# Derived Character Maps of Group Representations

Yuri Berest  Department of Mathematics, Cornell University, Ithaca, NY
14853-4201, USA<EMAIL_ADDRESS>and  Ajay C. Ramadoss  Department of
Mathematics, Indiana University, Bloomington, IN 47405, USA
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we construct and study derived character maps of finite-
dimensional representations of $\infty$-groups. As models for $\infty$-groups
we take homotopy simplicial groups, i.e. homotopy simplicial
${\mathfrak{G}}^{\rm op}$-algebras over the algebraic theory of groups (in the
sense of [4]). We define cyclic, symmetric and representation homology for
‘group algebras’ $k[\Gamma]$ over such groups and construct canonical trace
maps relating these homology theories. In the case of one-dimensional
representations, we show that our trace maps are of topological origin: they
are induced by natural maps of (iterated) loop spaces that are well studied in
homotopy theory. Using this topological interpretation, we deduce some
algebraic results about representation homology: in particular, we prove that
the symmetric homology of group algebras and one-dimensional representation
homology are naturally isomorphic, provided the base ring $k$ is a field of
characteristic zero. We also study the behavior of the derived character maps
of $n$-dimensional representations in the stable limit as $n\to\infty$, in
which case we show that they ‘converge’ to become isomorphisms.

##  1\. Introduction

If $\Gamma$ is a finite group and $k$ is a field of characteristic zero, every
finite-dimensional $k$-linear representation
$\varrho:\Gamma\to{\rm{GL}}_{n}(k)$ is semi-simple and determined (up to
equivalence) by its character: the trace function ${\rm{Tr}}_{n}:\langle
g\rangle\mapsto{\rm{Tr}}_{n}[\varrho(g)]\,$ defined on the set
$\langle\Gamma\rangle$ of conjugacy classes of elements of $\Gamma$. Moreover,
for each $n\geq 0$, there are only finitely many equivalence classes of such
representations. These well familiar facts from representation theory of
finite groups generalize to arbitrary groups by means of algebraic geometry.
For any discrete group $\Gamma$, the set of all $n$-dimensional
representations of $\Gamma$ can be naturally given the structure of an affine
algebraic variety (more precisely, an affine $k$-scheme)
${\rm{Rep}}_{n}(\Gamma)$ called the representation variety of $\Gamma$. The
equivalence classes of $n$-dimensional representations of $\Gamma$ are
classified by the orbits of the general linear group ${\rm{GL}}_{n}$ that acts
algebraically on ${\rm{Rep}}_{n}(\Gamma)$ by conjugation. The classes of semi-
simple representations correspond to the closed orbits111At least when
$\Gamma$ is finitely generated. and are parametrized by the affine quotient
scheme

${\rm{Rep}}_{n}(\Gamma)/\\!/{\rm{GL}}_{n}(k):={\rm{Spec}}\,\mathcal{O}[{\rm{Rep}}_{n}(\Gamma)]^{{\rm{GL}}_{n}}$

called the character variety of $\Gamma$. Now, the characters of
representations assemble into a linear map

(1.1) ${\rm{Tr}}_{n}(\Gamma):\
k\langle\Gamma\rangle\to\mathcal{O}[{\rm{Rep}}_{n}(\Gamma)]^{{\rm{GL}}_{n}}$

which is defined on the $k$-vector space $k\langle\Gamma\rangle$ spanned by
the conjugacy classes of elements of $\Gamma$. A well-known theorem of C.
Procesi [57] asserts that the characters of $\Gamma$, i.e. the images of the
map (1.1), generate $\mathcal{O}[{\rm{Rep}}_{n}(\Gamma)]^{{\rm{GL}}_{n}}$ as a
commutative $k$-algebra, and thus, by Nullstellensatz, detect the semi-simple
representations of $\Gamma$ when $k$ is algebraically closed. In general, the
equivariant geometry of ${\rm{Rep}}_{n}(\Gamma)$ is closely related to
representation theory of $\Gamma$, the geometric structure of
${\rm{GL}}_{n}$-orbits in ${\rm{Rep}}_{n}(\Gamma)$ determining the algebraic
structure of representations. Since the late 1980s this relation has been
extensively studied and exploited in many areas of mathematics, most notably
in geometric group theory and low-dimensional topology (see, for example, [47,
64]).

Derived algebraic geometry allows one to extend — and in some sense to
complete — this beautiful connection between representation theory and
geometry. The classical representation scheme ${\rm{Rep}}_{n}(\Gamma)$ admits
a natural derived extension ${\rm{DRep}}_{n}(\Gamma)$ called the derived
representation scheme222The first construction of this kind — the derived
moduli space $\bm{{\mathrm{R}}}{\rm Loc}_{G}(X)$ of $G$-local systems over a
pointed connected space $X$ — was introduced by Kapranov [40]. In recent
years, several other constructions have been developed in various frameworks
of derived algebraic geometry (most notably, in the work of Toën, Vezzosi,
Pridham and Pantev (see, e.g., [66, 56, 55, 54, 51, 50]). A comparison of
these constructions can be found in [12, Appendix]. The relation to our
present work is briefly discussed in the end of Section 3.2. of $\Gamma$. In
general, $\,{\rm{DRep}}_{n}(\Gamma)$ is represented by a simplicial
commutative $k$-algebra $\mathcal{O}[{\rm{DRep}}_{n}(\Gamma)]$ whose homotopy
groups $\,\pi_{i}\mathcal{O}[{\rm{DRep}}_{n}(\Gamma)]$, $\,i\geq 0$, are (non-
abelian) homological invariants of $\Gamma$ (or rather its classifying space
$B\Gamma$). Following [14, 13], we set

(1.2)
$\mathrm{HR}_{\ast}(\Gamma,{\rm{GL}}_{n}(k)):=\pi_{\ast}\mathcal{O}[{\rm{DRep}}_{n}(\Gamma)]$

and call (1.2) the $n$-dimensional representation homology of $\Gamma$. By
definition, $\,\mathrm{HR}_{\ast}(\Gamma,{\rm{GL}}_{n}(k))\,$ is a graded
commutative $k$-algebra, whose degree $0$ part is canonically isomorphic to
the coordinate ring of ${\rm{Rep}}_{n}(\Gamma)$:

(1.3)
$\mathrm{HR}_{0}(\Gamma,{\rm{GL}}_{n}(k))\cong\mathcal{O}[{\rm{Rep}}_{n}(\Gamma)]\
.$

Apart from groups, representation homology can be also defined for various
types of algebras (e.g., associative and Lie algebras, see [9, 10, 8, 7]) as
well as for topological spaces (see [14, 13, 12]). What is surprising perhaps
is that, in the case of groups and spaces, there is a simple construction of
representation homology that does not require the use of derived algebraic
geometry nor a heavy machinery of homotopical algebra. This construction
(discovered in [14]) plays a key role in the present paper: we will use it to
define representation homology for homotopy simplicial groups, which are
simple models for $\infty$-groups, a more general and (from the homotopy-
theoretic point of view) more natural concept than that of a discrete or
simplicial group (see [4]).

Now, returning to the character map (1.1), we observe that its domain — the
$k$-span $k\langle\Gamma\rangle$ of the conjugacy classes of elements of
$\Gamma$ — is isomorphic to the $0$-th cyclic homology of the group algebra
$k[\Gamma]$:

(1.4) $\mathrm{HC}_{0}(k[\Gamma])\cong k\langle\Gamma\rangle$

With isomorphisms (1.3) and (1.4), we can therefore rewrite (1.1) in the form

(1.5) ${\rm{Tr}}_{n}(\Gamma):\
\mathrm{HC}_{0}(k[\Gamma])\,\to\,\mathrm{HR}_{0}(\Gamma,{\rm{GL}}_{n}(k))^{{\rm{GL}}_{n}}$

This suggests that there might exist a natural extension of the map (1.1) to
higher cyclic homology with values in higher representation homology of
$\Gamma$:

(1.6) ${\rm{Tr}}_{n}(\Gamma)_{\ast}:\
\mathrm{HC}_{\ast}(k[\Gamma])\,\to\,\mathrm{HR}_{\ast}(\Gamma,{\rm{GL}}_{n}(k))^{{\rm{GL}}_{n}}$

We call (1.6) the derived character maps of $n$-dimensional representations of
$\Gamma$, and our first goal in this paper is to define and study these maps
for an arbitrary homotopy simplicial group $\Gamma$.

In the case of associative algebras, the derived character maps were
originally constructed in [9], using non-abelian homological algebra. This
construction was extended to Lie algebras in [8], where it was shown — among
other things — that derived character maps of Lie algebra representations are
Koszul dual to the classical Loday-Quillen-Tsygan maps [46, 67]. The case of
groups that we study in this paper is special for several reasons. First, as
mentioned above, in this case the representation homology admits a simple
construction that is similar to Connes’ well-known construction of cyclic
homology in terms of Tor-functors on the cyclic category. We will show that
behind this ‘similarity’ there is actually a connection: a simple formula for
the derived character maps (1.6) relating Tor-functors via classical
homological algebra (see Section 3.4 and, in particular, Definition 3.5).

Second, the cyclic homology of group algebras has a beautiful topological
interpretation that goes back to the work of Goodwillie, Burghelea,
Fiedorowicz and others (see [44, Chap. 7]). Specifically, there is a natural
isomorphism

(1.7)
$\mathrm{HC}_{*}(k[\Gamma])\,\cong\,\mathrm{H}_{*}(ES^{1}\times_{S^{1}}\\!\mathcal{L}(B\Gamma);\,k)\,,$

where the right-hand side is the $S^{1}$-equivariant homology of the free loop
space $\mathcal{L}(B\Gamma):={\rm{Map}}(S^{1},B\Gamma)$ of the classifying
space of $\Gamma$. In fact, (1.7) is just one on the list of several classical
isomorphisms relating algebraic homology theories associated with so-called
crossed simplicial groups [28] to (stable) homotopy theory:

$\displaystyle{\rm{HH}}_{*}(k[\Gamma])$ $\displaystyle\cong$
$\displaystyle\mathrm{H}_{*}(\mathcal{L}(B\Gamma);\,k)\,,$
$\displaystyle\mathrm{HC}_{*}(k[\Gamma])$ $\displaystyle\cong$
$\displaystyle\mathrm{H}_{*}(ES^{1}\times_{S^{1}}\\!\mathcal{L}(B\Gamma);\,k)\,,$
(1.8) $\displaystyle\mathrm{HS}_{*}(k[\Gamma])$ $\displaystyle\cong$
$\displaystyle\mathrm{H}_{*}(\Omega\,\Omega^{\infty}\Sigma^{\infty}\\!(B\Gamma);\,k)\,,$
$\displaystyle\mathrm{HB}_{*}(k[\Gamma])$ $\displaystyle\cong$
$\displaystyle\mathrm{H}_{*}(\Omega^{2}\Sigma(B\Gamma);\,k)\,,$
$\displaystyle\mathrm{HO}_{*}(k[\Gamma])$ $\displaystyle\cong$
$\displaystyle\mathrm{H}_{*}(E(\mathbb{Z}/2)_{+}\wedge_{\mathbb{Z}/2}\Omega\,\Omega^{\infty}\Sigma^{\infty}(B\Gamma);\,k)\,,$

where $\Omega$, $\Sigma$ and $\Omega^{\infty}\Sigma^{\infty}$ denote the based
loop, the (reduced) suspension, and the stable homotopy functors,
respectively. The first two of the above isomorphisms (for Hochschild and
cyclic homology) are well known: they were originally established in [30] and
[20], and their proofs appear in Loday’s textbook [44] (see also [45] for a
nice self-contained exposition). The last three (for the symmetric
$\mathrm{HS}_{*}$, braided $\mathrm{HB}_{*}$ and hyperoctahedral
$\mathrm{HO}_{*}$ homologies) are less known: they were discovered by
Fiedorowicz [27] in the early 1990s, but detailed proofs were published only
recently (see [2] and [34]).

The second (and perhaps, the main) goal of this paper is to extend the above
list of isomorphisms by adding to it representation homology. To be precise,
for any commutative ring $k$, let
$\mathrm{HR}_{\ast}(k[\Gamma]):=\mathrm{HR}_{\ast}(\Gamma,\mathbb{G}_{m}(k))$
denote the one-dimensional representation homology of $\Gamma$. We prove (see
Theorem 4.1 and Lemma 4.1):

######  Theorem 1.1.

For any homotopy simplicial group $\Gamma$, there is a natural isomorphism

(1.9)
$\mathrm{HR}_{\ast}(k[\Gamma])\,\cong\,\mathrm{H}_{\ast}(\Omega\,\mathrm{SP}^{\infty}\\!(B\Gamma);\,k)$

where $\,\mathrm{SP}^{\infty}\\!(B\Gamma)\,$ denotes the classical Dold-Thom
space of the classifying space of $\Gamma\,$.

Apart from the Hochschild and cyclic theories, most interesting on the list
(1) is the symmetric homology theory $\mathrm{HS}_{\ast}$ introduced in [27]
and studied in [2, 3]. Roughly speaking, $\mathrm{HS}_{\ast}$ is defined333See
Sections 3.3 and 4.2 for precise definitions of
$\mathrm{HC}_{\ast}(k[\Gamma])$ and $\mathrm{HS}_{\ast}(k[\Gamma])$ in the
context of homotopy simplicial groups. in the same way as
$\mathrm{HC}_{\ast}$, with Connes’ cyclic category $\Delta C$ replaced by the
symmetric category $\Delta S$, where the family of the symmetric groups
$\\{S^{\rm op}_{n+1}\\}_{n\geq 0}$ is used instead of the cyclic groups
$\\{C_{n+1}\\}_{n\geq 0}$. Now, the natural inclusions of groups
$C_{n+1}\,\hookrightarrow\,S_{n+1}$ extend to a functor $\iota:\Delta C^{\rm
op}\,\hookrightarrow\,\Delta S$, which, in turn, induces a natural map
$\,\mathrm{HC}_{*}(k[\Gamma])\to\mathrm{HS}_{*}(k[\Gamma])\,$. It turns out
that, with identifications (1), this last map is induced (on homology) by a
map of topological spaces

(1.10) ${\tt{CS}}_{B\Gamma}:\
ES^{1}\times_{S^{1}}\\!\mathcal{L}(B\Gamma)\,\to\,\Omega\,\Omega^{\infty}\Sigma^{\infty}\\!(B\Gamma)$

The map (1.10) is actually defined as a natural transformation ${\tt{CS}}_{X}$
on the (homotopy) category of all pointed spaces; it was originally
constructed in the paper [21], and its relation to symmetric homology was
noticed in [27]. We will refer to (1.10) as the Carlsson-Cohen map for
$B\Gamma$.

We can now state our second observation that provides a topological
interpretation of the derived character maps (1.6) for one-dimensional
representations. To shorten notation we will write the maps (1.6) for $n=1$ as

(1.11) ${\rm{Tr}}(\Gamma)_{\ast}:\
\mathrm{HC}_{\ast}(k[\Gamma])\,\to\,\mathrm{HR}_{\ast}(k[\Gamma])$

The next theorem encapsulates the main results of Section 4.3 (see Proposition
4.1 and Corollary 4.1), Section 5.2 (see Proposition 5.1) and Section 5.3 (see
Proposition 5.2).

######  Theorem 1.2.

With isomorphisms (1) and (1.9), the derived character maps (1.11) are induced
on homology by a natural map of topological spaces

(1.12) ${\tt{CR}}_{B\Gamma}:\
ES^{1}\times_{S^{1}}\\!\mathcal{L}(B\Gamma)\,\to\,\Omega\,\mathrm{SP}^{\infty}\\!(B\Gamma)$

The map (1.12) factors $($as a homotopy natural transformation$)$ through the
Carlsson-Cohen map (1.10):

(1.13)
$ES^{1}\times_{S^{1}}\\!\mathcal{L}(B\Gamma)\xrightarrow{{\tt{CS}}_{B\Gamma}}\Omega\,\Omega^{\infty}\Sigma^{\infty}\\!(B\Gamma)\xrightarrow{{\tt{SR}}_{B\Gamma}}\Omega\,\mathrm{SP}^{\infty}\\!(B\Gamma)$

where the induced map ${\tt{SR}}$ is the $($looped once$)$ canonical natural
transformation $\Omega^{\infty}\Sigma^{\infty}\to\mathrm{SP}^{\infty}\\!$
relating stable homotopy to $($reduced$)$ singular homology of pointed spaces.

Theorem 1.2 shows that, for any homotopy simplicial group $\Gamma$, the
derived character map (1.11) factors through symmetric homology, and the
induced map

(1.14) ${\tt{SR}}_{B\Gamma,\ast}:\
\mathrm{HS}_{\ast}(k[\Gamma])\,\to\,\mathrm{HR}_{\ast}(k[\Gamma])$

is determined by a map of spaces that is well known in topology. Using
topological results, we then conclude (see Corollary 5.1 and Remark 5.1):

######  Corollary 1.1.

If $k$ is a field of characteristic $0$, the map (1.14) is an isomorphism, at
least when $B\Gamma$ is a simply connected space.

The results stated above are all concerned with derived characters of one-
dimensional representations. For higher dimensional representations ($n>1$),
the maps (1.6) are more complicated: in particular, they do not seem to factor
through $\mathrm{HS}_{\ast}(k[\Gamma])$, and in general, the relation between
symmetric homology and representation homology remains mysterious. However,
when $n\to\infty$, things become more tractable. Assuming that $k$ is a field
of characteristic $0$, we can naturally pass to the projective limit:

$\mathrm{HR}_{\ast}(\Gamma,\,{\rm{GL}}_{\infty}(k))^{{\rm{GL}}_{\infty}}:=\,\varprojlim_{n}\,\mathrm{HR}_{\ast}(\Gamma,\,{\rm{GL}}_{n}(k))^{{\rm{GL}}_{n}}$

and construct the stable character maps

(1.15) ${\rm{Tr}}_{\infty}(\Gamma)_{\ast}:\
\overline{\mathrm{HC}}_{\ast}(k[\Gamma])\,\to\,\mathrm{HR}_{\ast}(\Gamma,\,{\rm{GL}}_{\infty}(k))^{{\rm{GL}}_{\infty}}\,,$

where $\overline{\mathrm{HC}}$ stands for the reduced cyclic homology. In this
case, we have the following result, the proof of which is parallel to [10] and
outlined in the last section of the paper (see Theorem 6.1).

######  Theorem 1.3.

Let $\Gamma$ be a homotopy simplicial group such that $B\Gamma$ is a simply
connected space of finite $($rational$)$ type. Then the stable character maps
(1.15) induce an algebra isomorphism

(1.16) $\Lambda{\rm{Tr}}_{\infty}(\Gamma)_{\ast}:\
\Lambda_{k}[\,\overline{\mathrm{HC}}_{\ast}(k[\Gamma])]\,\xrightarrow{\sim}\,\mathrm{HR}_{\ast}(\Gamma,{\rm{GL}}_{\infty})^{{\rm{GL}}_{\infty}}\,,$

where $\Lambda_{k}[\,\overline{\mathrm{HC}}_{\ast}(k[\Gamma])]$ is the graded
symmetric algebra generated by the reduced cyclic homology of $k[\Gamma]$.

We close this Introduction by mentioning one application of stable character
maps in derived Poisson geometry. If $\Gamma$ is a simplicial group model of a
simply-connected closed manifold $X$ of dimension $d$ (so that $X\simeq
B\Gamma$), then, by (1), we can identify
$\overline{\mathrm{HC}}_{\ast}(k[\Gamma])$ with the reduced
$S^{1}$-equivariant homology $\bar{\mathrm{H}}^{S^{1}}_{*}(\mathcal{L}(X);k)$
of the free loop space of $X$. Thanks to the work of Chas and Sullivan, the
latter is known to carry the so-called string topology Lie bracket, making the
symmetric algebra
$\Lambda_{k}[\,\overline{\mathrm{HC}}_{\ast}(k[\Gamma])]\cong\Lambda_{k}[\,\bar{\mathrm{H}}^{S^{1}}_{*}(\mathcal{L}(X);k)\,]$
a graded Poisson algebra. On the other hand, the representation homology ring
$\mathrm{HR}_{\ast}(\Gamma,{\rm{GL}}_{\infty})^{{\rm{GL}}_{\infty}}$ acquires
a $(2-d)$-shifted graded Poisson structure from the Poincaré duality pairing
on (the cohomology of) $X$. As an application of Theorem 1.3, we show that
under the isomorphism (1.16), these two Poisson structures agree: i.e., the
map (1.16) is an isomorphism of graded Poisson algebras (see Corollary 6.1).

