent to the functor obtained by
restricting $\mathtt{S}$ to the subcategory
${\mathtt{Epi}^{>1}}^{\operatorname{op}}$ of sets of cardinality greater than
$1$, and then taking derived left Kan extension back to
${\mathtt{Epi}}^{\operatorname{op}}$. By standard adjunctions, it follows that
there are equivalences

$\int^{\mathtt{Epi}}_{\mathrm{h}}\mathtt{S}^{>1}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}\simeq\int^{\mathtt{Epi}^{>1}}_{\mathrm{h}}\mathtt{S}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}\simeq(\mathop{\operatorname{hocolim}}_{\mathtt{Epi}^{>1}}{\mathfrak{G}_{m,1}^{m}})_{+}$

It follows that there is a homotopy cofibration sequence

$(\mathop{\operatorname{hocolim}}_{\mathtt{Epi}^{>1}}{{\mathfrak{G}_{m,1}^{m}}})_{+}\to
S^{0}\to\int^{\underline{t}\in\mathtt{Epi}}_{\mathrm{h}}\mathtt{I}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}$

By lemma 8.4 ${\mathcal{D}^{\mathrm{o}}_{m}}$ is the Grothendieck construction
on $\mathfrak{G}_{m,1}^{m}$. It follows by Lemma 8.3 that

$\mathop{\operatorname{hocolim}}_{\mathtt{Epi}^{>1}}{\mathfrak{G}_{m,1}^{m}}\simeq|\mathcal{D}^{\mathrm{o}}_{m}|.$

So we have a homotopy cofibration sequence

$|\mathcal{D}^{\mathrm{o}}_{m}|_{+}\to
S^{0}\to\int^{\underline{t}\in\mathtt{Epi}}_{\mathrm{h}}\mathtt{I}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}$

This implies that
$\int^{\underline{t}\in\mathtt{Epi}}_{\mathrm{h}}\mathtt{I}\wedge{\mathfrak{G}_{m,1}^{m}}_{+}\simeq|\mathcal{D}^{\mathrm{o}}_{m}|^{\diamond}$.
∎

Our next step is to show that $\mathcal{D}^{\mathrm{o}}_{m}$ can be replaced
with a smaller category, which we call the poset of unordered direct-sum
decompositions. First, the definition.

######  Definition 8.6.

Let $\mathcal{D}_{m}$ be the following category objects in topological spaces.
Its objects are unordered sets $\\{E_{i}\mid i\in I\\}$ of pairwise orthogonal
proper, non-trivial vector subspaces of $\mathbb{C}^{m}$, whose direct sum is
$\mathbb{C}^{m}$. There is a unique morphism $\\{E_{i}\mid i\in
I\\}\to\\{F_{j}\mid j\in J\\}$ if for each $i\in I$ there is a (necessarily
unique) $j\in J$ such that $E_{i}\subseteq F_{j}$. In keeping with recent
literature, the geometric realization of $\mathcal{D}_{m}$ will be denoted
$\mathcal{L}_{m}$, and its unreduced suspension is therefore
$\mathcal{L}_{m}^{\diamond}$.

As with $\mathcal{D}^{\mathrm{o}}_{m}$, there is a natural action of $U(m)$ on
$\mathcal{D}_{m}$ and both the sets of objects and morphisms of
$\mathcal{D}_{m}$ are topologized as unions of $U(m)$-orbits. We note that for
any two objects $P,Q$ of $\mathcal{D}^{\mathrm{o}}_{m}$ there is at most one
morphism from $P$ to $Q$. In other words, $\mathcal{D}^{\mathrm{o}}_{m}$ is a
topological preorder. By contrast, the category $\mathcal{D}_{m}$ is a
topological poset: it is the poset of isomorphism classes of
$\mathcal{D}^{\mathrm{o}}_{m}$. The category $\mathcal{D}_{m}$ will be
referred to as the category, or poset, of proper, unordered direct-sum
decompositions of $\mathbb{C}^{m}$.