The paper is organized as follows. In Section 2, we review basic facts from
abstract homotopy theory concerning homotopy colimits. The new result proved
in this section is Proposition 2.1, which we refer to as ‘Shapiro Lemma for
model categories’. This proposition provides a key step for proofs of main
theorems in Section 4 and may be of independent interest. In Section 3, after
reviewing basic theory of homotopy simplicial groups (Section 3.1), we define
representation homology (Section 3.2), and cyclic homology (Section 3.3) for
such groups and construct the derived character maps relating the two (Section
3.4). In Section 4, we prove Theorem 1.1 (Section 4.1) and then, after
defining symmetric homology for homotopy simplicial groups (Section 4.2), we
prove part of Theorem 1.2 (see Proposition 4.1 and Corollary 4.1 in Section
4.3). The proof of Theorem 1.2 is completed in Section 6, where we study the
maps (1.12) and (1.13) in topological terms, using Goodwillie homotopy
calculus and classical operads (see Proposition 5.1 and Proposition 5.2).
Finally, in Section 6, we describe the stabilization procedure for the derived
character maps as $n\to\infty$ and sketch the proofs of Theorem 1.3 and
Corollary 6.1. Each of the six sections begins with a short introduction that
provides more details about its contents.

### Acknowledgments

The work of the first author was partially supported by NSF grant DMS 1702372
and the Simons Collaboration Grant 712995. The second author was partially
supported by NSF grant DMS 1702323.

##  2\. Shapiro Lemma for Model Categories

In this section, we establish one general result in abstract homotopy theory
related to homotopy colimits that will provide a key step in the proof of our
main Theorem 1.1. We call this result (Proposition 2.1) ‘Shapiro Lemma for
model categories’ as it appears to be a non-abelian generalization (in the
context of model categories) of the classical Shapiro Lemma in homological
algebra. We begin with a brief overview of the theory of homotopy colimits.
The standard reference for this material is the last two chapters of
Hirschhorn’s book [36] but many results that we mention are classical and go
back to Bousfield-Kan [19] and Quillen [59]. Our exposition is inspired by
Cisinski’s beautiful paper [24] that treats homotopy colimits axiomatically by
the analogy with derived direct image functors in algebraic geometry (unlike
[24], however, we do not use the language of Grothendieck derivators). With
exception of Proposition 2.1, which (to the best of our knowledge) is new, all
results in this section are known.

###  2.1. Notation and conventions

Throughout this section, $\mathcal{M}$ will denote a fixed model category
which we assume to be cofibrantly generated and having all small limits and
colimits. Unless stated otherwise, ${\mathcal{A}}$, $\mathcal{B}$,
$\mathcal{C}$, $\ldots$ will denote small categories that we will use to index
diagrams in $\mathcal{M}$. For a small category ${\mathcal{A}}$, the category
of ${\mathcal{A}}$-diagrams in $\mathcal{M}$ (i.e. all functors
${\mathcal{A}}\to\mathcal{M}$) will be denoted $\mathcal{M}^{{\mathcal{A}}}$.
As usual, ${\rm Cat}$ will stand for the category of all small categories with
morphisms being arbitrary functors.

###  2.2. Homotopy colimits

For any small category ${\mathcal{A}}$, the category
$\mathcal{M}^{{\mathcal{A}}}$ has a projective (aka Bousfield-Kan) model
structure inherited from $\mathcal{M}$: the weak equivalences and fibrations
are defined in this model structure objectwise, while the cofibrations are
determined by the Lifting Axiom of model categories (specifically, as
morphisms having the left lifting property with respect to fibrations which
are also weak equivalences in $\mathcal{M}^{{\mathcal{A}}}$). Since
$\mathcal{M}$ is cofibrantly generated, such a model structure on
$\mathcal{M}^{{\mathcal{A}}}$ always exists and is cofibrantly generated (see
[36, Theorem 11.6.1]).

Any functor $f:{\mathcal{A}}\to\mathcal{B}$ (a morphism in ${\rm Cat}$)
defines the pullback functor on the diagram categories
$\,f^{\ast}:\mathcal{M}^{\mathcal{B}}\to\mathcal{M}^{{\mathcal{A}}}\,$, which
is obtained by restricting diagrams $\mathcal{B}\to\mathcal{M}$ along $f$.
This pullback functor preserves objectwise weak equivalences and fibrations
and — since $\mathcal{M}$ has small colimits – admits a left adjoint:

(2.1)
$f_{!}:\mathcal{M}^{{\mathcal{A}}}\rightleftarrows\mathcal{M}^{\mathcal{B}}:f^{\ast}$

defined on a diagram $X:{\mathcal{A}}\to\mathcal{M}$ as the left Kan extension
$\,f_{!}(X):={\rm Lan}_{f}(X)\,$ of $X$ along $f$. Thus, the functors (2.1)
form a Quillen pair between the model categories $\mathcal{M}^{{\mathcal{A}}}$
and $\mathcal{M}^{\mathcal{B}}$. Then, by Quillen’s Adjunction Theorem (see
[36, 8.5.8]), they admit total (left and right) derived functors

(2.2)
$\bm{L}f_{!}:{\mathtt{Ho}}(\mathcal{M}^{{\mathcal{A}}})\rightleftarrows{\mathtt{Ho}}(\mathcal{M}^{\mathcal{B}}):f^{\ast}$

that form an adjunction between the homotopy categories of diagrams induced by
(2.1).

The derived pushforward functor $\bm{L}f_{!}$ is called the homotopy left Kan
extension along $f$. It is a generalization of the classical homotopy colimit
functor
$\,\mathrm{hocolim}_{{\mathcal{A}}}:{\mathtt{Ho}}(\mathcal{M}^{{\mathcal{A}}})\to{\mathtt{Ho}}(\mathcal{M})\,$
that corresponds to the trivial map $\,{\mathcal{A}}\to\ast\,$, where “$\ast$”
denotes the one-point category (the terminal object in ${\rm Cat}$). In this
last case, we will use the classical notation writing ${\rm
hocolim}_{{\mathcal{A}}}(X)$ instead of $\bm{L}({\mathcal{A}}\to\ast)_{!}(X)$
for $X:{\mathcal{A}}\to\mathcal{M}$. We summarize the main properties of this
construction in the following theorem.

######  Theorem 2.1 (see [24]).

Let $\mathcal{M}$ be a model category with all small limits and colimits.

$(1)$ $2$-Functoriality: The pullback functors $f^{\ast}$ fit together to give
a strict, weakly product-preserving 2-functor $\,{\rm Cat}^{\rm op}\to{\rm
CAT}\,$ that takes a small category ${\mathcal{A}}\in{\rm Cat}$ to the
homotopy category ${\mathtt{Ho}}(\mathcal{M}^{{\mathcal{A}}})$. By adjunction,
this implies, in particular, the existence of natural weak equivalences

(2.3) $\bm{L}(fg)_{!}\simeq\bm{L}f_{!}\,\bm{L}g_{!}\ $

for any composable morphisms $f$ and $g$ in ${\rm Cat}$.

$(2)$ Reflexivity: For any ${\mathcal{A}}\in{\rm Cat}$, the functor
$i^{*}:{\mathtt{Ho}}(\mathcal{M}^{{\mathcal{A}}})\to{\mathtt{Ho}}(\mathcal{M}^{{{\mathcal{A}}}^{\delta}})$
corresponding to the inclusion of the underlying discrete subcategory
${\mathcal{A}}^{\delta}\subset{\mathcal{A}}$ is conservative, i.e. reflects
the weak equivalences in $\mathcal{M}^{{{\mathcal{A}}}^{\delta}}$.

$(3)$ Base change: For any $f:{\mathcal{A}}\to\mathcal{B}$ and any object
$b\in\mathcal{B}$, the $2$-commutativity of the fibre square

$\begin{diagram}$

induces a change-of-base natural transformation that is a natural weak
equivalence:

$\bm{L}p_{!}\,\pi^{\ast}\stackrel{{\scriptstyle\sim}}{{\to}}b^{\ast}\,\bm{L}f_{!}$

For a diagram $X:{\mathcal{A}}\to\mathcal{M}$, this simply says that

(2.4) $\bm{L}f_{!}X(b)\simeq{\rm hocolim}_{f\downarrow b}(\pi^{\ast}X)\ .$

######  Remark 2.1.

In terminology of [24] (cf. Def. 1.6, pp. 205-206), the properties $(1)$-$(3)$
of Theorem 2.1 can be summarized by saying that the 2-functor
${\mathtt{Ho}}(\mathcal{M}^{-}):{\rm Cat}^{\rm op}\to{\rm CAT}$ is a weak left
derivator (un dérivateur faible à gauche) associated to the model category
$\mathcal{M}$.

The properties of homotopy colimits listed in Theorem 2.1 are essentially
formal. The next result — called the Cofinality Theorem — gives a deeper
property of homotopy-theoretic nature that is very useful in computations. To
state this result we recall that a functor $\,f:{\mathcal{A}}\to\mathcal{B}\,$
is right homotopy cofinal if its comma-category $b\downarrow f$ under each
object $b\in\mathcal{B}$ is (weakly) contractible, i.e. $\,B(b\downarrow
f)\simeq{\rm pt}\,$. As an example, we point out that every right adjoint
functor is right homotopy cofinal: indeed, if
$\,f:{\mathcal{A}}\to\mathcal{B}\,$ admits a left adjoint, say
$g:\mathcal{B}\to{\mathcal{A}}$, then each comma-category $b\downarrow f$ has
an initial object (namely, $(b,\eta_{b})$, where $\eta_{b}:b\to fg(b)$ is the
unit of the adjunction evaluated at $b\in\mathcal{B}$), hence $b\downarrow f$
is contractible for any $b\in\mathcal{B}$.

######  Theorem 2.2 (Cofinality Theorem).

If $f:{\mathcal{A}}\to\mathcal{B}$ is right homotopy cofinal, then the natural
map

${\rm
hocolim}_{{\mathcal{A}}}(f^{\ast}X)\stackrel{{\scriptstyle\sim}}{{\to}}{\rm
hocolim}_{\mathcal{B}}(X)$

is a weak equivalence for any diagram $X:\,\mathcal{B}\to\mathcal{M}$.

For the proof of Theorem 2.2 we refer to [36, Theorem 19.6.7]. As an
application, we prove one simple lemma that we will need for our computations.
Given a functor $\,f:{\mathcal{A}}\to\mathcal{B}\,$, we recall that its fibre
category $f^{-1}(b)$ over an object $b\in\mathcal{B}$ is the subcategory of
${\mathcal{A}}$ consisting of all objects $a\in{\mathcal{A}}$ such that
$f(a)=b$ and all morphisms
$\varphi\in{\rm{Hom}}_{{\mathcal{A}}}(a,a^{\prime})$ such that
$f(\varphi)={\rm Id}_{b}$. Note that the fibre inclusion functor
$i:f^{-1}(b)\,\hookrightarrow\,{\mathcal{A}}$ factors through the comma-
category $f\downarrow b$ over $b$:

(2.5) $\begin{diagram}$

defining the ‘comparison’ functor

(2.6) $j:\ f^{-1}(b)\to f\downarrow b\,\qquad
a\mapsto(a,f(a)=b\stackrel{{\scriptstyle{\rm Id}}}{{\to}}b)$

Recall that a functor $f:{\mathcal{A}}\to\mathcal{B}$ is precofibred if (2.6)
has a left adjoint for every object $b\in\mathcal{B}$ (see [58, §1]).

######  Lemma 2.1.

If $f:{\mathcal{A}}\to\mathcal{B}$ is precofibred, then

$(\bm{L}f_{!}X)(b)\,\simeq\,{\rm hocolim}_{f^{-1}(b)}\,(i^{\ast}X)$

for any diagram $X:{\mathcal{A}}\to\mathcal{M}$.

###### Proof.

By assumption, the inclusion functor $\,j:f^{-1}(b)\to f\downarrow b\,$ is
right adjoint, hence right homotopy cofinal. By the base change formula (2.4)
and Cofinality Theorem 2.2, we conclude

$\displaystyle(\bm{L}f_{!}X)(b)$ $\displaystyle\simeq$ $\displaystyle{\rm
hocolim}_{f\downarrow b}(\pi^{\ast}X)$ $\displaystyle\simeq$
$\displaystyle{\rm hocolim}_{f^{-1}(b)}(j^{\ast}\pi^{\ast}X)$ $\displaystyle=$
$\displaystyle{\rm hocolim}_{f^{-1}(b)}((\pi j)^{\ast}X)$ $\displaystyle=$
$\displaystyle{\rm hocolim}_{f^{-1}(b)}(i^{\ast}X)$

where the last identification follows from (2.5). ∎

In practice, precofibred functors arise from the so-called Grothendieck
construction (see [65]). Given a functor $\,F:\mathcal{C}\to{\rm Cat}\,$
(i.e., a strict diagram of small catgories), its Grothendieck construction is
defined to be the small category $\,\mathcal{C}\\!\smallint\\!F$ with $\,{\rm
Ob}(\mathcal{C}\\!\smallint\\!F):=\\{(c,x):c\in\mathcal{C},\,x\in F(c)\\}\,$
and morphism sets

(2.7)
${\rm{Hom}}_{\mathcal{C}\\!\int\\!F}((c,x),\,(c^{\prime},x^{\prime})):=\\{(\varphi,f):\varphi\in{\rm
Hom}_{\mathcal{C}}(c,c^{\prime}),\,f\in{\rm
Hom}_{F(c^{\prime})}(F(\varphi)x,x^{\prime})\\}\ .$

The composition in $\mathcal{C}\\!\int\\!F$ is given by
$\,(\varphi,f)\circ(\varphi^{\prime},f^{\prime})=(\varphi\,\varphi^{\prime},\
f\,F(\varphi)f^{\prime})\,$. The category $\mathcal{C}\\!\int\\!F$ comes
equipped with a natural (forgetful) functor

$p:\,\mathcal{C}\\!\smallint\\!F\to\mathcal{C}\ ,\qquad(c,x)\mapsto c$

which is precofibred (in fact, cofibred) over $\mathcal{C}$. Notice that
$p^{-1}(c)=F(c)$ for any object $c\in\mathcal{C}$. Hence, by Lemma 2.1, for
any functor $\,X:\,\mathcal{C}\\!\int\\!F\to\mathcal{M}\,$,

(2.8) $(\bm{L}p_{!}X)(c)\simeq{\rm hocolim}_{F(c)}[X(c)]\ ,$

where $X(c):=i_{c}^{\ast}X$ is the restriction of $X$ to $F(c)$ via the
inclusion functor

$i_{c}:\,F(c)\to\mathcal{C}\\!\smallint\\!F\,,\qquad x\mapsto(c,x)\
,\quad(x\stackrel{{\scriptstyle f}}{{\to}}x^{\prime})\mapsto({\rm Id}_{c},f)\
.$

Note that, by 2-functoriality of homotopy Kan extensions (see (2.3)), (2.8)
implies the weak equivalence

(2.9) ${\rm hocolim}_{\mathcal{C}\\!\smallint\\!F}(X)\,\simeq\,{\rm
hocolim}_{c\in\mathcal{C}}({\rm hocolim}_{F(c)}X(c))$

which is known as Thomason’s formula for homotopy colimits over
$\mathcal{C}\\!\int\\!F$ (see [23, Theorem 26.8]).

An important special case arises when we apply the Grothendieck construction
to a set-valued functor $\,F:\mathcal{C}\to\mathrm{Set}\,$, regarding sets as
discrete categories (i.e. by embedding
$\,\mathrm{Set}\,\hookrightarrow\,\mathrm{Cat}$). In this case, the category
$\mathcal{C}\\!\int\\!F$ is usually denoted $\mathcal{C}_{F}$ and called the
category of elements of $F$ as its object set ${\rm Ob}(\mathcal{C}_{F})$ can
be identified with $\,\coprod_{c\in\mathcal{C}}F(c)\,$ (we will still write
the objects of $\mathcal{C}_{F}$ as pairs $(c,x)$, where $c\in\mathcal{C}$ and
$x\in F(c)$). The Hom-sets in $\mathcal{C}_{F}$ are given by
$\,{\rm{Hom}}_{\mathcal{C}_{F}}((c,x),\,(c^{\prime},x^{\prime}))=\\{\varphi\in{\rm
Hom}_{\mathcal{C}}(c,c^{\prime})\,:\,F(\varphi)x=x^{\prime}\\}\,$ (cf. (2.7)).
If we take $\mathcal{M}=s\mathrm{Set}$ to be the category of simplicial sets
(equipped with standard Quillen model structure) and apply Thomason’s formula
(2.9) to the trivial diagram $X:\mathcal{C}_{F}\to\ast$ in $\mathcal{M}$, then
for any functor $\,F:\mathcal{C}\to\mathrm{Set}\,$, we get

(2.10) $\mathrm{hocolim}_{\mathcal{C}}(F)\,\cong\,N_{*}(\mathcal{C}_{F})$

where $N_{\ast}(\mathcal{C}_{F})$ denotes the simplicial nerve of the category
$\mathcal{C}_{F}$. Formula (2.10) is known as the Bousfield-Kan construction
for homotopy colimits in $s\mathrm{Set}$ (see [19]).

###  2.3. Homotopy coends

Homotopy coends are special kinds of homotopy colimits defined for bifunctors,
i.e. the diagrams of the form $\mathcal{C}^{\rm
op}\times\mathcal{C}\to\mathcal{M}$. There is a broader range of techniques
for manipulating with such homotopy colimits, which makes them more accessible
for computations. The homotopy coends are defined in terms of the so-called
factorization category $\mathcal{F}(\mathcal{C})$ introduced by Quillen [59].
It can be described as the category of elements of the bifunctor ${\rm
Hom}:\,\mathcal{C}^{\rm op}\times\mathcal{C}\to{\rm Set}$ of the given
category $\mathcal{C}$:

(2.11) $\mathcal{F}(\mathcal{C}):=(\mathcal{C}^{\rm
op}\times\mathcal{C})\\!\smallint{\rm Hom}\ .$

We will be actually dealing with the opposite category
$\mathcal{F}(\mathcal{C})^{\rm op}$ which can be explicitly described as
follows: the objects of $\mathcal{F}(\mathcal{C})^{\rm op}$ are the morphisms
$\\{\varphi:c\to d\\}$ in $\mathcal{C}$, and the ${\rm Hom}$-sets are
commutative squares

(2.12) $\begin{diagram}$

i.e., ${\rm Hom}_{\mathcal{F}(\mathcal{C})^{\rm
op}}(\varphi,\varphi^{\prime})$ consists of the pairs of morphisms
$(\alpha,\beta)$ in $\mathcal{C}$ such that
$\varphi=\beta\varphi^{\prime}\alpha$, with compositions defined in the
obvious way. Note that $\,\mathcal{F}(\mathcal{C})^{\rm
op}\not\simeq\mathcal{F}(\mathcal{C}^{\rm op})\,$ in general. Now, there are
two natural functors

(2.13) $s^{\rm op}:\ \mathcal{F}(\mathcal{C})^{\rm
op}\to\mathcal{C}\,,\qquad(c\stackrel{{\scriptstyle\varphi}}{{\to}}d)\mapsto
c$ (2.14) $t^{\rm op}:\ \mathcal{F}(\mathcal{C})^{\rm op}\to\mathcal{C}^{\rm
op}\,,\qquad(c\stackrel{{\scriptstyle\varphi}}{{\to}}d)\mapsto d$

called the (opposite) source and target functors, respectively. We have

######  Lemma 2.2 (Quillen).

The functors (2.13) and (2.14) are both right homotopy cofinal.

###### Proof.

Since $\mathcal{F}(\mathcal{C})$ is defined by Grothendieck construction
(2.11), the canonical (forgetful) functor

$s\times t:\mathcal{F}(\mathcal{C})\to\mathcal{C}^{\rm op}\times\mathcal{C}$

is precofibred. It follows (cf. [59, Example, p. 94]) that both
$s:\mathcal{F}(\mathcal{C})\to\mathcal{C}^{\rm op}$ and
$t:\mathcal{F}(\mathcal{C})\to\mathcal{C}$ are precofibred. Hence the
inclusions $s^{-1}(c)\hookrightarrow s\downarrow c$ and
$t^{-1}(d)\hookrightarrow t\downarrow d$ induce weak equivalences of
classifying spaces

(2.15) $B(s^{-1}(c))\simeq B(s\downarrow c)\quad,\quad B(t^{-1}(d))\simeq
B(t\downarrow d)\ .$

On the other hand, by inspection, $s^{-1}(c)=c\downarrow\mathcal{C}$ and
$t^{-1}(d)=(\mathcal{C}\downarrow d)^{\rm op}$ are the slice and coslice
categories respectively. Since both $c\downarrow\mathcal{C}$ and
$(\mathcal{C}\downarrow d)^{\rm op}$ have initial objects, they are
contractible for all $c,d\in\mathcal{C}$. To complete the proof it remains to
note that $(c\downarrow s^{\rm op})=(s\downarrow c)^{\rm op}$ and
$(d\downarrow t^{\rm op})=(t\downarrow d)^{\rm op}$, where $s^{\rm op}$ and
$t^{\rm op}$ are the functors (2.13) and (2.14). Hence

$B(c\downarrow s^{\rm op})\,=\,B(s\downarrow c)^{\rm op}\simeq B(s\downarrow
c)\simeq B(s^{-1}(c))\simeq{\rm pt}$

and similarly $B(d\downarrow t^{\rm op})\simeq{\rm pt}$. This shows that
$s^{\rm op}$ and $t^{\rm op}$ are right homotopy cofinal. ∎

In view of Lemma 2.2, the Cofinality Theorem gives two natural weak
equivalences

(2.16) $s^{\ast}:{\rm hocolim}_{\mathcal{F}(\mathcal{C})^{\rm
op}}(s^{\ast}Y)\stackrel{{\scriptstyle\sim}}{{\to}}{\rm
hocolim}_{\mathcal{C}}(Y)$ (2.17) $t^{\ast}:{\rm
hocolim}_{\mathcal{F}(\mathcal{C})^{\rm
op}}(t^{\ast}X)\stackrel{{\scriptstyle\sim}}{{\to}}{\rm
hocolim}_{\mathcal{C}^{\rm op}}(X)$

for any diagrams $X:\mathcal{C}^{\rm op}\to\mathcal{M}$ and
$Y:\mathcal{C}\to\mathcal{M}$. These equivalences can be used to express
arbitrary homotopy colimits over $\mathcal{C}$ and $\mathcal{C}^{\rm op}$ as
homotopy coends which we introduce next. Set

$\pi^{\rm op}:=t^{\rm op}\times s^{\rm op}:\mathcal{F}(\mathcal{C})^{\rm
op}\to\mathcal{C}^{\rm op}\times\mathcal{C}$

and for a bifunctor $D:\mathcal{C}^{\rm op}\times\mathcal{C}\to\mathcal{M}$,
define its homotopy coend by

(2.18) $\int^{c\in\mathcal{C}}_{\bm{L}}D(c,c)\,:=\,{\rm
hocolim}_{\mathcal{F}(\mathcal{C})^{\rm op}}(\pi^{\ast}D)\,,$

where $\pi^{\ast}D:=D\circ\pi^{\rm op}:\mathcal{F}(\mathcal{C})^{\rm
op}\to\mathcal{M}$. This is indeed the (left) derived functor of the classical
coend functor which is usually denoted

(2.19) $\int^{c\in\mathcal{C}}D(c,c)\,:=\,{\rm
colim}_{\mathcal{F}(\mathcal{C})^{\rm op}}(\pi^{\ast}D)\ .$

The notation (2.18) is very convenient as it suggests the analogy with
(definite) integrals in Calculus. For example, for a bifunctor
$D:({\mathcal{A}}\times\mathcal{B})^{\rm e}\to M$ defined on a product of two
small categories $({\mathcal{A}}\times\mathcal{B})^{\rm e}:={\mathcal{A}}^{\rm
op}\times\mathcal{B}^{\rm op}\times{\mathcal{A}}\times\mathcal{B}$ there is a
natural weak equivalence

$\int^{(a,b)\in{\mathcal{A}}\times\mathcal{B}}_{\bm{L}}D(a,b;a,b)\simeq\int^{a\in{\mathcal{A}}}_{\bm{L}}\int^{b\in\mathcal{B}}_{\bm{L}}D(a,b;a,b)$

which is analogous to the classical Fubini Theorem in Calculus (and thus
called the Fubini Theorem for homotopy coends). Another useful formula that we
will need is

(2.20)
$\int_{\bm{L}}^{c\in\mathcal{C}}\bm{L}F[D(c,c)]\,\simeq\,\bm{L}F\left[\int_{\bm{L}}^{c\in\mathcal{C}}D(c,c)\right]$

where $F$ is a left Quillen functor between model categories. This formula is
a consequence of a more general (well-known) result that the derived functors
of left Quillen functors preserve homotopy colimits (for a short proof, see
e.g. [70, Proposition 3.15]).

We are now in a position to state the main result of this section.

######  Proposition 2.1 (Shapiro Lemma for model categories).

Let $\mathcal{M}$ be a model category, $\mathcal{C}$ a small category, and
$F:\mathcal{C}\to{\rm Set}$ any set-valued functor on $\mathcal{C}$. Then, for
any contravariant diagram $X:\,\mathcal{C}^{\rm op}\to\mathcal{M}$,

(2.21) ${\rm hocolim}_{\mathcal{C}_{F}^{\rm
op}}(p^{\ast}X)\,\simeq\,\int_{\bm{L}}^{c\in\mathcal{C}}X(c)\otimes F(c)$

where $\mathcal{C}_{F}$ is the category of elements of $F$, and “$\,\otimes$”
denotes the natural $($tensor$)$ action444That is, $\otimes$ is the bifunctor
$\mathcal{M}\times{\rm Set}\to\mathcal{M}$ defined by $A\otimes
S=\coprod_{S}A$, where $\coprod_{S}A$ is the coproduct of copies of $A$
indexed by the elements of $S$. of $\,{\rm Set}$ on $\mathcal{M}$.

For the proof of Proposition 2.1, we need the following observation.

######  Lemma 2.3.

For any set-valued functor $F:\mathcal{C}\to{\rm Set}$, the functor
$\mathcal{F}(p)^{\rm op}:\mathcal{F}(\mathcal{C}_{F})^{\rm
op}\to\mathcal{F}(\mathcal{C})^{\rm op}$ induced by the canonical projection
$p:\mathcal{C}_{F}\to\mathcal{C}$ is precofibred.

###### Proof.

The proof is by direct verification. To simplify the notation we set
$f:=\mathcal{F}(p)^{\rm op}$ and describe first the fibre category
$f^{-1}(\varphi)$ for $(\varphi:c\to d)\in\mathcal{F}(\mathcal{C})^{\rm op}$.
The objects of $f^{-1}(\varphi)$ are the morphisms in $\mathcal{C}_{F}$ of the
form $(c,x)\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}(d,y)$ such that
$y=F(\varphi)(x)$. We will write the object
$(c,x)\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}(d,F(\varphi)(x))$ of
$\mathcal{F}(\mathcal{C}_{F})^{\rm op}$ as $(\varphi,x)$. Thus,

${\rm Ob}(f^{-1}(\varphi))\,=\,\\{(\varphi,x):x\in F(c)\\}\ .$

Further, the morphisms $(\varphi,x)\to(\varphi,y)$ in $f^{-1}(\varphi)$ are
precisely the morphisms in $\mathcal{F}(\mathcal{C}_{F})^{\rm op}$, i.e.,
commutative diagrams of the form

$\begin{diagram}$

mapped to the identity by $f$. This last condition implies that $\alpha={\rm
Id}_{c}$ and $\beta={\rm Id}_{d}$. Hence,

${\rm
Hom}_{f^{-1}(\varphi)}\big{(}(\varphi,x),(\varphi,y)\big{)}=\begin{cases}\\{{\rm
Id}\\}\quad&\text{if }x=y\\\ \varnothing\quad&\text{ otherwise}\end{cases}\ .$

Hence, $f^{-1}(\varphi)\cong F(c)$, where the set $F(c)$ is viewed as a
discrete category.

Next, for $(\varphi:c\to d)\in\mathcal{F}(\mathcal{C})^{\rm op}$, we have

${\rm Ob}(f\downarrow\varphi)=\\{[\psi:k\to
l,z,\alpha,\beta]:(\psi,z)\in\mathcal{F}(\mathcal{C}_{F})^{\rm
op},(\alpha,\beta)\in{\rm Hom}_{\mathcal{F}(\mathcal{C}){\rm
op}}(\psi,\varphi)\\}\ .$

The morphisms
$[\psi,z,\alpha,\beta]\to[\psi^{\prime},z^{\prime},\alpha^{\prime},\beta^{\prime}]$
in $f\downarrow\varphi$ are the commutative diagrams in $\mathcal{C}_{F}$ of
the form

$\begin{diagram}$

such that the diagram

$\begin{diagram}$

commutes in $\mathcal{C}$. Hence, a morphism
$[\psi,z,\alpha,\beta]\to[\varphi,x,{\rm Id}_{c},{\rm Id}_{d}]$ in
$f\downarrow\varphi$ is a commutative diagram of the form

$\begin{diagram}$

Such a diagram exists if and only if $x=F(\alpha)(z)$, in which case it is
unique. Hence,

${\rm Hom}_{f\downarrow\varphi}([\psi,z,\alpha,\beta],[\varphi,x,{\rm
Id}_{c},{\rm Id}_{d}])\,=\,\begin{cases}\\{(\alpha,\beta)\\}&\text{ if
}x=F(\alpha)(z)\\\ \varnothing&\text{ otherwise}\ .\end{cases}$

Here, $(\alpha,\beta)$ is viewed as a morphism $(\psi,z)\to(\varphi,x)$ in
$\mathcal{F}(\mathcal{C}_{F})^{\rm op}$ rather then
$\mathcal{F}(\mathcal{C})^{\rm op}$.

Next, consider the assignment

$\Phi:f\downarrow\varphi\to
f^{-1}(\varphi)\,,\qquad[\psi,z,\alpha,\beta]\mapsto(\varphi,F(\alpha)(z))\ .$

If $(\gamma,\delta):(\psi,z)\to(\psi^{\prime},z^{\prime})$ is a morphism in
$f\downarrow\varphi$, then $z^{\prime}=F(\gamma)(z)$ and
$\alpha^{\prime}\circ\gamma=\alpha$. Hence, letting $\Phi$ map
$(\gamma,\delta)$ to the identity on $(\varphi,F(\alpha)(z))$ makes $\Phi$ a
functor. We then note that

${\rm Hom}_{f^{-1}(\varphi)}(\Phi([\psi,z,\alpha,\beta]),(\varphi,x))\,=\,{\rm
Hom}_{f^{-1}(\varphi)}((\varphi,F(\alpha)(z)),(\varphi,x))=\begin{cases}\\{{\rm
Id}\\}&\text{ if }x=F(\alpha)(z)\\\ \varnothing&\text{ otherwise}\end{cases}\
.$

Hence, there is a natural bijection

${\rm Hom}_{f^{-1}(\varphi)}(\Phi([\psi,z,\alpha,\beta]),(\varphi,x))\cong{\rm
Hom}_{f\downarrow\varphi}([\psi,z,\alpha,\beta],[\varphi,x,{\rm Id}_{c},{\rm
Id}_{d}])\,,$

showing that $\Phi$ is left adjoint to the canonical inclusion

$f^{-1}(\varphi)\hookrightarrow
f\downarrow\varphi\,,\qquad(\varphi,x)\mapsto[\varphi,x,{\rm Id}_{c},{\rm
Id}_{d}]\ .$

This shows that $f$ is precofibred, as desired. ∎

###### Proof of Proposition 2.1.

By formula (2.17) (applied to the category $\mathcal{C}_{F}$), there is a
natural weak equivalence

$t^{\ast}:{\rm hocolim}_{\mathcal{F}(\mathcal{C}_{F})^{\rm
op}}(t^{\ast}p^{\ast}X)\,\xrightarrow{\sim}\,{\rm
hocolim}_{\mathcal{C}_{F}^{\rm op}}(p^{\ast}X)$

where

$t^{\ast}p^{\ast}X:\ \mathcal{F}(\mathcal{C}_{F})^{\rm
op}\,\xrightarrow{t^{\rm op}}\,\mathcal{C}_{F}^{\rm op}\,\xrightarrow{p^{\rm
op}}\,\mathcal{C}^{\rm op}\,\xrightarrow{X}\,\mathcal{M}\ .$

On the other hand, by definition (2.18),

$\int_{\bm{L}}^{c\in\mathcal{C}}X(c)\otimes F(c)\,=\,{\rm
hocolim}_{\mathcal{F}(\mathcal{C})^{\rm op}}[\pi^{\ast}(X\otimes F)]$

where

$\pi^{\ast}(X\otimes F):\ \mathcal{F}(\mathcal{C})^{\rm
op}\,\xrightarrow{\pi^{\rm op}}\,\mathcal{C}^{\rm
op}\times\mathcal{C}\,\xrightarrow{X\times F}\,\mathcal{M}\times{\rm
Set}\,\xrightarrow{\otimes}\,\mathcal{M}\ .$

To prove the desired proposition we thus need to show that

(2.22) ${\rm hocolim}_{\mathcal{F}(\mathcal{C}_{F})^{\rm
op}}(t^{\ast}p^{\ast}X)\,\simeq\,{\rm hocolim}_{\mathcal{F}(\mathcal{C})^{\rm
op}}[\pi^{\ast}(X\otimes F)]$

By Theorem 2.1(1) (see (2.3)), it suffices to show that there is an weak
equivalence of $\mathcal{F}(\mathcal{C})^{\rm op}$-diagrams

(2.23) $\bm{L}f_{!}(t^{\ast}p^{\ast}X)\,\simeq\,\pi^{\ast}(X\otimes F)\,,$

where $f:\,\mathcal{F}(\mathcal{C}_{F})^{\rm
op}\to\mathcal{F}(\mathcal{C})^{\rm op}$ is the functor induced by the
canonical projection $p:\mathcal{C}_{F}\to C\,$. Thanks to Lemma 2.3, we can
use Lemma 2.1 to evaluate the homotopy Kan extension in (2.23) in terms of
homotopy colimits over fibre categeories. Specifically, for any
$\varphi\in\mathcal{F}(\mathcal{C})^{\rm op}$, we have

$\bm{L}f_{!}(t^{\ast}p^{\ast}X)(\varphi)\,\simeq\,{\rm
hocolim}_{f^{-1}(\varphi)}(i^{\ast}t^{\ast}p^{\ast}X)$

where $i:f^{-1}(\varphi)\hookrightarrow\mathcal{F}(\mathcal{C}_{F})^{\rm op}$.
In the proof of Lemma 2.3, we have described the fibre category
$f^{-1}(\varphi)$: namely, $\,f^{-1}(\varphi)$ is isomorphic to the discrete
category $F(c)$ for any $(\varphi:c\to d)\in\mathcal{F}(\mathcal{C})^{\rm
op}$. Now, since both sides of (2.21) are invariant under weak equivalences in
$\mathcal{M}^{\mathcal{C}^{\rm op}}\\!$, we may assume that $X$ is an
objectwise cofibrant diagram in $\mathcal{M}$. Then, since
$\,i^{\ast}t^{\ast}p^{\ast}X=i^{\ast}f^{\ast}t^{\ast}X=(fi)^{\ast}t^{\ast}X=t^{\ast}X(\varphi)=X(d)\,$,
we have

${\rm
hocolim}_{f^{-1}(\varphi)}(i^{\ast}t^{\ast}p^{\ast}X)\simeq\coprod_{F(c)}X(d)\,$

which is precisely the value of $\,\pi^{\ast}(X\otimes F)\,$ at $\varphi$.
Thus, $\bm{L}f_{!}(t^{\ast}p^{\ast}X)\varphi\simeq\pi^{\ast}(X\otimes
F))\varphi$ in $\mathcal{M}$ for all $\varphi\in\mathcal{F}(\mathcal{C})^{\rm
op}$. By Theorem 2.1(2), this implies (2.23). Summing up, we have constructed
the pullback-pushforward diagram

$\begin{diagram}$

each arrow in which is a weak equivalence. This shows that the objects in both
sides of (2.21) are weakly equivalent in $\mathcal{M}$ as claimed by the
proposition. ∎

In the special case, if we take $\mathcal{M}=\mathrm{Ch}(\mathrm{Mod}_{k})$ to
be the category of chain complexes of $k$-modules equipped with standard
projective model structure (see [37, 2.3.11]), Proposition 2.1 implies the
following classical result in homological algebra.

######  Lemma 2.4 (Shapiro Lemma).

Let $k$ be a commutative ring, $\mathcal{C}$ a small category, and
$\mathrm{Mod}_{k}(\mathcal{C}^{\rm op})$ the $($abelian$)$ category of
$\,\mathcal{C}^{\rm op}$-diagrams of $k$-modules. Then, for any
$X\in\mathrm{Mod}_{k}(\mathcal{C})$ and any set-valued functor
$F:\,\mathcal{C}\to\mathrm{Set}$, there is a natural isomorphism

${\rm Tor}^{\mathcal{C}_{F}}_{\ast}(p^{\ast}X,k)\,\cong\,{\rm
Tor}^{\mathcal{C}}_{\ast}(X,k[F])$

where $\,k[F]:\ \mathcal{C}\,\xrightarrow{F}\,{\rm
Set}\,\xrightarrow{k[\mbox{--}\,]}\,{\rm Mod}_{k}$ is the $k$-linear functor
generated by $F$.

Lemma 2.4 appears in [44, Appendix C.12], where it is proved in the special
case $X=k$ the trivial module (the constant $\,\mathcal{C}^{\rm op}$-diagram
valued at $k$); in the general form, it is stated, for example, in [25].

##  3\. Representation and Cyclic Homology of Homotopy Simplicial Groups

In this section, we define representation homology of groups with coefficients
in a commutative Hopf algebra ${\mathcal{H}}$, following the approach of [13,
14]. Taking ${\mathcal{H}}=\mathcal{O}(G)$, where $G$ is an affine algebraic
group, we then construct the derived character maps for $G$-representations of
$\Gamma$. In the case when $G={\rm{GL}}_{n}$, these maps specialize to the
character maps (1.6) announced in the Introduction. We will work with homotopy
simplicial groups (in the sense of Badzioch [4]), which is a more general and
flexible setting than that of the usual (strict) simplicial groups used in
[13, 14]. In Section 3.1, we define the classifying spaces for such groups,
and in Section 3.3, the cyclic bar construction and cyclic homology, both of
which may be of independent interest. We begin by reviewing the main results
of [4] specializing to the algebraic theory of (discrete) groups.

###  3.1. Homotopy simplicial groups

Let $\mathfrak{G}$ denote the small category whose objects $\langle n\rangle$
are the finitely generated free groups $\mathbb{F}_{n}={\mathbb{F}}\langle
x_{1},x_{2},\ldots,x_{n}\rangle$, one for each $n\geq 0$ (with convention that
$\langle 0\rangle$ is the trivial group), and the morphisms are arbitrary
group homomorphisms. Every discrete group $\Gamma$ defines a contravariant
functor $\underline{\Gamma}:\mathfrak{G}^{\rm op}\to{\rm Set}$, $\langle
n\rangle\mapsto\Gamma^{n}$, which is simply the restriction of the Yoneda
functor ${\rm Hom}(\mbox{--},\Gamma):\mathrm{Gr}^{\rm op}\to{\rm Set}$ to
$\mathfrak{G}\subset\mathrm{Gr}$. More generally, every simplicial group
$\Gamma\in\mathrm{sGr}$ (i.e. a simplicial object in $\mathrm{Gr}$) defines a
functor

(3.1) $\underline{\Gamma}:\mathfrak{G}^{\rm
op}\to\mathrm{sSet}\,,\qquad\langle n\rangle\mapsto\Gamma^{n}\,,$

where $\Gamma^{n}$ denotes the product of $n$ copies of the underlying
simplicial set of $\Gamma$ in the category $\mathrm{sSet}$. The functors (3.1)
can be characterized by the property of being product-preserving. To make it
precise, observe that the category $\mathfrak{G}$ carries a (strict) monoidal
structure $\amalg:\mathfrak{G}\times\mathfrak{G}\to\mathfrak{G}$ given by the
coproduct (free product) of free groups: $\langle n\rangle\amalg\langle
m\rangle=\langle n+m\rangle$. The opposite category $\mathfrak{G}^{\rm op}$ is
thus equipped with the dual monoidal structure which we simply denote by
$\Pi:\mathfrak{G}^{\rm op}\times\mathfrak{G}^{\rm op}\to\mathfrak{G}^{\rm
op}$. Every object $\langle n\rangle^{\rm o}\in\mathfrak{G}^{\rm op}$ comes
equipped with $n$ natural projections:

(3.2) $p_{n,k}:\ \langle n\rangle^{\rm o}\to\langle 1\rangle^{\rm o}\,,\qquad
1\leqslant k\leqslant n\,,$

that correspond to the canonical inclusions $i_{n,k}:\,\langle
1\rangle\hookrightarrow\langle n\rangle$, $\,x_{1}\mapsto x_{k}$, in
$\mathfrak{G}$. We say that a functor $\mathcal{F}:\mathfrak{G}^{\rm
op}\to\mathrm{sSet}$ is product-preserving if the maps induced by (3.2)

(3.3) $\mathcal{F}(p_{n}):=\prod_{k=1}^{n}\mathcal{F}(p_{n,k}):\
\mathcal{F}\langle n\rangle\,\to\,(\mathcal{F}\langle 1\rangle)^{n}$

are isomorphisms in $\mathrm{sSet}$ for all $n\geqslant 0$. It is easy to show
that assigning to a simplicial group $\Gamma\in\mathrm{sGr}$ the functor (3.1)
defines an equivalence of categories

(3.4) $\mathrm{sGr}\xrightarrow{\sim}\mathrm{sSet}^{\mathfrak{G}^{\rm
op}}_{\otimes}$

where $\mathrm{sSet}^{\mathfrak{G}^{\rm op}}_{\otimes}$ denotes the full
subcategory of product-preserving functors in the diagram category
$\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$. We will use (3.4) to identify
$\mathrm{sGr}=\mathrm{sSet}^{\mathfrak{G}^{\rm op}}_{\otimes}$, thus regarding
the simplicial groups as functors of the form (3.1). Now, the homotopy
simplicial groups are obtained by replacing the assumption that the maps (3.3)
are isomorphisms in $\mathrm{sSet}$ with that of being weak equivalences,
which is a more natural condition from the point of view of homotopy theory.
Precisely,

######  Definition 3.1 (Badzioch [4]).

A homotopy simplicial group is a functor $\mathcal{F}:\,\mathfrak{G}^{\rm
op}\to\mathrm{sSet}$ that is weakly product-preserving in the sense that the
maps (3.3) are weak equivalences in $\mathrm{sSet}$ for all $n\geqslant 0$
(with convention that $\mathcal{F}\langle 0\rangle\simeq{\rm pt}$).