There is a topological functor
$q\colon\mathcal{D}^{\mathrm{o}}_{m}\to\mathcal{D}_{m}$, which forgets the
order of the components.

######  Proposition 8.7.

The natural functor $q\colon\mathcal{D}^{\mathrm{o}}_{m}\to\mathcal{D}_{m}$
induces an equivalence of geometric realizations
$|\mathcal{D}^{\mathrm{o}}_{m}|\xrightarrow{\simeq}|\mathcal{D}_{m}|$

###### Proof.

We are going to use Quillen's theorem A. We need a version of it that is valid
for topological categories. There are several such versions scattered in the
literature, we will use [EbRW, Theorem 4.7]. According to this theorem, it is
enough if we prove the following

  1. 1.

For every object $\Lambda$ of $\mathcal{D}_{m}$, the classifying space of the
over category $q/\Lambda$ is contractible.

  2. 2.

The map from the morphism space of $\mathcal{D}^{\mathrm{o}}_{m}$ to the
object space of $\mathcal{D}^{\mathrm{o}}_{m}$, that sends every morphism to
its target, is a fibration (in the language of [EbRW], this means that
$\mathcal{D}^{\mathrm{o}}_{m}$ is right fibrant).

  3. 3.

The map from the space of objects of the over category $q/\mathcal{D}_{m}$ to
the space of objects of $\mathcal{D}_{m}$, that sends a morphism to its
target, is a fibration. Here $q/\mathcal{D}_{m}$ is the category of arrows in
$\mathcal{D}_{m}$ of the form $q(\Theta)\to\Lambda$, where $\Theta$ is an
object of $\mathcal{D}^{\mathrm{o}}_{m}$.

For part (1), let $\Lambda=\\{F_{i}\mid i\in I\\}$ be an object of
$\mathcal{D}_{m}$, i.e., an unordered collection of pairwise orthogonal non-
trivial subspaces of $\mathbb{C}^{m}$ whose direct sum is $\mathbb{C}^{m}$.
Let $t$ be the number of elements of $I$ and choose a bijection
$I\cong\\{1,\ldots,t\\}$. Then $\widetilde{\Lambda}=(F_{1},\ldots,F_{t})$ is a
choice of lift of $\Lambda$ to an object of $\mathcal{D}^{\mathrm{o}}_{m}$. An
object of $q/\Lambda$ consists of an object $\Theta=(E_{1},\ldots,E_{s})$ such
that each $E_{i}$ is a subspace of $F_{j}$ for some (necessarily unique) $j$.
It follows that there exists a unique surjection
$\alpha\colon\\{1,\ldots,s\\}\twoheadrightarrow\\{1,\ldots,t\\}$ such that
$E_{i}\subset F_{\alpha(i)}$ for all $i$. This means that there is a unique
morphism from $\Theta$ to $\widetilde{\Lambda}$ in $q/\Lambda$. Thus the
category $q/\Lambda$ has a (not necessarily unique) terminal object, and
therefore its classifying space is contractible.

For part (2), using the identification of $\mathcal{D}^{\mathrm{o}}_{m}$ with
the Grothendieck construction $\mathtt{Epi}^{>1}\wr\mathfrak{G}_{m,1}^{m}$
(Lemma 8.4), the map from the space of morphisms of
$\mathcal{D}^{\mathrm{o}}_{m}$ to the space of objects of
$\mathcal{D}^{\mathrm{o}}_{m}$ which sends each morphism to its target, has
the following form