The category of homotopy simplicial groups (i.e. the full subcategory of all
weakly product-preserving functors in $\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$)
does not carry any model structure as it is not closed under colimits.
Instead, as suggested in [4], one can put a new model structure on the diagram
category $\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$ in which the homotopy
simplicial groups are exhibited as fibrant objects (cf. [4, Proposition 5.5]).
We call this model structure the Badzioch model structure and denote it by
$\mathrm{sGr}^{h}$. To be precise, $\mathrm{sGr}^{h}$ is defined by localizing
(i.e. taking the left Boufield localization of) the standard projective model
structure on $\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$ with respect to the
collection of maps $S=\\{i_{n}:\amalg_{k=1}^{n}{\rm
Hom}_{\mathfrak{G}}(\mbox{--},\langle 1\rangle)\to{\rm
Hom}_{\mathfrak{G}}(\mbox{--},\langle n\rangle)\\}_{n\geq 0}$ induced by the
inclusions $i_{n,k}:\,\langle 1\rangle\to\langle n\rangle$:

$\mathrm{sGr}^{h}\,:=\,\mathcal{L}_{S}(\mathrm{sSet}^{\mathfrak{G}^{\rm op}})\
.$

By definition, the underlying category of $\mathrm{sGr}^{h}$ is that of
$\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$ but the class of weak equivalences in
$\mathrm{sGr}^{h}$ — called $S$-local weak equivalences — is larger: it
includes the set $S$ in addition to all (objectwise) weak equivalences of
diagrams in $\mathrm{sSet}^{\mathfrak{G}^{\rm op}}\\!\\!$. There is a
canonical localization functor
$\mathcal{L}_{S}:\mathrm{sSet}^{\mathfrak{G}^{\rm op}}\to\mathrm{sGr}^{h}$
that takes a diagram $\Gamma\in\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$ to its
functorial fibrant replacement in the model structure $\mathrm{sGr}^{h}$. In
this way, one can make any diagram in $\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$
a homotopy simplicial group. On the other hand, the model category of (strict)
simplicial groups $\mathrm{sGr}$ is related to $\mathrm{sGr}^{h}$ by a Quillen
adjunction:

(3.5) $K:\,\mathrm{sGr}^{h}\rightleftarrows\mathrm{sGr}\,:J$

which is obtained by localizing (at $S$) the Quillen adjunction
$\,K:\mathrm{sSet}^{\mathfrak{G}^{\rm op}}\rightleftarrows\mathrm{sGr}:J\,$
between $\mathrm{sGr}$ and the model category of all diagrams
$\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$. In particular, the right adjoint
functor in (3.5) is given by the inclusion $\,J(\Gamma)=\underline{\Gamma}\,$
(see (3.1)), while the left adjoint — called the Badzioch rigidification
functor — is described explicitly in Lemma 3.1 below. Now, the main result of
[4] reads:

######  Theorem 3.1 (Badzioch).

The adjunction (3.5) is a Quillen equivalence.

######  Remark 3.1.

Theorem 3.1 was proved in [4] (see loc. cit., Theorem 6.4) for an arbitrary
one-sorted algebraic theory (not only the theory of groups). It was extended
to all multi-sorted theories in [17], and further to limit theories and to
diagrams in model categories other than $\mathrm{sSet}$ in [61].

Next, recall that there is a classical adjunction — called the Kan loop group
construction [39] — that relates the model category $\mathrm{sGr}$ of (strict)
simplicial groups to that of (reduced) simplicial sets:

(3.6)
$\mathbb{G}\,:\,\mathrm{sSet}_{0}\rightleftarrows\mathrm{sGr}\,:\,\bar{W}$

The left adjoint $\mathbb{G}$ is called the Kan loop group functor, and the
right adjoint $\bar{W}$ is the classifying complex functor on simplicial
groups. The properties of these functors are well known and discussed in
detail, for example, in [29, Chapter V] (see also [14, Section 2.2]). Here, we
mention only two important facts: first, the pair (3.6) is a Quillen
equivalence, both $\mathbb{G}$ and $\bar{W}$ being homotopy invariant functors
(see [29, V.6.4]). Second, for any reduced simplicial set $X$, there is a weak
homotopy equivalence (see [29, V.5.11])

(3.7) $|\mathbb{G}(X)|\simeq\Omega|X|$

where $\Omega|X|$ is the (Moore) based loop space of the geometric realization
of $X$. The equivalence (3.7) clarifies the topological meaning of the Kan
loop group functor $\mathbb{G}$ (and justifies its name). Combining now
Badzioch’s Theorem 3.1 with Kan’s construction, we get natural equivalences of
homotopy categories

(3.8)
${\mathtt{Ho}}(\mathrm{sGr}^{h})\,\xrightarrow{\bm{L}K}\,{\mathtt{Ho}}(\mathrm{sGr})\,\xrightarrow{\bar{W}}\,{\mathtt{Ho}}(\mathrm{sSet}_{0})\,\xrightarrow{|\,\mbox{--}\,|}\,{\mathtt{Ho}}({\rm
Top}_{0,\ast})$

induced by the above indicated functors. This leads us to the following
definition.

######  Definition 3.2.

For a homotopy simplicial group $\Gamma\in\mathrm{sGr}^{h}$, we define its
classifying space $B\Gamma$ by

(3.9) $B\Gamma\,:=\,|\bar{W}\bm{L}K(\Gamma)|$

where $\bm{L}K:{\mathtt{Ho}}(\mathrm{sGr}^{h})\to{\mathtt{Ho}}(\mathrm{sGr})$
is the derived rigidification functor (see (3.11)).

Note that if $\Gamma$ is a (strict) simplicial group, i.e.
$\Gamma=J(\Gamma)\,$, then $B\Gamma\cong|\bar{W}\Gamma|$, since $\bm{L}K\circ
J\cong{\rm Id}$. Thus the above definition is a natural extension of Kan’s
definition of classifying spaces for simplicial groups (which is, in turn, an
extension of the classical definition of $B\Gamma$ for ordinary discrete
groups).

We conclude this section by giving a simple formula for the Badzioch
rigidification functor that did not seem to appear explicitly in [4].

######  Lemma 3.1.

The functor $K:\mathrm{sGr}^{h}\to\mathrm{sGr}$ in (3.5) is given by the coend

(3.10) $K(\Gamma)=\int^{\langle n\rangle\in\mathfrak{G}}\Gamma\langle
n\rangle\otimes\mathbb{F}\langle n\rangle\,,$

where
$\mathbb{F}:\mathfrak{G}\hookrightarrow\mathrm{Gr}\hookrightarrow\mathrm{sGr},\,\langle
n\rangle\mapsto\mathbb{F}_{n}$, is the canonical inclusion functor, and
$\,\otimes:\mathrm{sSet}\times\mathrm{sGr}\to\mathrm{sGr}\,$ is the standard
simplicial tensor action on the category of simplicial groups.

It follows from Lemma 3.1 that the derived functor $\bm{L}K$ can be written as
the homotopy coend

(3.11) $\bm{L}K(\Gamma)=\int^{\langle
n\rangle\in\mathfrak{G}}_{\bm{L}}\Gamma\langle
n\rangle\otimes\mathbb{F}\langle n\rangle\ .$

For the proof of Lemma 3.1 and formula (3.11) (in the general setting of [4])
we refer to our forthcoming paper [11].

###  3.2. Representation homology

Let $k$ be a commutative ring. Recall that, for a small category
$\mathcal{C}$, we denote by ${\rm Mod}_{k}(\mathcal{C})$ and ${\rm
Mod}_{k}(\mathcal{C}^{\rm op})$ the categories of all covariant and
contravariant functors from $\mathcal{C}$ to ${\rm Mod}_{k}$, respectively. It
is well known that these are abelian categories with sufficiently many
projective and injective objects. Recall also (see, e.g., [44, Appendix C.10])
that there is a natural bi-additive functor

$\mbox{--}\otimes_{\mathcal{C},k}\mbox{--}\,:\,{\rm Mod}_{k}(\mathcal{C}^{\rm
op})\times{\rm Mod}_{k}(\mathcal{C})\to{\rm Mod}_{k}$

called the functor tensor product. Explicitly, for
$\mathcal{M}:\mathcal{C}\to\mathrm{Mod}_{k}$ and $\mathcal{N}:\mathcal{C}^{\rm
op}\to\mathrm{Mod}_{k}$, it is defined by

(3.12)
$\mathcal{N}\otimes_{\mathcal{C},k}\mathcal{M}\,:=\,\big{[}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{c\in\mathcal{C}}\mathcal{N}(c)\otimes_{k}\mathcal{M}(c)\big{]}/R$

where $R$ is the $k$-submodule spanned by elements of the form
$\mathcal{N}(\varphi)x\otimes y-x\otimes\mathcal{M}(\varphi)y$ for all
$x\in\mathcal{N}(c^{\prime})$, $y\in\mathcal{M}(c)$ and $\varphi\in{\rm
Hom}_{\mathcal{C}}(c,c^{\prime})$. The functor (3.12) is right exact (with
respect to each argument), preserves sums, and is left balanced. Its classical
(left) derived functors with respect to each argument are canonically
isomorphic and their common value is denoted by ${\rm
Tor}^{\mathcal{C}}_{\ast}(\mathcal{N},\mathcal{M})$. More generally, we can
extend the bifunctor (3.12) to chain complexes of $\mathcal{C}$-modules, i.e.
the categories ${\rm Ch}({\rm Mod}_{k}(\mathcal{C}^{\rm op}))$ and ${\rm
Ch}({\rm Mod}_{k}(\mathcal{C}))$, and define

(3.13) ${\rm Tor}^{\mathcal{C}}_{\ast}(\mathcal{N},\mathcal{M})\,:=\,{\rm
H}_{\ast}(\mathcal{N}\otimes^{\bm{L}}_{\mathcal{C},k}\mathcal{M})$

for any $\mathcal{N}\in{\rm Ch}({\rm Mod}_{k}(\mathcal{C}^{\rm op}))$ and
$\mathcal{M}\in{\rm Ch}({\rm Mod}_{k}(\mathcal{C}))$. Note that
$\mathcal{N}\otimes^{\bm{L}}_{\mathcal{C},k}\mathcal{M}$ is an object in the
(unbounded) derived category $\mathcal{D}(k)=\mathcal{D}({\rm Mod}_{k})$ of
$k$-modules, and (3.13) is just the usual hyper-Tor functor on chain
complexes. Next, observe that there is a natural functor

(3.14) $\mathrm{sSet}^{\mathcal{C}^{\rm
op}}\,\xrightarrow{\,k[\,\mbox{--}\,]\,}\,{\rm sMod}_{k}({\mathcal{C}}^{\rm
op})\stackrel{{\scriptstyle N}}{{\,\hookrightarrow\,}}\,{\rm Ch}({\rm
Mod}_{k}(\mathcal{C}^{\rm op}))$

transforming the $\mathcal{C}^{\rm op}$-diagrams in $\mathrm{sSet}$
(simplicial presheaves on $\mathcal{C}$) to chain complexes over ${\rm
Mod}_{k}(\mathcal{C}^{\rm op})$. Here $N$ stands for the classical Dold-Kan
normalization functor that identifies simplicial objects in ${\rm
Mod}_{k}(\mathcal{C}^{\rm op})$ with non-negatively graded chain complexes in
${\rm Ch}({\rm Mod}_{k}(\mathcal{C}^{\rm op}))$. Abusing notation, we will
write the functor (3.14) simply as $\,k[\,\mbox{--}\,]\,$.

We are now in a position to define representation homology of homotopy
simplicial groups with coefficients in commutative Hopf algebras. We recall
the well-known fact (see, e.g., [60, Proposition 14.1.6]) that every such
algebra ${\mathcal{H}}$ defines a covariant functor (a left
$\mathfrak{G}$-module) by the rule

(3.15) $\underline{{\mathcal{H}}}:\mathfrak{G}\to{\rm Mod}_{k}\ ,\quad\langle
n\rangle\mapsto{\mathcal{H}}^{\otimes n}\ .$

In particular if $G$ is an affine algebraic group (e.g., $G={\rm GL}_{n}(k)$)
with coordinate ring ${\mathcal{H}}=\mathcal{O}(G)$, then (3.15) can be
written in the form $\langle n\rangle\mapsto\mathcal{O}[{\rm Rep}_{G}(\langle
n\rangle)]$ which makes the functoriality clear.

######  Definition 3.3.

The representation homology of a homotopy simplicial group
$\Gamma\in\mathrm{sGr}^{h}$ with coefficients in ${\mathcal{H}}$ is defined by

${\rm HR}_{\ast}(\Gamma,{\mathcal{H}})\,:=\,{\rm
Tor}^{\mathfrak{G}}_{\ast}(k[\Gamma],\underline{{\mathcal{H}}})\ .$

In the special case when $G$ is an affine algebraic group over $k$ and
${\mathcal{H}}=\mathcal{O}(G)$, we simply write ${\rm HR}_{\ast}(\Gamma,G)$
instead of ${\rm HR}_{\ast}(\Gamma,\mathcal{O}(G))\,$.

The next lemma shows that the above definition agrees with the Badzioch model
structure on $\mathrm{sGr}^{h}$.

######  Lemma 3.2.

If two homotopy simplicial groups $\Gamma$ and $\Gamma^{\prime}$ are weakly
equivalent in $\mathrm{sGr}^{h}$, then

(3.16)
$\mathrm{HR}_{*}(\Gamma,{\mathcal{H}})\,\cong\,\mathrm{HR}_{*}(\Gamma^{\prime},{\mathcal{H}})$

for any commutative Hopf algebra ${\mathcal{H}}$.

###### Proof.

By [4, Proposition 5.6], if two homotopy simplicial groups $\Gamma$ and
$\,\Gamma^{\prime}$ are $S$-locally weakly equivalent, then their underlying
diagrams are, in fact, weakly equivalent in $\mathrm{sSet}^{\mathfrak{G}^{\rm
op}}$. It therefore suffices to show that (3.16) holds for any objectwise weak
equivalent diagrams $\Gamma,\Gamma^{\prime}:\,\mathfrak{G}^{\rm
op}\to\mathrm{sSet}$. To this end, observe that the linearization functor

(3.17) $k[\,\mbox{--}\,]\,:\,\mathrm{sSet}^{\mathfrak{G}^{\rm op}}\to{\rm
sMod}_{k}(\mathfrak{G}^{\rm op})$

is left Quillen with respect to the projective model structures (its right
adjoint is the forgetful functor). Since the weak equivalences in
$\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$ are defined objectwise and the model
structure on $\mathrm{sSet}$ is cofibrant, being left Quillen, the functor
(3.17) is actually homotopy invariant: i.e., it maps weakly equivalent objects
in $\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$ to weakly equivalent objects in
${\rm sMod}_{k}(\mathfrak{G}^{\rm op})$, which, in turn, are transformed by
the normalization functor $N$ to quasi-isomorphic complexes in ${\rm Ch}({\rm
Mod}_{k}(\mathfrak{G}^{\rm op}))$. Thus if $\Gamma\simeq\Gamma^{\prime}$ in
$\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$, then
$\,k[\Gamma]\otimes^{\bm{L}}_{\mathfrak{G},k}\underline{{\mathcal{H}}}\,\simeq\,k[\Gamma^{\prime}]\otimes^{\bm{L}}_{\mathfrak{G},k}\underline{{\mathcal{H}}}\,$
in $\mathcal{D}(k)$, which implies (3.16). ∎

######  Remark 3.2.

The result of Lemma 3.2 does not extend to arbitrary objects in
$\mathrm{sGr}^{h}$, since the functor (3.17) does not preserve $S$-local weak
equivalences in general. The latter can be seen easily by looking at (3.17)
evaluated at representable simplicial presheaves in
$\mathrm{sSet}^{\mathfrak{G}^{\rm op}}$.

An important consequence of Lemma 3.2 is that the representation homology of a
homotopy simplicial group $\Gamma$ depends only on the homotopy type of its
classifying space $B\Gamma$ (Definition 3.2). In fact, we have

(3.18) ${\rm HR}_{\ast}(\Gamma,{\mathcal{H}})\,\cong\,{\rm
HR}_{\ast}(B\Gamma,{\mathcal{H}})$

where the ‘${\rm HR}$’ in the right-hand side stands for representation
homology of topological spaces as defined in [14], using a (non-abelian)
derived representation functor (see loc. cit., Definition 3.1). Indeed, by
Badzioch’s results (cf. [4, Theorem 3.1]), every homotopy simplicial group
$\Gamma$ is weakly equivalent to a strict one, say $\Gamma^{\prime}$; hence

(3.19) $B\Gamma\,\simeq\,B\Gamma^{\prime}\,\simeq\,\bar{W}\Gamma^{\prime}\ .$

On the other hand, by [14, Theorem 4.2], $\,{\rm
HR}_{\ast}(\Gamma^{\prime},{\mathcal{H}})\cong{\rm
HR}_{\ast}(\bar{W}\Gamma^{\prime},{\mathcal{H}})\,$, which together with
(3.19) and the isomorphism (3.16) of Lemma 3.2 implies (3.18).

We conclude this section by briefly explaining how our approach (Definition
3.3) relates to derived algebraic geometry (DAG). For a model of DAG, we will
take the simplicial presheaf model developed in [66]. Given a homotopy
simplicial group $\Gamma\in\mathrm{sGr}^{h}$ and an affine algebraic group
(scheme) $G$ over $k$ with coordinate algebra ${\mathcal{H}}=\mathcal{O}(G)$,
we introduce the derived representation scheme of $\Gamma$ in $G$:

(3.20)
${\rm{DRep}}_{G}(\Gamma):=\bm{\mathrm{R}}{\rm{Spec}}\,(k[\Gamma]\otimes^{\bm{L}}_{\mathfrak{G}}\mathcal{O}(G))\
.$

Here ‘$\,\bm{\mathrm{R}}{\rm{Spec}}\,$’ denotes the Toën-Vezzosi derived
Yoneda functor that assigns to a (homotopy) simplicial commutative algebra $A$
— a derived ring in terminology of [66] — the simplicial presheaf (prestack)

$\bm{\mathrm{R}}{\rm{Spec}}(A):\,\mathrm{dAff}_{k}^{\rm
op}:=\mathrm{sComm}_{k}\ \to\ \mathrm{sSet}\ ,\quad
B\,\mapsto\,{\mathrm{Map}}(QA,\,B)\ ,$

where $QA$ is a cofibrant replacement of $A$ and ‘$\mathrm{Map}$’ is the
simplicial mapping space (function complex) in $\mathrm{sComm}_{k}$. The
prestack $\bm{\mathrm{R}}{\rm{Spec}}(A)$ satisfies the descent condition for
étale hypercoverings and hence defines a derived stack (which is a derived
affine scheme in the sense of [66]). On the other hand, for any pointed space
(simplicial set) $X$, we can define the pointed mapping stack
$\bm{\mathrm{Map}}_{*}(X,BG)$ to be the homotopy fibre of the canonical map in
the (homotopy) category of derived stacks:

(3.21) $\bm{\mathrm{Map}}_{*}(X,BG):={\rm
hofib}\,[\bm{\mathrm{Map}}(X,BG)\,\to\,BG]\ ,$

where $\bm{\mathrm{Map}}(X,BG)$ stands for the (unpointed) derived mapping
stack defined in [66, 2.2.6.2]. This last mapping stack is a basic object of
derived algebraic geometry that plays an important role in applications (see,
e.g., [51]). Now, its relation to representation homology is clarified by the
following

######  Proposition 3.1 (see [12]).

There is a $($weak$)$ equivalence of derived stacks

${\rm{DRep}}_{G}(\Gamma)\,\simeq\,\bm{\mathrm{Map}}_{*}(B\Gamma,BG)$

For a detailed proof of Proposition 3.1 and more explanations we refer to [12,
Appendix A.1].

###  3.3. Cyclic homology

We now define cyclic homology for homotopy simplicial groups. To this end, we
will associate to each $\Gamma\in\mathrm{sGr}^{h}$ a cyclic module $k[B^{\rm
cyc}\Gamma]$ that generalizes the classical cyclic bar construction
$C_{\ast}(k[\Gamma])$ when $\Gamma$ is an ordinary discrete group.

Let $\Delta$ denote the standard (co)simplicial category whose objects are
finite ordered sets $[n]=\\{0<1<\cdots<n\\}$ and morphisms are (nonstrictly)
order preserving maps. The category $\Delta$ is generated by two families of
maps $d^{i}_{n}:[n-1]\to[n]$ ($0\leqslant i\leqslant n$, $n\geqslant 1$) and
$s^{j}_{n}:[n+1]\to[n]$ ($0\leqslant j\leqslant n$, $n\geqslant 0$), called
the (co)face and (co)degeneracy maps respectively. These maps satisfy the
standard (co)simplicial relations listed, for example, in [44, Appendix B.3].
Connes’ cyclic category $\Delta C$ is a natural extension of $\Delta$ that has
the same objects and is generated by the morphisms of $\Delta$ and the cyclic
maps $\tau_{n}:[n]\to[n]$, $n\geqslant 0$, satisfying $\tau_{n}^{n+1}={\rm
Id}$ (see [44, 6.11]). In fact, $\Delta C$ can be characterized by the two
properties:

  1. (Cyc1)

For each $n\geqslant 0$, ${\rm Aut}_{\Delta C}([n])\cong C_{n+1}\,$, where
$C_{n+1}=\mathbb{Z}/(n+1)\mathbb{Z}\,$,

  2. (Cyc2)

Any morphism $f:[n]\to[m]$ in $\Delta C$ can be factored uniquely as
$f=g\circ\varphi$, where $g\in{\rm Hom}_{\Delta}([n],[m])$ and $\varphi\in{\rm
Aut}_{\Delta C}([n])\,$,

These properties show that $\Delta C$ is a crossed simplicial category
associated to the family of cyclic groups $\\{C_{n+1}\\}_{n\geqslant 0}$ (see
[44, 6.3.0]).

Now, if $\Gamma$ is an ordinary discrete group, there is a natural functor

(3.22) $B^{\rm cyc}_{\ast}\Gamma\,:\,\Delta C^{\rm op}\to\mathrm{Set}$

called the cyclic bar construction of $\Gamma$ that has the property that
$k[B^{\rm cyc}_{\ast}\Gamma]\cong C_{\ast}(k[\Gamma])$, where
$C_{\ast}(k[\Gamma])$ is the standard cyclic module associated to $k[\Gamma]$
as an associative $k$-algebra. Explicitly, the functor (3.22) is defined by
(see [44, 7.3.10])

$\displaystyle d_{i}(g_{0},\ldots,g_{n})$ $\displaystyle=$
$\displaystyle\begin{cases}(g_{0},\ldots,g_{i-1},g_{i}g_{i+1},\ldots,g_{n})&0\leqslant
i<n\\\ (g_{n}g_{0},g_{1},\ldots,g_{n-1})&i=n\end{cases}$ $\displaystyle
s_{j}(g_{0},\ldots,g_{n})$ $\displaystyle=$
$\displaystyle(g_{0},\ldots,g_{j},1,g_{j+1},\ldots,g_{n})$ $\displaystyle
t_{n}(g_{0},\ldots,g_{n})$ $\displaystyle=$
$\displaystyle(g_{n},g_{0},g_{1},\ldots,g_{n-1})$

where $(g_{0},\ldots,g_{n})\in\Gamma^{n+1}$. Clearly, $\Gamma\mapsto B^{\rm
cyc}_{\ast}\Gamma$ gives a functor $B^{\rm
cyc}_{\ast}:\mathrm{Gr}\to\mathrm{Set}^{\Delta C^{\rm op}}$. If we identify
$\mathrm{Gr}=\mathrm{Set}^{\mathfrak{G}^{\rm op}}_{\otimes}$ as in (3.1), then
it turns out that $B^{\rm cyc}_{\ast}$ coincides with the pull-back functor
for a certain natural map $\Psi_{\rm cyc}:\Delta C\to\mathfrak{G}$ in ${\rm
Cat}$. Specifically,

(3.23) $\Psi_{\rm cyc}:\Delta C\to\mathfrak{G}$

is defined on objects by

$\Psi_{\rm cyc}([n])\,:=\,\langle n+1\rangle\,=\,\mathbb{F}\langle
x_{0},\ldots,x_{n}\rangle$

and on morphisms by the following formulas

(3.24) $\left.\begin{aligned} \Psi_{\rm cyc}(d^{i}_{n})&:\langle
n\rangle\xrightarrow{}\langle
n+1\rangle\,,\qquad(x_{0},x_{1},\ldots,x_{n-1})\mapsto\begin{cases}(x_{0},\ldots,x_{i-1},x_{i}x_{i+1},\ldots,x_{n})\
,&0\leq i<n\\\ (x_{n}x_{0},x_{1},\ldots,x_{n-1})\ ,&i=n\end{cases}\\\
\Psi_{\rm cyc}(s^{j}_{n})&:\langle n+2\rangle\xrightarrow{}\langle
n+1\rangle\,,\qquad(x_{0},\ldots,x_{n+1})\mapsto(x_{0},\ldots,x_{j},1,x_{j+1},\ldots,x_{n})\
,\\\ \Psi_{\rm cyc}(\tau_{n})&:\langle n+1\rangle\xrightarrow{}\langle
n+1\rangle\,,\qquad(x_{0},x_{1},\ldots,x_{n})\mapsto(x_{n},x_{0},x_{1},\ldots,x_{n-1})\
.\\\ \end{aligned}\right.$

where $(x_{0},x_{1},\ldots,x_{n})$ is an ordered sequence of generators of the
free group $\mathbb{F}\langle x_{0},\ldots,x_{n}\rangle$.

######  Lemma 3.3.

For any discrete group $\Gamma$ there is a natural isomorphism of cyclic sets

$B^{\rm cyc}\Gamma\,\cong\,\Psi^{\ast}_{\rm cyc}(\underline{\Gamma})$

where $\underline{\Gamma}:\mathfrak{G}^{\rm op}\to\mathrm{Set}$ is the functor
corresponding to $\Gamma$ under the identification (3.1).

###### Proof.

Straightforward. ∎

######  Remark 3.3.

The functor (3.23) was defined in [16] on a slightly larger – the so-called
epicyclic – category $\Delta\Psi$, which is an extension of $\Delta C$
describing the Adams operations on cyclic modules.

Lemma 3.3 motivates the following definition.

######  Definition 3.4.

For a homotopy simplicial group $\Gamma\in\mathrm{sGr}^{h}$, we define its
cyclic bar construction by

(3.25) $B^{\rm cyc}\Gamma\,:=\,\Psi^{\ast}_{\rm cyc}(\Gamma):\ \Delta C^{\rm
op}\to\mathrm{sSet}$

and its cyclic homology by

(3.26) ${\rm HC}_{\ast}(k[\Gamma])\,:=\,{\rm{Tor}}^{\Delta C^{\rm
op}}_{\ast}(k,\,k[B^{\rm cyc}\,\Gamma])\,\cong\,{\rm{Tor}}^{\Delta
C}_{\ast}(k[B^{\rm cyc}\,\Gamma],\,k)$

The same argument as in (the proof of) Lemma 3.2 shows that ${\rm
HC}_{\ast}(k[\Gamma])$ depends only on the homotopy type of $\Gamma$ in the
Badzioch model category $\mathrm{sGr}^{h}$, and hence, on the homotopy type of
its classifying space $B\Gamma$. In view of Lemma 3.3, the above definition of
${\rm HC}_{\ast}(k[\Gamma])$ for $\Gamma$ an ordinary discrete group coincides
with the classical (Connes’) definition of cyclic homology of group algebras
(see [44, 6.2.8]).

###  3.4. Derived character maps

Next, we will construct a family of natural transformations relating the
cyclic homology to representation homology of a homotopy simplicial group
$\Gamma$. In the special case when ${\mathcal{H}}=\mathcal{O}({\rm{GL}}_{n})$,
this family contains a distinguished element determined by the usual trace
${\rm{Tr}}_{n}$ that gives the derived character map (1.6) announced in the
Introduction. With our current definitions of representation and cyclic
homology the construction is actually very simple. It is based on two lemmas.
The first one is a standard result of homological algebra that simply exhibits
the naturality of derived tensor products (3.13).

######  Lemma 3.4.

Let $f:{\mathcal{A}}\to\mathcal{B}$ be a functor between small categories. For
any complexes $\mathcal{N}\in\mathrm{Ch}(\mathrm{Mod}_{k}\,\mathcal{B}^{\rm
op})$ and $\mathcal{M}\in\mathrm{Ch}(\mathrm{Mod}_{k}\,\mathcal{B})$, there is
a natural map
$\,f^{*}\mathcal{N}\otimes^{\bm{L}}_{{\mathcal{A}},k}f^{*}\mathcal{M}\to\mathcal{N}\otimes^{\bm{L}}_{\mathcal{B},k}\mathcal{M}\,$
in the derived category $\mathcal{D}(k)$ of $k$-modules that induces

$f^{*}:\
{\rm{Tor}}^{{\mathcal{A}}}_{\ast}(f^{*}\mathcal{N},\,f^{*}\mathcal{M})\,\to\,{\rm{Tor}}^{\mathcal{B}}_{\ast}(\mathcal{N},\,\mathcal{M})$

To apply this lemma in our situation we recall that every commutative Hopf
$k$-algebra ${\mathcal{H}}$ defines the covariant functor
$\underline{{\mathcal{H}}}:\mathfrak{G}\to\mathrm{Mod}_{k}$ by formula (3.15).
Restricting this functor via the morphism (3.23) gives rise to a cocyclic
$k$-module that we denote

$B_{\rm cyc}{\mathcal{H}}:=\Psi_{\rm cyc}^{\ast}(\underline{{\mathcal{H}}}):\
\Delta C\to{\rm Mod}_{k}$

On the other hand, by Definition 3.4, $\,\Psi_{\rm
cyc}^{*}(k[\Gamma])=k[B^{\rm cyc}(\Gamma)]\,$ for any homotopy simplicial
group $\Gamma$. Thus, by Lemma 3.4, the functor $\Psi_{\rm cyc}$ induces a
canonical map

(3.27) $\Psi^{*}_{\rm cyc}:\ {\rm{Tor}}^{\Delta C}_{\ast}(k[B^{\rm
cyc}\,\Gamma],\,B_{\rm cyc}{\mathcal{H}})\ \to\
{\rm{Tor}}^{\mathfrak{G}}_{*}(k[\Gamma],\,\underline{{\mathcal{H}}})$

The target of this map is precisely $\mathrm{HR}_{*}(\Gamma,{\mathcal{H}})$
(see Definition 3.3), while the domain differs from
$\mathrm{HC}_{*}(k[\Gamma])$ in the second argument of ‘Tor’ (cf. Definition
3.4). To connect the two Tor’s we will use the following lemma which we state
in the language of affine algebraic groups.

######  Lemma 3.5.

Let $G$ be an affine algebraic group defined over $k$, and let
$\mathcal{O}(G)$ be its coordinate algebra. There is a natural isomorphism

(3.28) ${\rm{Hom}}_{\mathrm{Mod}_{k}(\Delta C)}(k,\ B_{\rm
cyc}[\mathcal{O}(G)])\,\cong\,\mathcal{O}(G)^{G}$

where $\mathcal{O}(G)^{G}$ denotes the invariant subalgebra of
$\mathcal{O}(G)$ under the adjoint $G$-action.

###### Proof.

For $m\geq 0$, denote by $\pi_{m}:[0]\to[m]$ the composition of maps
$\,d^{0}_{m}\,d^{0}_{m-1}\,\ldots\,d^{0}_{1}\,$ in $\Delta C$. It follows from
(3.24) that $\Psi_{\rm cyc}(\pi_{m}):\langle 1\rangle\to\langle m+1\rangle$ is
the homomorphism of groups taking the generator $x$ of ${\mathbb{F}}\langle
x\rangle$ to the product of generators $x_{0}x_{1}\ldots x_{m}$ in
${\mathbb{F}}\langle x_{0},\ldots,x_{m}\rangle$. The corresponding map
$[B_{\rm
cyc}\mathcal{O}(G)](\pi_{m}):\mathcal{O}(G)\to\mathcal{O}(G)^{\otimes(m+1)}$
can thus be identified with the the $m$-fold coproduct in $\mathcal{O}(G)$:

(3.29) $\Delta_{G}^{(m)}:\ \mathcal{O}(G)\to\mathcal{O}(G^{m+1})\,,\quad
P\mapsto\big{[}(g_{0},g_{1},\ldots,g_{m})\mapsto
P(g_{0}\,g_{1}\,\ldots\,g_{m})\big{]}\ .$

Now, it is easy to check that, for a fixed $P\in\mathcal{O}(G)^{G}$, the maps
$\Delta_{G}^{(m)}(P):\,k\to\mathcal{O}(G^{m+1})$ taking $1\in k$ to
$\Delta_{G}^{(m)}(P)$ assemble to a morphism of cocylic modules
$\,\Delta_{G}(P):\ k\,\to\,B_{\rm cyc}[\mathcal{O}(G)]\,$, the commutativity
with cyclic operators $\tau_{m}$ being a consequence of the $G$-invariance of
$P$. We claim that the assignment $\,P\mapsto\Delta_{G}(P)\,$ defines a
$k$-linear isomorphism

(3.30) $\Delta_{G}:\
\mathcal{O}(G)^{G}\,\xrightarrow{\sim}\,{\rm{Hom}}_{\mathrm{Mod}_{k}(\Delta
C)}(k,\ B_{\rm cyc}[\mathcal{O}(G)])\ .$

The inverse of (3.30) can be constructed as follows. Let
$\varphi\in{\rm{Hom}}_{\mathrm{Mod}_{k}(\Delta C)}(k,\ B_{\rm
cyc}[\mathcal{O}(G)])$. Note that, for all $[m]\in\Delta C$, its components
$\varphi_{[m]}:\,k\to\mathcal{O}(G)^{\otimes(m+1)}\cong\mathcal{O}(G^{m+1})$
are $k$-linear maps. Define $T\varphi:=\varphi_{[0]}(1)\in\mathcal{O}(G)$,
where $1\in k$. Since $\varphi$ is a natural transformation,

$\varphi_{[m]}(1)=\\{[B_{\rm
cyc}\mathcal{O}(G)](\pi_{m})\\}(\varphi_{[0]}(1))=\\{[B_{\rm
cyc}\mathcal{O}(G)](\pi_{m})\\}(T\varphi)=\Delta^{m}(T\varphi)\ ,$

where $\Delta^{m}$ is defined in (3.29). Similarly, applying $B_{\rm
cyc}[\mathcal{O}(G)]$ to the cyclic operators $\tau_{m}$ in $\Delta C$, we
have

$\varphi_{[m]}(1)=\\{[B_{\rm
cyc}\mathcal{O}(G)](\tau_{m})\\}(\varphi_{[m]}(1))\,,$

from which it follows that $T\varphi(g_{0}\ldots
g_{m})=T\varphi(g_{m}g_{0}\ldots g_{m-1})$ for all $g_{0},\ldots,g_{m}\in G$.
This is equivalent to the assertion that $T\varphi\in\mathcal{O}(G)^{G}$. Thus
$T$ defines a $k$-linear map

${\rm{Hom}}_{\mathrm{Mod}_{k}(\Delta C)}(k,\ B_{\rm
cyc}[\mathcal{O}(G)])\to\mathcal{O}(G)^{G}\,,\qquad\varphi\mapsto T\varphi\ .$

It is clear from its construction that the above map is the inverse of (3.30).
∎

We can now make the following definition.

######  Definition 3.5.

Let $\Gamma\in\mathrm{sGr}^{h}$ be a homotopy simplicial group. For an affine
algebraic group $G$ and an ${\rm{Ad}}\,G$-invariant polynomial
$P\in\mathcal{O}(G)^{G}$, we define the derived $G$-character map of
$\,\Gamma\,$ associated to $P$ by

(3.31) $\chi_{G,P}(\Gamma)_{\ast}:\
\mathrm{HC}_{*}(k[\Gamma])\,\xrightarrow{(\Delta_{G}P)_{*}}\,{\rm{Tor}}^{\Delta
C}_{\ast}(k[B^{\rm cyc}\,\Gamma],\,B_{\rm
cyc}[\mathcal{O}(G)])\,\xrightarrow{\Psi_{\rm
cyc}^{*}}\,\mathrm{HR}_{\ast}(\Gamma,G)\ ,$

where $(\Delta_{G}P)_{*}$ is a linear map induced by the map of cocyclic
modules $\Delta_{G}P:\,k\to B_{\rm cyc}[\mathcal{O}(G)]$ (see (3.29) and
(3.30)), and $\Psi_{\rm cyc}^{*}$ is the map (3.27) defined for
${\mathcal{H}}=\mathcal{O}(G)$.

Explicitly, if we choose a projective resolution
$\,Q\xrightarrow{\sim}k[\Gamma]\,$ of $k[\Gamma]$ in the (abelian) category
$\mathrm{Mod}_{k}(\mathfrak{G}^{\rm op})$, applying the functor $\Psi_{\rm
cyc}^{*}$ gives a projective resolution $\,\Psi_{\rm
cyc}^{*}Q\xrightarrow{\sim}k[B^{\rm cyc}\,\Gamma]\,$ of the cyclic module
$k[B^{\rm cyc}\,\Gamma]$ in $\mathrm{Mod}_{k}(\Delta C^{\rm op})$. The map
(3.31) is then induced by a map of chain complexes

(3.32) $\chi_{G,P}(\Gamma)_{\ast}:\ (\Psi_{\rm cyc}^{*}Q)\otimes_{\Delta
C}k\,\to\,Q\otimes_{\mathfrak{G}}\mathcal{O}(G)$

which, in turn, is induced by the following map (see (3.12))

(3.33)
$\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{[m]\in\Delta
C}\,Q\langle m+1\rangle\ \to\
\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{\langle
n\rangle\in\mathfrak{G}}\,Q\langle n\rangle\otimes\mathcal{O}(G)^{\otimes n}\
,\quad v_{m+1}\,\mapsto\,v_{m+1}\otimes\Delta_{G}^{(m)}(P)\ ,$

where $v_{m+1}\in Q\langle m+1\rangle$ and
$\Delta_{G}^{(m)}(P)\in\mathcal{O}(G)^{\otimes(m+1)}$ is defined by (3.29).

In the special case when $G={\rm{GL}}_{n}(k)$ and
$P={\rm{Tr}}_{n}\in\mathcal{O}({\rm{GL}}_{n})$ is the usual trace function on
$(n\times n)$-matrices, we denote the map (3.31) by

(3.34) ${\rm{Tr}}_{n}(\Gamma)_{\ast}:\
\mathrm{HC}_{*}(k[\Gamma])\,\to\,\mathrm{HR}_{\ast}(\Gamma,{\rm{GL}}_{n}(k))\,,$

and call it the derived character map of $n$-dimensional representations of
$\,\Gamma$. In the rest of the paper, we will study the maps
${\rm{Tr}}_{n}(\Gamma)_{\ast}$ in two extreme cases: $\,n=1\,$ and
$\,n=\infty\,$. In the first case, we will give a topological realization of
${\rm{Tr}}(\Gamma)_{\ast}:={\rm{Tr}}_{1}(\Gamma)_{*}$ by showing that this map
is induced on homology by a natural map of topological spaces; in the second
case, we will show that
${\rm{Tr}}_{\infty}(\Gamma)_{\ast}:=\varprojlim\,{\rm{Tr}}_{n}(\Gamma)_{\ast}\,$
extends to an isomorphism between the graded symmetric algebra generated by
$\overline{\mathrm{HC}}_{*}(k[\Gamma])$ and the ${\rm{GL}}_{\infty}$-invariant
subalgebra of the stable representation homology
$\mathrm{HR}_{*}(\Gamma,{\rm{GL}}_{\infty}(k))$. We close this section with a
general remark linking the above construction to earlier work.

######  Remark 3.4.

If $\Gamma$ is an ordinary discrete or (strict) simplicial group, then
$k[\Gamma]$ is naturally a simplicial associative $k$-algebra. By (a monoidal
version of) the classical Dold-Kan correspondence (see [62]), we can therefore
view $k[\Gamma]$ as a differential-graded (DG) associative $k$-algebra. For
such algebras (defined over a field $k$ of characteristic $0$), the derived
character maps of $n$-dimensional representations were constructed in [9]. One
can show that these maps agree with (3.34) in the case of group algebras,
although the comparison is not entirely trivial as the methods used in [9] and
the present paper are quite different. We will address this question in our
forthcoming paper [11] in a greater generality.

##  4\. Topological Realization of Derived Character Maps

In this and next sections, we will prove our main results (Theorem 1.1 and
Theorem 1.2) stated in the Introduction. Here we will construct the required
spaces and maps simplicially: in terms of homotopy colimits of small diagrams
of simplicial sets and associated natural maps. Then, in the next section, we
will reproduce these maps in topological terms, using Goodwillie homotopy
calculus and topological operads. The connection between the two approaches
seems instructive and deserves a further investigation.