$\coprod_{\underline{s}\twoheadrightarrow\underline{t}\in\mathtt{Epi}^{>1}}\coprod_{\underset{m_{i}>0,\Sigma
m_{i}=m\\}}{\\{(m_{1},\ldots,m_{s})\mid}}U(m)/\prod_{i=1}^{s}U(m_{i})\to\coprod_{\underline{t}\in\mathtt{Epi}^{>1}}\coprod_{\underset{n_{i}>0,\Sigma
n_{j}=m\\}}{\\{(n_{1},\ldots,n_{t})\mid}}U(m)/\prod_{j=1}^{t}U(n_{j})$ (22)

where for every surjective function
$\underline{s}\twoheadrightarrow\underline{t}$, the space
$U(m)/\prod_{i=1}^{s}U(m_{i})$ is sent to $U(m)/\prod_{j=1}^{s}U(n_{j})$,
where for each $j=1,\ldots,t$, $n_{j}=\Sigma_{i\in\alpha^{-1}(j)}m_{i}$, by
the canonical quotient map associated with the sub-conjugation of
$\prod_{i=1}^{s}U(m_{i})$ into $\prod_{j=1}^{s}U(n_{j})$ induced by $\alpha$.
The map (22) is clearly a fibration. Indeed, it is a $U(m)$ equivariant map
between disjoint union of $U(m)$-orbits, and such a map is necessarily a
fibration.

Finally, the proof of part (3) is similar to that of part (2). Since
$\mathcal{D}_{m}$ is the poset of isomorphism classes of the pre-order
$\mathcal{D}^{\mathrm{o}}_{m}$, the space of objects of the category
$q/\mathcal{D}_{m}$ is the quotient of the space of morphisms of
$\mathcal{D}^{\mathrm{o}}_{m}$ by the action of the groupoid of isomorphisms
of the target. Similarly, the space of objects of $\mathcal{D}_{m}$ is the
quotient of the space of objects of $\mathcal{D}^{\mathrm{o}}_{m}$ by the
groupoid of isomorphisms. This means that we have the following map

$\left(\coprod_{\underline{s}\twoheadrightarrow\underline{t}}\coprod_{\underset{m_{i}>0,\Sigma
m_{i}=m\\}}{\\{(m_{1},\ldots,m_{s})\mid}}U(m)/\prod_{i=1}^{s}U(m_{i})\right)_{\operatorname{Iso}(t)}\to\left(\coprod_{\underline{t}}\coprod_{\underset{n_{i}>0,\Sigma
n_{j}=m\\}}{\\{(n_{1},\ldots,n_{t})\mid}}U(m)/\prod_{j=1}^{t}U(n_{j})\right)_{\operatorname{Iso}(t)}$

The action of the groupoid of isomorphisms of the variable $\underline{t}$
respects the action of $U(m)$. It follows that the resulting map is still a
$U(m)$-equivariant map between disjoint union of orbits, and therefore is a
fibration. ∎

Propositions 8.5 and 8.7, together with equation (21) give us an equivalence

$\mathbb{S}^{k,l}_{m}\simeq
U(k)/U(k-lm)_{+}\wedge_{U(m)}\Sigma^{\infty}\mathcal{L}_{m}^{\diamond}.$ (23)

We also want to describe the composition morphisms
$\mathbb{S}^{k,l}_{m}\wedge\mathbb{S}^{j,k}_{n}\to\mathbb{S}^{j,l}_{mn}$. We
begin by observing that tensor product induces a natural map
$\mathcal{L}^{\diamond}_{m}\wedge\mathcal{L}^{\diamond}_{n}\to\mathcal{L}^{\diamond}_{mn}$
as follows. Suppose that $\mathcal{E}=\\{E_{i}\mid i\in I\\}$ and
$\mathcal{F}=\\{F_{j}\mid j\in J\\}$ are direct-sum decompositions of
$\mathbb{C}^{m}$ and $\mathbb{C}^{n}$ respectively. Then
$\mathcal{E}\otimes\mathcal{F}:=\\{E_{i}\otimes F_{j}\mid(i,j)\in I\times
J\\}$ is a direct-sum decomposition of
$\mathbb{C}^{m}\otimes\mathbb{C}^{n}\cong\mathbb{C}^{mn}$. Note that if at
least one of $\mathcal{E}$, $\mathcal{F}$ is a proper decomposition (i.e., has
more than one component) then $\mathcal{E}\otimes\mathcal{F}$ is a proper
decomposition as well. This means that the tensor product induces a map
$\mathcal{L}^{\diamond}_{m}\wedge\mathcal{L}^{\diamond}_{n}\to\mathcal{L}^{\diamond}_{mn}$
as desired. Note that this map is equivariant with respect to the tensor
product homomorphisms $U(m)\times U(n)\to U(mn)$.

Next, we extend it to a map

$\left(U(k)/U(k-lm)_{+}\wedge_{U(m)}\mathcal{L}_{m}^{\diamond}\right)\wedge\left(U(j)/U(j-kn)_{+}\wedge_{U(n)}\mathcal{L}_{n}^{\diamond}\right)\to\\\
\to U(j)/U(j-lnm)_{+}\wedge_{U(nm)}\mathcal{L}^{\diamond}_{nm}.$

Now recall that
$U(k)/U(k-lm)\cong\operatorname{Inj}(\mathbb{C}^{lm},\mathbb{C}^{k})$ and
$U(j)/U(j-kn)\cong\operatorname{Inj}(\mathbb{C}^{kn},\mathbb{C}^{j})$. Given
homomorphisms $g\in\operatorname{Inj}(\mathbb{C}^{lm},\mathbb{C}^{k})$ and
$f\in\operatorname{Inj}(\mathbb{C}^{kn},\mathbb{C}^{j})$, we may form the
homomorphism $f\circ
g^{n}\colon{\mathbb{C}^{lmn}}\hookrightarrow\mathbb{C}^{j}$. Clearly, this
defines a map $U(k)/U(k-lm)\times U(j)/U(j-kn)\to U(j)/U(j-lnm)$. This map is
equivariant with respect to the tensor product homomorphism $U(m)\times
U(n)\to U(mn)$. Combining it with the map
$\mathcal{L}^{\diamond}_{m}\wedge\mathcal{L}^{\diamond}_{n}\to\mathcal{L}^{\diamond}_{mn}$
defined earlier, we obtain the desired map.

We are ready to state the main theorem of the paper.

######  Theorem 8.8.

There is an equivalence

$\mathbb{S}^{k,l}_{m}\simeq\Sigma^{\infty}U(k)/U(k-lm)_{+}\wedge_{U(m)}\mathcal{L}_{m}^{\diamond}\cong\Sigma^{\infty}\operatorname{Inj}(\mathbb{C}^{ml},\mathbb{C}^{k})_{+}\wedge_{U(m)}\mathcal{L}_{m}^{\diamond}.$

Under this equivalence, the composition product
$\mathbb{S}^{k,l}_{m}\wedge\mathbb{S}^{j,k}_{n}\to\mathbb{S}^{j,l}_{mn}$
corresponds to the map

$\left(\operatorname{Inj}\left(\mathbb{C}^{ml},\mathbb{C}^{k}\right)_{+}\wedge_{U(m)}\mathcal{L}_{m}^{\diamond}\right)\wedge\left(\operatorname{Inj}\left(\mathbb{C}^{nk},\mathbb{C}^{j}\right)_{+}\wedge_{U(n)}\mathcal{L}_{n}^{\diamond}\right)\to\\\
\to\operatorname{Inj}\left(\mathbb{C}^{nml},\mathbb{C}^{j}\right)_{+}\wedge_{U(nm)}\mathcal{L}^{\diamond}_{nm}.$

that were defined above.

###### Proof.

We already proved the formula for $\mathbb{S}^{k,l}_{m}$ (equation 23). It
remains to check the statement about the composition product. Recall that
$\mathbb{S}^{k,l}_{m}$ is the stabilization of the functor $G_{k,l}^{m}$. The
composition product is determined by the natural transformation
$\mathfrak{G}_{k,l}^{m}(v)\wedge\mathfrak{G}_{j,k}^{n}(u)\to\mathfrak{G}_{j,l}^{mn}(u\times
v)$. An analysis of this composition map shows that it is induced by disjoint
union of maps of the form