###  4.1. The space $X_{\Gamma}$

Recall that $\mathfrak{G}$ denotes the skeleton of the category of finitely
generated free groups. There is a natural abelianization functor

(4.1) $\underline{\mathbb{Z}}\,:\,\mathfrak{G}\to\mathrm{Set}\,,\qquad\langle
n\rangle\mapsto\mathbb{Z}^{n}\ ,$

that takes the free group $\langle n\rangle={\mathbb{F}}_{n}$ to (the
underlying set of) its abelianization $\langle n\rangle_{\rm
ab}=\mathbb{Z}^{n}$. As in Section 2, we can form the category of elements of
(4.1), using the Grothendieck construction:

(4.2)
$\mathfrak{G}_{\mathbb{Z}}\,:=\,\mathfrak{G}\\!\smallint\underline{\mathbb{Z}}$

The objects of $\mathfrak{G}_{\mathbb{Z}}$ are given explicitly by

${\rm Ob}(\mathfrak{G}_{\mathbb{Z}})\,=\,\\{(\langle
n\rangle;k_{1},\ldots,k_{n})):\langle
n\rangle\in\mathfrak{G}\,,\,(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}\\}$

and the morphism sets are

${\rm Hom}_{\mathfrak{G}_{\mathbb{Z}}}((\langle
n\rangle;k_{1},\ldots,k_{n}),(\langle
m\rangle;l_{1},\ldots,l_{m}))\,=\,\\{\varphi\in{\rm
Hom}_{\mathfrak{G}}(\langle n\rangle,\langle m\rangle):\ \varphi_{\rm
ab}(k_{1},\ldots,k_{n})=(l_{1},\ldots,l_{m})\\}$

Note that the abelianized map $\,\varphi_{\rm
ab}:\mathbb{Z}^{n}\to\mathbb{Z}^{m}\,$ above is represented by an integral
$(m\times n)$-matrix, $\varphi_{\rm ab}\in{\mathbb{M}}_{m\times
n}(\mathbb{Z})$, and its action on $n$-tuples of integers is simply given by
matrix multiplication. The category (4.2) comes together with the canonical
(forgetful) functor

(4.3) $p:\mathfrak{G}_{\mathbb{Z}}\to\mathfrak{G}\,,\qquad(\langle
n\rangle;k_{1},\ldots,k_{n})\mapsto\langle n\rangle\ .$

Given a homotopy simplicial group $\Gamma:\mathfrak{G}^{\rm
op}\to\mathrm{sSet}$, we now define

(4.4) $X_{\Gamma}\,:=\,|\,\mathrm{hocolim}_{\mathfrak{G}_{\mathbb{Z}}^{\rm
op}}(p^{\ast}\Gamma)\,|\,,$

where $p^{\ast}$ is the pullback functor $\mathrm{sSet}^{\mathfrak{G}^{\rm
op}}\to\mathrm{sSet}^{\mathfrak{G}_{\mathbb{Z}}^{\rm op}}$ associated to
(4.3). The relation of the space (4.4) to representation homology becomes
clear from the following observation.

######  Lemma 4.1.

For any $\Gamma$ and any commutative ring $k$, there is a natural isomorphism

(4.5) ${\rm
H}_{\ast}(X_{\Gamma},k)\,\cong\,\mathrm{HR}_{\ast}(\Gamma,\mathbb{G}_{m}(k))$

###### Proof.

We have the sequence of natural isomorphisms

$H_{\ast}(X_{\Gamma},k)\,\cong\,{\rm Tor}^{\mathfrak{G}_{\mathbb{Z}}^{\rm
op}}_{\ast}(k,\,k[p^{\ast}\Gamma])\,\cong\,{\rm
Tor}^{\mathfrak{G}_{\mathbb{Z}}}_{\ast}(k[p^{\ast}\Gamma],\,k)\,=\,{\rm
Tor}^{\mathfrak{G}_{\mathbb{Z}}}_{\ast}(p^{\ast}k[\Gamma],\,k)\,\cong\,{\rm
Tor}^{\mathfrak{G}}_{\ast}(k[\Gamma],\,k[\mathbb{Z}])\ ,$

where the first two are standard (see, e.g., [44, Appendix C.10]) and the last
one follows from the classical Shapiro Lemma (see Lemma 2.4). To complete the
proof it remains to note that $k[\mathbb{Z}]$ can be identified with
$\mathcal{O}[{\mathbb{G}}_{m}(k)]$ as a commutative Hopf algebra. ∎

As in the Introduction, we shorten notation for one-dimensional representation
homology, writing

(4.6)
$\mathrm{HR}_{\ast}(k[\Gamma])\,:=\,\mathrm{HR}_{\ast}(\Gamma,\mathbb{G}_{m}(k))\
.$

Our next goal is to identify the homotopy type of the space $X_{\Gamma}$ in
terms of the classifying space of $\Gamma$. The following theorem is one of
the main results of the present paper.

######  Theorem 4.1.

For any homotopy simplicial group $\Gamma$, there is a weak equivalence in
${\rm Top}_{\ast}\\!:$

(4.7) $X_{\Gamma}\,\simeq\,\Omega\,{\rm SP}^{\infty}(B\Gamma)$

where $B\Gamma$ is the classifying space of $\,\Gamma$ $($see Definition
3.2$)$.

Before proving this theorem, we recall a few basic facts about the Dold-Thom
space and related constructions (see, e.g., [35, Chap. 4.K]). For any pointed
connected CW complex $X$, the Dold-Thom space ${\rm SP}^{\infty}(X)$ is
defined as the infinite symmetric product: namely,

(4.8) ${\rm SP}^{\infty}(X)\,=\,\varinjlim_{n}\,{\rm SP}^{n}(X)$

where ${\rm SP}^{n}(X):=X^{n}\\!/S_{n}$ with $S_{n}$ acting on $X^{n}$ the
natural way (by permuting the factors). The maps ${\rm SP}^{n}(X)\to{\rm
SP}^{n+1}(X)$ along which the inductive limit (4.8) is taken are induced by
the natural inclusion

$X^{n}\hookrightarrow
X^{n+1}\,,\qquad(x_{1},\ldots,x_{n})\mapsto(x_{1},\ldots,x_{n},\ast)\ ,$

where ‘$\ast$’ stands for the basepoint of $X$. The Dold-Thom Theorem asserts
that, for all $i\geq 1$, there are isomorphisms of abelian groups

$\pi_{i}[{\rm SP}^{\infty}(X)]\,\cong\,H_{i}(X,\mathbb{Z})\,,$

that are natural in pointed connected CW complexes $X$. In fact, this
classical theorem provides a topological realization for the Hurewicz
homomorphisms in the sense that the natural map of spaces

(4.9) $X={\rm SP}^{1}(X)\hookrightarrow{\rm SP}^{\infty}(X)$

induces the homomorphisms of groups:
$\,\pi_{i}(X)\to\mathrm{H}_{i}(X,\mathbb{Z})\,$ for all $i\geqslant 1$.

Now, let ${\mathcal{F}}X$ denote the homotopy fibre of the inclusion map (4.9)
so that we have a homotopy fibration sequence

(4.10) ${\mathcal{F}}X\to X\to{\rm SP}^{\infty}(X)\ .$

There is an alternative way to obtain this fibration sequence, using Kan’s
simplicial group model $\mathbb{G}(X)$ of the space555Abusing notation, we
will identify a pointed connected space $X$ with a reduced simplicial set that
represents $X$, i.e. $X=|X|$. $X$. Namely (see, e.g., [5, Section 7]), (4.10)
arises from the short exact sequence of simplicial groups

(4.11) $1\to\mathbb{G}_{2}(X)\to\mathbb{G}{(X)}\to\mathbb{A}(X)\to 1$

by applying the classifying space functor $\,B=|\bar{W}(\mbox{--})|\,$. Here
$\,\mathbb{G}_{2}(X):=[\mathbb{G}(X),\mathbb{G}(X)]\,$ denotes the commutator
subgroup of the Kan loop group $\mathbb{G}(X)$ and $\mathbb{A}(X)$ its
abelianization:

(4.12) $\mathbb{A}(X)\,:=\,(\mathbb{G}{X})_{\rm
ab}\,:=\,\mathbb{G}(X)/\mathbb{G}_{2}(X)$

Thus, we have $\,{\rm SP}^{\infty}(X)\simeq
B\mathbb{A}(X)=|\bar{W}\mathbb{A}(X)|\,$, which, by Kan’s Theorem (see (3.7)),
implies

(4.13) $\Omega\,{\rm
SP}^{\infty}(X)\,\simeq\,\Omega\,|\bar{W}\mathbb{A}(X)|\,\simeq\,|\mathbb{G}\bar{W}\mathbb{A}(X)|\,\simeq\,|\mathbb{A}(X)|\
.$

Note that for any reduced simplicial set $X$,
$\,{\mathbb{A}}(X)\cong\tilde{\mathbb{Z}}[X]\,$ is just the reduced free
simplicial abelian group generated by $X$. After these preliminary remarks we
can proceed with

###### Proof of Theorem 4.1.

As a first step we apply Proposition 2.1 to express the homotopy colimit (4.4)
as a homotopy coend:

(4.14) $\mathrm{hocolim}_{\mathfrak{G}_{\mathbb{Z}}^{\rm
op}}(p^{\ast}\Gamma)\,\simeq\,\int_{\bm{L}}^{\langle
n\rangle\in\mathfrak{G}}\\!\Gamma\langle
n\rangle\times\underline{\mathbb{Z}}^{n}$

Next, observe that the bifunctor

(4.15) $\Gamma\times\underline{\mathbb{Z}}:\ \mathfrak{G}^{\rm
op}\times\mathfrak{G}\,\to\,\mathrm{sSet}\ ,\quad(\langle n\rangle,\langle
m\rangle)\,\mapsto\,\Gamma\langle n\rangle\times\mathbb{Z}^{m}\ ,$

that appears in the homotopy coend (4.14) can be factored as

$\mathfrak{G}^{\rm
op}\times\mathfrak{G}\,\xrightarrow{\,\Gamma\,\otimes\,{\mathbb{F}}\,}\,\mathrm{sGr}\,\xrightarrow{\,(\,\mbox{--}\,)_{\rm
ab}\,}\,{\rm sAb}\,\xrightarrow{{\rm forget}}\,\mathrm{sSet}$

where the first arrow is precisely the bifunctor
$\,\Gamma\,\otimes\,{\mathbb{F}}\,$ that appears in formula (3.10) of Lemma
3.1, expressing the rigidification functor $K$. This last bifunctor takes an
object $(\langle n\rangle,\langle m\rangle)\in\mathfrak{G}^{\rm
op}\times\mathfrak{G}$ to the simplicial group $\,\amalg_{\Gamma\langle
n\rangle}\,{\mathbb{F}}_{m}\,$, which is given, in each simplicial degree, by
a free product of copies of the free group ${\mathbb{F}}_{m}$ indexed by the
components of the simplicial set $\Gamma\langle n\rangle$. Hence
$\,\Gamma\,\otimes\,{\mathbb{F}}\,$ is an objectwise cofibrant diagram in
$\mathrm{sGr}$, and therefore

(4.16) $\bm{L}(\Gamma\,\otimes\,{\mathbb{F}})_{\rm
ab}\,\simeq\,(\Gamma\,\otimes\,{\mathbb{F}})_{\rm
ab}\,\cong\,\Gamma\times\underline{\mathbb{Z}}\ ,$

where $\bm{L}(\,\mbox{--}\,)_{\rm ab}$ stands for the (left) derived functor
of the abelianization functor $\,(\,\mbox{--}\,)_{\rm
ab}:\,\mathrm{sGr}\to{\rm sAb}$. Since the abelianization functor is left
Quillen, its derived functor commutes with homotopy coends (see (2.20)).
Hence, combining (3.11) with (4.16), we get

(4.17) $\bm{L}[\bm{L}K(\Gamma)]_{\rm ab}\,\simeq\,\int^{\langle
n\rangle\in\mathfrak{G}}_{\bm{L}}\bm{L}(\Gamma\langle
n\rangle\otimes\mathbb{F}\langle n\rangle)_{\rm ab}\,\simeq\,\int^{\langle
n\rangle\in\mathfrak{G}}_{\bm{L}}(\Gamma\langle
n\rangle\,\otimes\,{\mathbb{F}}\langle n\rangle)_{\rm
ab}\,\simeq\,\int^{\langle n\rangle\in\mathfrak{G}}_{\bm{L}}\Gamma\langle
n\rangle\times\underline{\mathbb{Z}}^{n}$

On the other hand, $\,\bm{L}[\bm{L}K(\Gamma)]_{\rm
ab}\simeq[\mathbb{G}{\bar{W}}\bm{L}K(\Gamma)]_{\rm
ab}=\mathbb{A}(\bar{W}\bm{L}K(\Gamma))$, hence, by (4.13), we have

(4.18) $|\,\bm{L}[\bm{L}K(\Gamma)]_{\rm
ab}\,|\,\simeq\,|\,\mathbb{A}(\bar{W}\bm{L}K(\Gamma))\,|\,\simeq\,\Omega\,{\rm
SP}^{\infty}(B\Gamma)$

Combining now (4.14), (4.17) and (4.18), we get the desired equivalence
$X_{\Gamma}\simeq\Omega\,{\rm SP}^{\infty}(B\Gamma)\,$. ∎

Note that Theorem 4.1 combined with Lemma 4.1 implies Theorem 1.1 stated in
the Introduction.

###  4.2. Symmetric homology

In Section 3.3, we defined cyclic homology of homotopy simplicial groups by
associating to each $\Gamma\in\mathrm{sGr}^{h}$ a cyclic bar construction
$B^{\rm cyc}\Gamma:\Delta C^{\rm op}\to\mathrm{sSet}$ (see Definition 3.4). In
this section, we introduce an analogue of this construction for symmetric
groups. Recall that the symmetric crossed simplicial category $\Delta S$ is
defined to be an extension of $\Delta$ that has the same objects as $\Delta$
(and $\Delta C$) with morphisms characterized by the two properties (cf. [44,
6.1.4]):

  1. (Sym1)

For each $n\geqslant 0$, ${\rm Aut}_{\Delta S}([n])\cong S_{n+1}^{\rm op}\,$,
where $S_{n+1}$ is the $(n+1)$-th symmetric group.

  2. (Sym2)

Any morphism $f:[n]\to[m]$ in $\Delta S$ can be factored uniquely as the
composite $f=g\circ\sigma$ with $g\in{\rm Hom}_{\Delta}([n],[m])$ and
$\sigma\in{\rm Aut}_{\Delta S}([n])\cong S_{n+1}^{\rm op}$.

There is an inclusion functor (a morphism in ${\rm Cat}$):

(4.19) $\iota:\ \Delta C^{\rm
op}\,\stackrel{{\scriptstyle\sim}}{{\to}}\,\Delta C\,\hookrightarrow\,\Delta
S\ ,$

where the first arrow is an isomorphism of categories (called Connes’ duality)
and the second one is induced by the natural inclusion of groups
$C_{n+1}\hookrightarrow S_{n+1}$(cf. [44, 6.1.11]). Explicitly, the functor
(4.19) is given on objects by $\iota([n])=[n]$ and on generators by the
following formulas

(4.20) $\left.\begin{aligned} \iota(d^{n}_{i})&=\begin{cases}s^{i}_{n-1}\
,&0\leq i<n\\\ s^{0}_{n-1}\circ(n,0,1,\ldots,n-1)\ ,&i=n\end{cases}\\\
\iota(s^{n}_{j})&=d^{j+1}_{n+1}\\\
\iota(t_{n})&=(n,0,1,\ldots,n-1)\end{aligned}\right.$

where $d^{n}_{i}:[n]\to[n-1]$, $\,s^{n}_{j}:[n]\to[n+1]$ and $t_{n}:[n]\to[n]$
denote the generators of $\Delta C^{\rm op}$ dual (opposite) to the generators
$d^{i}_{n}$, $s^{j}_{n}$ and $\tau_{n}$ of $\Delta C$, respectively.

######  Lemma 4.2.

The functor $\,\Psi_{\rm cyc}^{\rm op}:\Delta C^{\rm op}\to\mathfrak{G}^{\rm
op}$ defined by (3.23), (3.24) extends through $\iota$, giving a commutative
diagram of small categories

(4.21) $\begin{diagram}$

###### Proof.

In order to construct the functor $\Psi_{\rm sym}$ it is convenient to use the
following notation for morphisms in $\Delta S$ (cf. [2, Section 1.1]). Any
morphism $\,f:\,[n]\xrightarrow{\sigma}[n]\xrightarrow{g}[m]\,$ in $\Delta S$
can be written uniquely as a ‘tensor product’ of $m+1$ noncommutative
monomials $\,X_{0},\,X_{1},\,\ldots,X_{m}\,$ in $n+1$ formal variables
$\\{x_{0},x_{1},\ldots,x_{n}\\}$:

(4.22) $f=X_{0}\otimes X_{1}\otimes\ldots\otimes X_{m}$

where each $X_{i}$ is the product $\,x_{i_{1}}x_{i_{2}}\ldots x_{i_{r}}\,$ of
$r=|f^{-1}(i)|$ variables whose indices $i_{k}$ occur in the fibre $f^{-1}(i)$
and that are ordered in the same way as numbers in
$\\{\sigma(0),\ldots,\sigma(n)\\}$, i.e.
$\sigma(i_{1})<\sigma(i_{2})<\ldots<\sigma(i_{r})$. For example, if
$f:[4]\to[3]\,$ is given by the composition $g\circ\sigma$ in $\Delta S$,
where $g\in{\rm{Hom}}_{\Delta}([4],[3])$ is defined by $g(0)=g(1)=0$,
$g(2)=g(3)=1$ and $g(4)=3$ and $\sigma\in{\rm{Aut}}_{\Delta S}([4])=S_{5}^{\rm
op}$ is the permutation

$\sigma=\left(\\!\begin{array}[]{ccccc}0&1&2&3&4\\\ 1&0&4&2&3\\\
\end{array}\\!\right)$

then $f$ is represented by $\,x_{1}x_{0}\otimes x_{3}x_{4}\otimes 1\otimes
x_{2}\,$. The composition of morphisms $f_{1}\circ f_{2}$ is defined by a
natural substitution rule: for example, if $f_{1}:[3]\to[3]$ and
$f_{2}:[4]\to[3]$ in $\Delta S$ are represented by

$f_{1}=1\otimes x_{0}\otimes 1\otimes x_{3}x_{2}x_{1}\ ,\quad
f_{2}=x_{2}x_{1}\otimes x_{4}\otimes 1\otimes x_{0}x_{3}\ ,$

then $f_{1}\circ f_{2}:\,[4]\to[3]$ can be computed as

$\left.\begin{aligned} f_{1}\circ f_{2}&=(1\otimes X_{0}\otimes 1\otimes
X_{3}X_{2}X_{1})\circ(\underbrace{x_{2}x_{1}}_{X_{0}}\otimes\underbrace{x_{4}}_{X_{1}}\otimes\underbrace{1}_{X_{2}}\otimes\underbrace{x_{0}x_{3}}_{X_{3}})\\\
&=1\otimes x_{2}x_{1}\otimes 1\otimes(x_{0}x_{3})\cdot
1\cdot(x_{4})\,=\,1\otimes x_{2}x_{1}\otimes 1\otimes
x_{0}x_{3}x_{4}\end{aligned}\right.$

With this notation, we define the functor

(4.23) $\Psi_{\rm sym}:\ \Delta S\to\mathfrak{G}^{\rm op}$

on objects by

$\Psi_{\rm sym}([n])=\langle n+1\rangle$

and on morphisms by the following formula: if $f\in{\rm{Hom}}_{\Delta
S}([n],[m])$ is represeented by

$f=(x_{i_{1}}\ldots x_{i_{r}})\otimes\cdots\otimes(x_{k_{1}}\ldots x_{k_{s}})\
,$

then

(4.24) $\Psi_{\rm sym}(f):\langle m+1\rangle\to\langle n+1\rangle\,,\qquad
X_{0}\mapsto x_{i_{1}}\ldots x_{i_{r}}\,,\ \ldots\ ,\,X_{m}\mapsto
x_{k_{1}}\ldots x_{k_{s}}\ ,$

where $\langle m+1\rangle=\mathbb{F}\langle X_{0},\ldots,X_{m}\rangle$ and
$\langle n+1\rangle=\mathbb{F}\langle x_{0},\ldots,x_{n}\rangle$. Note that
the maps (4.20) can be rewritten in this tensor notation as

$\left.\begin{aligned}
\iota(d^{n}_{i})&=\begin{cases}x_{0}\otimes\ldots\otimes x_{i-1}\otimes
x_{i}x_{i+1}\otimes x_{i+2}\otimes\ldots\otimes x_{n}\ ,&0\leq i<n\\\
x_{n}x_{0}\otimes x_{1}\otimes\ldots\otimes x_{n-1}\ ,&i=n\end{cases}\\\
\iota(s^{n}_{j})&=x_{0}\otimes\ldots\otimes x_{j}\otimes 1\otimes
x_{j+1}\otimes\ldots\otimes x_{n}\\\ \iota(\tau_{n})&=x_{n}\otimes
x_{0}\otimes x_{1}\otimes\ldots\otimes x_{n-1}\end{aligned}\right.$

The commutativity of (4.21) can now be checked by a trivial calculation that
we leave to the reader. ∎

Having in hand the functor $\Psi_{\rm sym}:\ \Delta S\to\mathfrak{G}^{\rm
op}\,$, we can now define a symmetric bar construction in the same way as we
defined the cyclic bar construction in Definition 3.4.