$\operatorname{Inj}(\mathbb{C}^{(m_{1}+\cdots+m_{t})l},\mathbb{C}^{k})/_{\prod_{j=1}^{t}U(m_{j})}\times\operatorname{Inj}(\mathbb{C}^{(n_{1}+\cdots+n_{s})k},\mathbb{C}^{j})/_{\prod_{i=1}^{s}U(n_{i})}\to\\\
\to{\operatorname{Inj}(\mathbb{C}^{(\sum_{i=1,j=1}^{i=s,j=t}n_{i}m_{j})l},\mathbb{C}^{j})/_{\prod_{i=1,j=1}^{i=s,j=t}U(n_{i}m_{j})}}$

that sends $(q,p)$ to $p\circ q^{n}$, where $n=n_{1}+\cdots+n_{s}$. Note that
the decomposition of $\mathbb{C}^{mn}$ associated with the target of this map
is the tensor product of the given decompositions of $\mathbb{C}^{m}$ and
$\mathbb{C}^{n}$, just as was claimed. This induces the claimed map of
spectra. ∎

##  9 Some calculations of $\mathbb{S}^{k,l}$

In this section we calculate the spectra $\mathbb{S}^{k,l}$ in some cases, and
also prove that the map $\mathbb{S}^{k,l}\to ku$ is an isomorphism on
$\pi_{0}$. Our main tool is Theorem 8.8, which expresses the subquotients of
the rank filtration in terms of the complexes $\mathcal{L}_{m}^{\diamond}$. To
use it, we need to know something about the complexes
$\mathcal{L}_{m}^{\diamond}$. So let us begin by reviewing some of the rather
remarkable properties of these complexes that were uncovered in [Ar1, AL1,
BJL+, AL3]. The following proposition lists the relevant facts.

######  Proposition 9.1.

  1. 1.

The space $\mathcal{L}_{m}^{\diamond}$ is rationally contractible for $m>1$.

  2. 2.

The space $\mathcal{L}_{m}^{\diamond}$ is (integrally) contractible unless $m$
is a prime power.

  3. 3.

If $m=p^{k}$ with p a prime and $k>0$, then $\mathcal{L}_{p^{k}}^{\diamond}$
is $p$-local, and has chromatic type $k$.

###### Proof.

Except for the statement about the chromatic type, this is [AL1, proposition
9.6], which in turn relies on [Ar1]. The statement about the chromatic type is
part of [Ar2, Theorem 2.2].

The proofs in [Ar2] are based on a rather deep connection between
$\mathcal{L}_{m}^{\diamond}$ and the calculus of functors. Since part (1)
plays a prominent role in our applications, we indicate an independent, more
direct way to prove this part. The space $\mathcal{L}_{m}^{\diamond}$ is
equivalent to the total homotopy cofiber of the following $m-1$-dimensional
cubical diagram. Suppose
$U=\\{i_{1},\ldots,i_{k}\\}\subseteq\\{2,\ldots,m\\}$, with
$i_{1}>\ldots>i_{k}$. Let $\mathcal{X}(U)$ be the space of chains of
decompositions of $\mathbb{C}^{m}$ of the form
$(\Lambda_{1}<\cdots<\Lambda_{k})$ where each $\Lambda_{j}$ has
$i_{j}$-components. If $U$ is empty then $\mathcal{X}(U)=*$. Note that in
general $\mathcal{X}(U)$ is a disjoint union of $U(m)$-orbits. The assignment
$U\mapsto\mathcal{X}(U)$ defines a diagram indexed on the opposite of the
poset of subsets of $\\{2,\ldots,m\\}$, i.e., an $m-1$-dimensional cubical
diagram. It is elementary to show that $\mathcal{L}_{m}^{\diamond}$ is
equivalent to the total homotopy cofiber of the cube $\mathcal{X}$. For
example, in the case $m=3$, $\mathcal{X}$ is the following square of
$U(3)$-orbits.

$\begin{array}[]{ccc}U(3)/\Sigma_{2}\wr U(1)\times U(1)&\to&U(3)/U(2)\times