######  Definition 4.1.

For a homotopy simplicial group $\Gamma\in\mathrm{sGr}^{h}$, its symmetric bar
construction is the functor

(4.25) $B_{\rm sym}\Gamma\,:=\,\Psi^{\ast}_{\rm sym}\Gamma:\Delta
S\to\mathrm{sSet}$

ans its symmetric homology is defined by

(4.26) ${\rm HS}_{\ast}(k[\Gamma])\,:=\,{\rm Tor}^{\Delta S}_{\ast}(k,k[B_{\rm
sym}\Gamma])\ .$

######  Remark 4.1.

The same argument as (in the proof of) Lemma 3.2 shows that ${\rm
HS}_{\ast}(k[\Gamma])$ depends only on the homotopy type of $\Gamma$ in
$\mathrm{sGr}^{h}$ and hence on the homotopy type of the space $B\Gamma$.

######  Remark 4.2.

For $\Gamma$ an ordinary discrete group, the definition (4.25) agrees with
Fiedorowicz’s original definition of the symmetric bar construction (see [27]
and also [2]). In this case, formula (4.26) defines the symmetric homology of
the group algebra $k[\Gamma]$. Note that, unlike $B^{\rm cyc}\Gamma$ (see
(3.25)), the functor $B_{\rm sym}\Gamma:\Delta S\to\mathrm{sSet}$ is covariant
on $\Delta S$ (which we emphasize by writing “sym” as a subscript).

######  Remark 4.3.

To study symmetric homology it is often convenient to work with the augmented
symmetric category $\Delta S_{+}$ which is defined by adding to $\Delta S$ the
initial object $[-1]$ and morphisms $[-1]\to[n]$, one for each $n\geqslant-1$
(see [2]). It is easy to see that the map $\Psi_{\rm sym}$ defined in Lemma
4.2 extends to $\Delta S_{+}$:

(4.27) $\Psi_{{\rm sym},+}:\ \Delta S_{+}\to\mathfrak{G}^{\rm op}$

by letting $\Psi_{{\rm sym},+}([-1]):=\langle 0\rangle$. Now, the category
$\Delta S_{+}$ is isomorphic to the category of so-called finite associative
sets, ${\mathcal{F}}({\rm as})$, introduced in [52] (see also [60, Section
15.4] for a detailed discussion) . The latter is known to be a permutative
category (PROP) that describes the associative unital algebras (see [53] and
also [60]). It opposite category ${\mathcal{F}}({\rm as})^{\rm op}$ describes
the coassociative counital coalgebras. If we identify $\,\Delta
S_{+}={\mathcal{F}}({\rm as})\,$, the restriction functor $\Psi_{{\rm
sym},+}^{\ast}:\,{\rm Mod}_{k}(\mathfrak{G})\to{\rm Mod}_{k}[{\mathcal{F}}(\rm
as)^{\rm op}]$ associated to the opposite of (4.27) takes commutative Hopf
algebras viewed as functors (3.15) on $\mathfrak{G}$ to the underlying
coassociative coalgebras viewed as functors on ${\mathcal{F}}(\rm as)^{\rm
op}$. In other words, the morphism $\Psi_{{\rm sym},+}^{\rm op}$ is isomorphic
to a morphism of PROPs: $\,{\mathcal{F}}(\rm as)^{\rm op}\to\mathfrak{G}\,$
that “forgets” the algebra structure on commutative Hopf algebras.

###  4.3. Symmetric homology vs representation homology

Recall that in Section 3.4, we constructed the derived character map
${\rm{Tr}}(\Gamma)_{\ast}$ relating the cylic homology of $\Gamma$ to its
(one-dimensional) representation homology:

(4.28) ${\rm{Tr}}(\Gamma)_{\ast}:\
\mathrm{HC}_{\ast}(k[\Gamma])\to\mathrm{HR}_{\ast}(k[\Gamma])$

On the other hand, as a consequence of Lemma 4.2, we have a restriction map

(4.29) $\iota^{\ast}:\ \mathrm{HC}_{\ast}(k[\Gamma])={\rm Tor}^{\Delta C^{\rm
op}}_{\ast}(k,\,k[B^{\rm cyc}\Gamma])\,\to\,{\rm{Tor}}^{\Delta
S}_{\ast}(k,\,k[B_{\rm sym}\Gamma])={\rm HS}_{\ast}(k[\Gamma])\ .$

induced by the isomorphism of cyclic spaces

(4.30) $B^{\rm cyc}\Gamma\,\cong\,\iota^{\ast}B_{\rm sym}\Gamma$

The next proposition shows that the derived character map (4.28) factors
through (4.29), thus relating representation homology to symmetric homology.

######  Proposition 4.1.

For any homotopy simplicial group $\Gamma\in\mathrm{sGr}^{h}\\!$, there is a
natural map

(4.31) $\tilde{\Psi}_{\rm sym}^{\ast}:\ {\rm
HS}_{\ast}(k[\Gamma])\,\to\,\mathrm{HR}_{\ast}(k[\Gamma])$

such that

(4.32) $\begin{diagram}$

###### Proof.

As our notation suggests, the map (4.31) is actually induced by a morphism
$\tilde{\Psi}_{\rm sym}$ in $\mathrm{Cat}$. We construct $\tilde{\Psi}_{\rm
sym}$ by lifting the functor $\Psi_{\rm sym}$ of Lemma 4.2 to the (opposite)
category of elements of the abelianization functor (4.1):

(4.33) $\begin{diagram}$

The existence of such a lifting is a consequence of the following observation.
Consider the composition of functors

(4.34) $\begin{diagram}$

that takes an object $[n]\in\Delta S$ to the abelian group $\mathbb{Z}^{n+1}$.
If we represent a morphism $f:[n]\to[m]$ in $\Delta S$ using the tensor
notation (4.22), then $\Psi_{\rm sym}^{\rm op}(f)_{\rm
ab}:\mathbb{Z}^{m+1}\to\mathbb{Z}^{n+1}$, the value of (4.34) on $f$, is
represented by an integral $(n+1)\times(m+1)$-matrix whose rows are indexed by
$0\leqslant i\leqslant n$ and columns by $0\leqslant j\leqslant m$, and the
$j$-th column consists entirely of $0$’s and $1$’s, with the $1$’s occurring
in positions indicated by the elements of $f^{-1}(j)$. For example, if
$f:[4]\to[3]$ in $\Delta S$ is represented by the product $x_{1}x_{0}\otimes
x_{3}x_{4}\otimes 1\otimes x_{2}$, then $\Psi_{\rm sym}^{\rm op}(f)_{\rm
ab}:\,\mathbb{Z}^{4}\to\mathbb{Z}^{5}$ is given by

$\Psi_{\rm sym}^{\rm op}(f)_{\rm ab}=\left(\begin{array}[]{cccc}1&0&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ 0&1&0&0\\\ 0&1&0&0\\\ \end{array}\\!\\!\right)\ .$

Observe that for any morphism $f$ in $\Delta S$ the matrix $\Psi_{\rm
sym}^{\rm op}(f)_{\rm ab}$ thus defined has exactly one nonzero entry in each
row and that entry is $1$. Hence $\Psi_{\rm sym}^{\rm op}(f)_{\rm ab}$ maps
the vector $(1,1,\ldots,1)^{t}\in\mathbb{Z}^{m+1}$ to the vector
$(1,1,\ldots,1)^{t}\in\mathbb{Z}^{n+1}$. This shows that there is a well-
defined functor

(4.35) $\tilde{\Psi}_{\rm sym}:\ \Delta S\to\mathfrak{G}_{\mathbb{Z}}^{\rm
op}\ ,\quad[n]\mapsto(\langle n+1\rangle;1,1,\ldots,1)\ ,$

that makes the diagram (4.33) commutative. It follows from (4.33) that

$k[B_{\rm sym}\Gamma]=\Psi_{\rm sym}^{\ast}(k[\Gamma])={\tilde{\Psi}}_{\rm
sym}^{\ast}(k[p^{\ast}\Gamma])$

Hence, by Lemma 3.4, the functor (4.35) induces a natural map

(4.36) ${\rm HS}_{\ast}(k[\Gamma])={\rm{Tor}}^{\Delta S}_{\ast}(k,k[B_{\rm
sym}\Gamma])\xrightarrow{{\tilde{\Psi}}_{\rm
sym}^{\ast}}{\rm{Tor}}^{\mathfrak{G}^{\rm
op}_{\mathbb{Z}}}_{\ast}(k,\,k[p^{*}\Gamma])\,.$

We claim that if the target of the map (4.36) is identified with the
representation homology of $k[\Gamma]$ via the Shapiro Isomorphism (see Lemma
2.4), then the required factorization property (4.32) holds. To verify this we
fix a projective resolution $Q\xrightarrow{\sim}k[\Gamma]$ of $k[\Gamma]$ in
$\mathrm{Mod}_{k}(\mathfrak{G}^{\rm op})$. Then
$p^{*}(Q)\xrightarrow{\sim}p^{*}k[\Gamma]=k[p^{*}\Gamma]$ gives a projective
resolution of $k[p^{*}\Gamma]$ in
$\mathrm{Mod}_{k}(\mathfrak{G}_{\mathbb{Z}}^{\rm op})$, and the Shapiro
Isomorphism

${\rm{Tor}}^{\mathfrak{G}_{\mathbb{Z}}}_{\ast}(k[p^{*}\Gamma],\,k)\xrightarrow{\sim}{\rm{Tor}}^{\mathfrak{G}}_{\ast}(k[\Gamma],\,p_{!}(k))$

is induced by the composition

$p^{*}(Q)\otimes_{\mathfrak{G}_{\mathbb{Z}}}k\xrightarrow{{\rm{Id}}\otimes\varepsilon_{k}}p^{*}(Q)\otimes_{\mathfrak{G}_{\mathbb{Z}}}p^{*}p_{!}(k)\xrightarrow{p^{*}}Q\otimes_{\mathfrak{G}}p_{!}(k)$

where the first map is given by the adjunction unit
$\,\varepsilon:{\rm{Id}}\to p^{*}p_{!}\,$ and the second is the restriction
map via $p$. Explicitly, using the definition (3.12) of functor tensor
products, we can represent the above composite map as

(4.37)
$\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{(\langle
n\rangle;\,k_{1},\ldots,k_{n})\in\mathfrak{G}_{\mathbb{Z}}}Q\langle n\rangle\
\to\
\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{\langle
n\rangle\in\mathfrak{G}}Q\langle n\rangle\otimes k[\mathbb{Z}^{n}]\
,\quad(v_{n})_{(\langle
n\rangle;\,k_{1},\ldots,k_{n})\in\mathfrak{G}_{\mathbb{Z}}}\mapsto(v_{n}\otimes(k_{1},\ldots,k_{n}))_{\langle
n\rangle\in\mathfrak{G}}$

where $v_{n}\in Q\langle n\rangle$ and the subscripts denote the indices of
the corresponding components of direct sums. Now, using the same resolution
$Q$, we can write explicitly the composition of maps (4.29) and (4.36):

$\Psi^{\ast}_{\rm cyc}(Q)\otimes_{\Delta
C}k\,\xrightarrow{\iota^{*}}\,\Psi^{\ast}_{\rm sym}(Q)\otimes_{\Delta S^{\rm
op}}k\,\xrightarrow{{\tilde{\Psi}}_{\rm
sym}^{\ast}}\,p^{*}(Q)\otimes_{\mathfrak{G}_{\mathbb{Z}}}k$

At the level of chain complexes, this last composition is induced by the map

(4.38)
$\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{[m]\in\Delta
C}Q\langle m+1\rangle\ \to\
\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{[m]\in\Delta
S^{\rm op}}Q\langle m+1\rangle\ \to\
\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{(\langle
n\rangle;k_{1},\ldots k_{n})\in\mathfrak{G}_{\mathbb{Z}}}Q\langle n\rangle$

that takes $\,(v_{m+1})_{[m]\in\Delta C}\,\mapsto\,(v_{m+1})_{[m]\in\Delta
S^{\rm op}}\,\mapsto\,(v_{m+1})_{(\langle
m+1\rangle;\,1,1,\ldots,1)\in\mathfrak{G}_{\mathbb{Z}}}\,$. Combining (4.37)
and (4.38), we see that the resulting map

$\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{[m]\in\Delta
C}Q\langle m+1\rangle\ \to\
\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.75}{$\bigoplus$}}}}_{\langle
n\rangle\in\mathfrak{G}}Q\langle n\rangle\otimes k[\mathbb{Z}^{n}]\
,\quad(v_{m+1})_{[m]\in\Delta C}\ \mapsto\
(v_{m+1}\otimes(1,1,\ldots,1))_{\langle m+1\rangle\in\mathfrak{G}}$

coincides exactly with the map (3.33) representing the derived character
$\chi_{{\rm{GL}}_{1},{\rm{Tr}}_{1}}(\Gamma)_{\ast}={\rm{Tr}}(\Gamma)_{\ast}$.
This finishes the proof of the proposition. ∎

######  Remark 4.4.

The proof of Proposition 4.1 shows that, apart from (4.35), any functor of the
form

(4.39) $\tilde{\Psi}^{(m)}_{\rm sym}:\ \Delta
S\to\mathfrak{G}_{\mathbb{Z}}^{\rm op}\ ,\quad[n]\mapsto(\langle
n+1\rangle;\,m,m,\ldots,m)\ ,$

where $m\in\mathbb{Z}$ is a fixed integer, satisfies the lifting property
(4.33). It is easy to see that there are no other solutions to this lifting
problem. Among (4.39) the functor $\tilde{\Psi}^{(0)}_{\rm sym}$ corresponding
to $m=0$ is the only one that factors through $\mathfrak{G}^{\rm op}$:
$\,\tilde{\Psi}^{(0)}_{\rm sym}=s^{\rm op}\circ{\Psi}_{\rm sym}$, where
$s:\mathfrak{G}\,\hookrightarrow\,\mathfrak{G}_{\mathbb{Z}}$ is the ‘zero’
section of $p$.

Next, we observe that the linear maps factoring ${\rm{Tr}}(\Gamma)_{*}$ in
(4.32) arise (on homology) from the natural maps of topological spaces induced
by the functors (4.19) and (4.35) (cf. Lemma 4.1):

(4.40) $|{\rm hocolim}_{\Delta C^{\rm op}}\,(B^{\rm
cyc}\Gamma)|\,\xrightarrow{\iota^{\ast}}\,|{\rm hocolim}_{\Delta S}\,(B_{\rm
sym}\Gamma)|\,\xrightarrow{\tilde{\Psi}_{\rm sym}^{\ast}}\,|{\rm
hocolim}_{\mathfrak{G}_{\mathbb{Z}}^{\rm op}}\,(p^{\ast}\Gamma)|$

(Here, abusing notation, we denote these topological maps by the same symbols
as the corresponding linear maps.) By Theorem 4.1, we know that

(4.41) $|{\rm hocolim}_{\mathfrak{G}_{\mathbb{Z}}^{\rm
op}}(p^{\ast}\Gamma)|\,\simeq\,\Omega\,{\rm SP}^{\infty}(B\Gamma)$

On the other hand, by theorems of Goodwillie (see [44, Theorem 7.2.4]) and
Fiedorowicz [27] (see [2, Section 5.3]),

(4.42) $\displaystyle|{\rm hocolim}_{\Delta C^{\rm op}}(B^{\rm cyc}\Gamma)|$
$\displaystyle\simeq$ $\displaystyle
ES^{1}\times_{S^{1}}\mathcal{L}(B\Gamma)\,,$ (4.43) $\displaystyle|{\rm
hocolim}_{\Delta S}(B_{\rm sym}\Gamma)|$ $\displaystyle\simeq$
$\displaystyle\Omega\Omega^{\infty}\Sigma^{\infty}(B\Gamma)\,,$

where $\mathcal{L}(B\Gamma):={\rm Map}(S^{1},B\Gamma)$ and
$\,\Omega^{\infty}\Sigma^{\infty}(B\Gamma):=\mathrm{hocolim}_{n\to\infty}\,\Omega^{n}\Sigma^{n}(B\Gamma)\,$
denote the free loop space and the infinite loop space of $B\Gamma$,
respectively. Combining (4.40) with equivalences (4.41), (4.42) and (4.43), we
can thus refine the result of Proposition 4.1 as follows:

######  Corollary 4.1.

The derived character map

${\rm{Tr}}(\Gamma)_{\ast}:\mathrm{HC}_{\ast}(k[\Gamma])\,\xrightarrow{\iota^{*}}\,{\rm
HS}_{\ast}(k[\Gamma])\,\xrightarrow{\tilde{\Psi}_{\rm
sym}^{\ast}}\,\mathrm{HR}_{\ast}(k[\Gamma])$

is induced on homology by a natural map of topological spaces in
${\mathtt{Ho}}(\mathrm{Top}_{*})$:

(4.44)
$ES^{1}\times_{S^{1}}\mathcal{L}(B\Gamma)\,\xrightarrow{{\tt{CS}}_{B\Gamma}}\,\Omega\Omega^{\infty}\Sigma^{\infty}(B\Gamma)\,\xrightarrow{{\tt{SR}}_{B\Gamma}}\,\Omega{\rm
SP}^{\infty}(B\Gamma)$

In the next section, we will describe the maps ${\tt{CS}}$ and ${\tt{SR}}$ in
topological terms in two ways: using the classical ‘little cubes’ operads and
the Goodwillie calculus of homotopy functors.

###  4.4. Generalization to monoids

All results of this section generalize naturally to (simplicial) monoids. We
briefly outline this generalization as we will need it in Section 5.3. Instead
of $\mathfrak{G}$, we start with the category $\mathfrak{M}\subset{\rm Mon}$
whose objects are finitely generated free monoids666Abusing notation, we will
use the same symbols to denote the objects of $\mathfrak{M}$ and
$\mathfrak{G}$. $\langle n\rangle$, one for each $n\geq 0$. In this case, the
abelianization functor reads

$\underline{\mathbb{N}}:\,\mathfrak{M}\to\mathrm{Set}\,,\qquad\langle
n\rangle\mapsto\mathbb{N}^{n}$

where $\mathbb{N}$ is the set of natural numbers, i.e. the underlying set of
the free abelian monoid of rank one. The associated category of elements
$\mathfrak{M}_{\mathbb{N}}\,:=\,\mathfrak{M}\\!\smallint\underline{\mathbb{N}}$
has an explicit description similar to that of $\mathfrak{G}_{\mathbb{Z}}$:
its objects are $(\langle n\rangle;\,k_{1},\ldots,k_{n}))$, where $\langle
n\rangle$ is the free monoid on $n$ generators and
$(k_{1},\ldots,k_{n})\in\mathbb{N}^{n}$. Any simplicial monoid $M$ gives a
functor $M:\mathfrak{M}^{\rm op}\to\mathtt{sSet}$ that restricts to
$\mathfrak{M}_{\mathbb{N}}^{\rm op}$ via the canonical projection
$p:\,\mathfrak{M}_{\mathbb{N}}\to\mathfrak{M}$. The analogue (generalization)
of Theorem 4.1 says:

######  Proposition 4.2.

For any simplicial monoid $M$, there is a weak equivalence in
$\mathrm{Top}_{*}$:

(4.45) $|{\rm hocolim}_{\mathfrak{M}_{\mathbb{N}}^{\rm
op}}(p^{\ast}M)|\simeq\Omega\,{\rm SP}^{\infty}(BM)\,,$

where $BM$ is the classifying space of $M$.

###### Proof.

The same argument as in the proof of Theorem 4.1 — based on Proposition 2.1 —
shows

${\rm hocolim}_{\mathfrak{M}_{\mathbb{N}}^{\rm
op}}(p^{\ast}M)\simeq\bm{L}(M)_{\rm ab}$

where $\,\bm{L}(\,-\,)_{\rm ab}\,$ denotes the derived abelianization functor
on simplicial monoids. To compute this last functor, instead of Kan loop
group, we will use the 2-sided (simplicial) bar resolution (5.22):
$\,B_{*}(\underline{\mathcal{C}}_{1},\underline{\mathcal{C}}_{1},M)\xrightarrow{\sim}M\,$
in $\mathtt{sSet}_{*}$, where $\underline{\mathcal{C}}_{1}$ is the monad
associated to the (simplicial analogue of) little $1$-cube operad (see
(5.24)). Since $(\underline{\mathcal{C}}_{1}(X))_{\rm
ab}=\underline{\mathcal{C}}_{0}(X)$, we have

$|\bm{L}(M)_{\rm
ab}|\simeq|B_{*}(\underline{\mathcal{C}}_{1},\underline{\mathcal{C}}_{1},M)_{\rm
ab}|\simeq|B_{*}(\underline{\mathcal{C}}_{0},\underline{\mathcal{C}}_{1},M)|\simeq\Omega\,{\rm
SP}^{\infty}(BM)\,,$

where the last equivalence is a result of Lemma 5.1 below (see (5.27)). ∎

The relation between monoids and groups is determined by the canonical (group
completion) functor $l:\mathfrak{M}\to\mathfrak{G}$. This last functor extends
naturally to a functor
$\tilde{l}:\mathfrak{M}_{\mathbb{N}}\to\mathfrak{G}_{\mathbb{Z}}$, and the
maps $\Psi_{\rm sym}:\Delta S\to\mathfrak{G}^{\rm op}$ and $\tilde{\Psi}_{\rm
sym}:\ \Delta S\to\mathfrak{G}_{\mathbb{Z}}^{\rm op}$ defined by (4.23) and
(4.35) factor through $l$ and $\tilde{l}$ respectively, giving the commutative
diagram

(4.46) $\begin{diagram}$

As a consequence of Proposition 4.2, we get

######  Corollary 4.2.

For any homotopy simplicial group $\Gamma\in\mathrm{sGr}^{h}$, there is a weak
equivalence

$|{\rm hocolim}_{\mathfrak{M}_{\mathbb{N}}^{\rm
op}}(p^{\ast}l^{\ast}\Gamma)|\,\simeq\,\Omega\,{\rm SP}^{\infty}(B\Gamma)$

###### Proof.

Apply Proposition 4.2 to the simplicial group $\bm{L}K(\Gamma)$ viewed as a
simplicial monoid. ∎

######  Remark 4.5.

Corollary 4.2 can be also deduced from Theorem 4.1 if we notice that the
natural map

${\rm hocolim}_{\mathfrak{M}_{\mathbb{N}}^{\rm
op}}(p^{\ast}l^{\ast}\Gamma)\,\xrightarrow{\sim}\,{\rm
hocolim}_{\mathfrak{G}_{\mathbb{Z}}^{\rm op}}(p^{\ast}\Gamma)$

is a weak equivalence for any $\Gamma$. This last fact follows from Theorem
2.2, the assumptions of which hold thanks to the known properties of the group
completion functor (cf. [14, Lemma 3.2]).

##  5\. Topological character maps via Goodwillie calculus and operads

In this section, we will describe the maps ${\tt{CS}}$ and ${\tt{SR}}$
explicitly in topological terms, using Goodwillie calculus and classical
operads. The latter approach is based on ideas of Fiedorowicz [27] that were
developed by Ault in [2]. The former is inspired by results of Biedermann and
Dwyer that appeared in [18]. The interpretation in terms of Goodwillie
derivatives leads to a natural nonlinear (polynomial) generalization of
topological character maps that deserves a further study (see Section 5.4).

###  5.1. Goodwillie homotopy calculus

Goodwillie calculus provides a universal approximation (“Taylor
decomposition”) of basic homotopy functors that arise in topology in terms of
polynomial functors. This method — introduced by T. Goodwillie in the series
of papers [31, 32, 33] — has been studied extensively in recent years and has
found many interesting applications (see, e.g., the survey papers [1] and
[42]).

Recall that by a homotopy functor we mean a functor on topological spaces that
preserves weak homotopy equivalences. A homotopy functor
$F:\mathrm{Top}_{*}\to\mathrm{Top}_{*}$ is called $n$-excisive (or polynomial
of degree $\leq n$) if it takes any strongly coCartesian $(n+1)$-dimensional
cubical diagram in $\mathrm{Top}_{*}$ to a Cartesian diagram (cf. [1,
Definition 1.1.2]). For $n=0$, this simply means that $F$ is homotopically
constant: i.e. $F(X)\simeq F(\ast)$ for any $X\in{\rm Top}_{\ast}$. For $n=1$,
this is the usual Mayer-Vietoris property: a functor $F$ is $1$-excisive if
and only if it maps homotopy pushout squares to homotopy pullback squares in
${\rm Top}_{\ast}$ (see [1, Example 1.1.4]). For $n>1$, $F$ enjoys a higher
dimensional version of the Mayer-Vietoris property that reduces to the usual
one inductively in $n$.

The main construction of Goodwillie calculus can be described as follows (cf.
[33, Theorem 1.8]).

######  Theorem 5.1 (Goodwillie).

For any homotopy functor $\,F:{\rm Top}_{\ast}\to{\rm Top}_{\ast}\,$ on
pointed spaces, there exists a natural tower of functors $($fibrations$)$
under $F$:

(5.1) $\begin{diagram}$

satisfying the following properties: for all $n\geq 0$,

  1. $(1)$

$\,P_{n}F:\,{\rm Top}_{\ast}\to{\rm Top}_{\ast}$ is an $n$-excisive functor,

  2. $(2)$

$\delta_{n}:\,F\to P_{n}F$ is the universal weak natural transformation to an
$n$-excisive functor.

The last property needs an explanation. By a weak natural transformation
$\,\delta:F\to P$ one means a pair (‘zig-zag’) of natural transformations
$\,F\xrightarrow{\delta^{\prime}}G\xleftarrow{\delta^{\prime\prime}}P\,$,
where $\delta^{\prime\prime}$ is a natural weak equivalence, i.e.
$\,\delta_{X}^{\prime\prime}:G(X)\xleftarrow{\sim}P(X)\,$ is a weak homotopy
equivalence for all spaces $X\in\mathrm{Top}_{*}$. Note that if $F$ and $P$
are homotopy functors, a weak natural transformation $\,\delta:F\to P$ induces
a well-defined natural tranformation between the corresponding functors on the
homotopy category ${\mathtt{Ho}}(\mathrm{Top}_{*})$. Property (2) of Theorem
5.1 then says that the weak natural transformation $\,\delta_{n}:\,F\to
P_{n}F$ is homotopically initial among all natural transformations from $F$ to
$n$-excisive functors.

Given a homotopy functor $F:\mathrm{Top}_{*}\to\mathrm{Top}_{*}$, we define
its $n$-th layer to be the homotopy fibre

(5.2) $D_{n}F(X)\,:=\,{\rm
hofib}\\{P_{n}F(X)\xrightarrow{p_{n}}P_{n-1}F(X)\\}$

where $p_{n}$ is the canonical projection at the $n$-th stage of the
Goodwillie tower (5.1). A remarkable fact discovered in [33] (see [1, Example
1.2.4]) is that, all layers of a homotopy functor $F$ are naturally infinite
loop spaces. More precisely, for each $n\geqslant 0$, there is a spectrum
$\partial_{n}F$ equipped with a (naïve) action of the symmetric group $S_{n}$
such that

(5.3)
$D_{n}F(X)\,\simeq\,\Omega^{\infty}(\partial_{n}F\wedge(\Sigma^{\infty}X)^{\wedge
n})_{hS_{n}}\,,$

where $\,(\Sigma^{\infty},\,\Omega^{\infty})\,$ are the suspension spectrum
and the infinite loop space functors, respectively. The spectrum
$\partial_{n}F$ is called the $n$-th Goodwillie derivative of $F$ (at the
basepoint $\ast$).

###  5.2. The map ${\tt{CS}}$

Recall that, by Corollary 4.1, the derived character map
${\rm{Tr}}(\Gamma)_{\ast}$ is induced by the composition of natural maps in
${\mathtt{Ho}}(\mathrm{Top}_{*})$:

(5.4)
$ES^{1}\times_{S^{1}}\mathcal{L}(X)\,\xrightarrow{{\tt{CS}}_{X}}\,\Omega\Omega^{\infty}\Sigma^{\infty}(X)\,\xrightarrow{{\tt{SR}}_{X}}\,\Omega{\rm
SP}^{\infty}(X)$

where $X=B\Gamma$. Since the classifying space functor on homotopy simplicial
groups induces an equivalence
$\,{\mathtt{Ho}}(\mathrm{sGr}^{h})\cong{\mathtt{Ho}}({\rm Top}_{0,\ast})\,$,
the maps (5.4) are defined on (the homotopy types of) all pointed connected
spaces. To analyze these maps we introduce the notation:

$\Theta(X)\,:=\,ES^{1}\times_{S^{1}}\mathcal{L}(X)\,=\,ES^{1}\times_{S^{1}}{\rm
Map}(S^{1},X)$

and define $\bar{\Theta}:\,{\rm Top}_{\ast}\to{\rm Top}_{\ast}$ by

(5.5)
$\bar{\Theta}(X)\,:=\,\Theta(X)/\Theta(\ast)\,\cong\,ES^{1}\times_{S^{1}}\mathcal{L}(X)/BS^{1}\,\cong\,ES^{1}_{+}\wedge_{S^{1}}\mathcal{L}(X)$

Note that (5.5) is a reduced homotopy functor, so that
$\,P_{0}\bar{\Theta}(X)\simeq\bar{\Theta}(\ast)=\\{\ast\\}\,$ and
$\,P_{1}\bar{\Theta}(X)\cong D_{1}\bar{\Theta}(X)$ for any space
$X\in\mathrm{Top}_{*}$ (see (5.2)).

The next proposition shows that the natural transformation ${\tt{CS}}$ in
(5.4), relating cyclic to symmetric homology, essentially coincides with the
first Goodwillie layer of the functor (5.5). We deduce this from results of
Carlsson and Cohen [21] by elaborating on a remark of Fiedorowicz [27].

######  Proposition 5.1.

The map ${\tt{CS}}$ in (5.4) is represented by

$\begin{diagram}$

where the right vertical arrow is a natural weak equivalence and $\delta_{1}$
is the 1-st layer of the functor (5.5).

###### Proof.

As noticed in [27, Remark 1.4]), the map ${\tt{CS}}_{X}$ factors in the
homotopy category as

(5.6) $ES^{1}\times_{S^{1}}\mathcal{L}(X)\xrightarrow{{\rm
can}}ES^{1}_{+}\wedge_{S^{1}}\mathcal{L}(X)\xrightarrow{f_{X}}\Omega\Omega^{\infty}\Sigma^{\infty}(X)$

where $f_{X}$ is a certain natural map constructed in [21]. We review the
construction of $f_{X}$ and compare it to a well-known general formula for the
first Goodwillie layer of a reduced homotopy functor.

First, we recall a standard stabilization construction due to Waldhausen [68].
For a pointed space $X$, denote by $CX=X\wedge I$ and $\Sigma X=X\wedge S^{1}$
the reduced cone and the reduced suspension of $X$, respectively. The latter
can be obtained by glueing two copies of the former along a common base which
is identified with $X$: this yields the natural pushout square in
$\mathrm{Top}_{*}$

(5.7) $\begin{diagram}$

Applying the given functor $F$ to (5.7) and taking the homotopy pullback along
the maps $j_{0}$ and $j_{1}$ induces a natural map

(5.8) $F(X)\to\mathrm{hocolim}\,[\,F(CX)\xrightarrow{}F(\Sigma
X)\xleftarrow{}F(CX)\,]$

Since the functor $F$ is homotopic and reduced, we have $\,F(CX)\simeq
F(\ast)\simeq\\{\ast\\}\,$, which implies that the homotopy colimit in (5.8)
is equivalent to $\Omega F(\Sigma X)$. In this way, we get a natural map
$s:F(X)\to\Omega F(\Sigma X)$. This last map can be iterated any number of
times:

(5.9) $s_{n}:\ F(X)\,\to\,\Omega^{n}F(\Sigma^{n}X)\ ,\quad n\geq 0\ ,$

and eventually stabilized, defining the map

(5.10) $s_{\infty}:\
F(X)\,\to\,\varinjlim_{n}\,\Omega^{n}F(\Sigma^{n}X)\,=\,\Omega^{\infty}F\Sigma^{\infty}(X)$

In particular, (5.10) exists for our functor $F=\bar{\Theta}$, see (5.5).

Next, for each $n\geqslant 0$, define
$\,\Sigma^{n}X\to\bar{\Theta}\Sigma^{n}(\Sigma X)\,$ to be the composition of
the following natural maps

$\Sigma^{n}X\stackrel{{\scriptstyle\varepsilon}}{{\to}}\Omega\Sigma(\Sigma^{n}X)=\Omega(\Sigma^{n+1}X)\hookrightarrow\mathcal{L}(\Sigma^{n+1}X)\simeq
ES^{1}\times\mathcal{L}(\Sigma^{n+1}X)\twoheadrightarrow
ES^{1}_{+}\wedge_{S^{1}}\mathcal{L}(\Sigma^{n+1}X)=\bar{\Theta}\Sigma^{n}(\Sigma
X)\,,$

where $\varepsilon:{\rm{Id}}\to\Omega\Sigma$ is the adjunction unit of
$(\Sigma,\Omega)$. Looping $n$ times then yields an inductive system of maps

(5.11)
$i_{n}:\,\Omega^{n}\Sigma^{n}X\to\Omega^{n}\bar{\Theta}\Sigma^{n}(\Sigma X)\
,\quad\forall\,n\geq 0\ ,$

which, by [21, Lemma 4.1], induce in the limit a homotopy equivalence

(5.12) $i_{\infty}:\
\Omega^{\infty}\Sigma^{\infty}X\,\xrightarrow{\sim}\,\Omega^{\infty}\bar{\Theta}\Sigma^{\infty}(\Sigma
X)$

Finally, we note the following canonical identifications

(5.13)
$\displaystyle\Omega^{\infty}\bar{\Theta}\Sigma^{\infty}(X)\,:=\,\varinjlim_{n}\,\Omega^{n}\bar{\Theta}\Sigma^{n}(X)\,=\,\varinjlim_{n}\,\Omega^{n+1}\bar{\Theta}\Sigma^{n+1}(X)\,=$
$\displaystyle=\,\varinjlim_{n}\,\Omega[\Omega^{n}\bar{\Theta}\Sigma^{n}(\Sigma
X)]\cong\Omega\varinjlim_{n}\,[\Omega^{n}\bar{\Theta}\Sigma^{n}(\Sigma
X)]=\Omega\,\Omega^{\infty}\bar{\Theta}\Sigma^{\infty}(\Sigma X)$

The Carlsson-Cohen map $f_{X}$ that appears in (5.6) can now be represented by
the zig-zag of natural transformations

(5.14)
$\bar{\Theta}(X)\xrightarrow{s_{\infty}}\Omega^{\infty}\bar{\Theta}\Sigma^{\infty}(X)\stackrel{{\scriptstyle\eqref{ident}}}{{\cong}}\Omega\,\Omega^{\infty}\bar{\Theta}\Sigma^{\infty}(\Sigma
X)\xleftarrow{\Omega i_{\infty}}\Omega\,\Omega^{\infty}\Sigma^{\infty}(X)\,,$

where the leftmost arrow is the Waldhausen stabilization map (5.10) for
$\bar{\Theta}$ and the rightmost arrow is a natural weak equivalence induced
by (5.12). To complete the proof it remains to note that
$\,P_{1}(F)\simeq\Omega^{\infty}F\Sigma^{\infty}$ for any reduced homotopy
functor $F$, and the universal natural transformation $\delta_{1}:F\to
P_{1}F=D_{1}F$ coincides (up to homotopy) with the stabilization map (5.10)
(see, e.g. [42, Example 5.3]). ∎

###  5.3. The map ${\tt{SR}}$

We now turn to the second map ${\tt{SR}}_{X}$ in (5.4) that relates symmetric
homology to representation homology. In this section, we construct this map
topologically by a method similar to that of Proposition 5.1; its relation to
Goodwillie calculus will be discussed in Section 5.4. Our starting point is
the well-known fact that the Dold-Thom functor
$\mathrm{SP}^{\infty}:\,\mathrm{Top}_{*}\to\mathrm{Top}_{*}$ factors through
the category of abelian topological monoids: in fact,
$\mathrm{SP}^{\infty}(X)$ is the free abelian topological monoid generated by
the space $X$ (see, e.g., [49]). This implies that $\mathrm{SP}^{\infty}$ is a
linear (i.e., $1$-excisive) functor. The latter can be seen directly as
follows. Consider the natural maps (5.9) for the functor
$F=\mathrm{SP}^{\infty}$ constructed in the proof of Proposition 5.1:

(5.15) $s_{n}:\
\mathrm{SP}^{\infty}(X)\,\to\,\Omega^{n}\mathrm{SP}^{\infty}\Sigma^{n}(X)\
,\quad n\geq 0\ ,$

The maps (5.15) are all weak equivalences, which follows immediately from the
commutative diagrams

$\begin{diagram}$

where the vertical arrows are isomorphisms by the Dold-Thom Theorem. Thus, in
the limit, we get

(5.16) $s_{\infty}:\
\mathrm{SP}^{\infty}(X)\,\xrightarrow{\sim}\,\Omega^{\infty}\,\mathrm{SP}^{\infty}\,\Sigma^{\infty}(X)\
,$

showing that $\,\mathrm{SP}^{\infty}\simeq P_{1}(\mathrm{SP}^{\infty})\simeq
D_{1}(\mathrm{SP}^{\infty})\,$, whence the linearity of
$\mathrm{SP}^{\infty}$.

On the other hand, for all $n\geq 0$, we have canonical maps
$\,\Sigma^{n}X\to\mathrm{SP}^{\infty}(\Sigma^{n}X)\,$ inducing the Hurewicz
homomorphisms, see (4.9)). Applying loop functors to these maps yield an
inductive system of maps

(5.17) $i_{n}:\
\Omega^{n}\Sigma^{n}(X)\,\to\,\Omega^{n}\mathrm{SP}^{\infty}\Sigma^{n}(X)\
,\quad n\geq 0\ ,$

which in the limit, induces

(5.18) $i_{\infty}:\
\Omega^{\infty}\Sigma^{\infty}(X)\,\to\,\Omega^{\infty}\mathrm{SP}^{\infty}\Sigma^{\infty}(X)$

Unlike the analogous map (5.12) for the functor $\bar{\Theta}$, (5.18) is not
a weak equivalence in general. Nevertheless, looping it once and combining
with (5.16), we get the pair of natural transformations

(5.19) $\Omega\Omega^{\infty}\Sigma^{\infty}(X)\,\xrightarrow{\Omega
i_{\infty}}\,\Omega\Omega^{\infty}\mathrm{SP}^{\infty}\Sigma^{\infty}(X)\,\xleftarrow{\Omega
s_{\infty}}\,\Omega\mathrm{SP}^{\infty}(X)$

where the rightmost one is a natural weak equivalence. Our goal is to prove
the following analogue of Proposition 5.1.

######  Proposition 5.2.

The map ${\tt{SR}}$ is represented by the weak natural transformation (5.19).
Thus, in the homotopy category, ${\tt{SR}}_{X}$ is equivalent to the map

(5.20) $(\Omega s_{\infty})^{-1}\,(\Omega i_{\infty}):\
\Omega\Omega^{\infty}\Sigma^{\infty}(X)\ \to\ \Omega\mathrm{SP}^{\infty}(X)$

which is the $($looped once$)$ canonical natural transformation relating
stable homotopy to $($reduced$)$ singular homology of pointed spaces.

To prove this proposition we will reinterpret the map (5.20) in terms of
(topological) operads. The standard reference for the background material that
we need is [48] (for a brief introduction, see also [60, Chapter 12]). Recall
that an operad $\mathcal{C}$ in $\mathrm{Top}_{*}$ is a collection of pointed
spaces $\\{\mathcal{C}(j)\\}_{j\geq 0}$ with $\mathcal{C}(0):=\\{\ast\\}$ such
that each $\mathcal{C}(j)$ carries a right $S_{j}$-action and there are
composition laws
$\mathcal{C}(k)\times\mathcal{C}(j_{1})\times\ldots\times\mathcal{C}(j_{k})\to\mathcal{C}(j_{1}+\ldots+j_{k})$
satisfying natural associativity and unitality conditions. If $\mathcal{C}$ is
an operad, a $\mathcal{C}$-space is a pointed space $X$ equipped with an
action of $\mathcal{C}$, which is given by a sequence of $S_{j}$-equivariant
maps $\,\theta_{j}:\mathcal{C}(j)\times X^{j}\to X\,$, with
$\theta_{0}:\mathcal{C}(0)\,\hookrightarrow\,X$ being the basepoint inclusion,
that satisfy associativity and unitality conditions compatible with those of
$\mathcal{C}$. Every operad $\mathcal{C}$ determines a monad
$\,\underline{\mathcal{C}}$ on $\mathrm{Top}_{*}$ (i.e., a monoid with respect
to ‘$\circ$’ in the category of endofunctors
$\mathrm{Top}_{*}\to\mathrm{Top}_{*}$) in such a way that the notion of a
$\mathcal{C}$-space is equivalent to that of
$\underline{\mathcal{C}}$-algebra. Explicitly, given an operad $\mathcal{C}$,
the corresponding monad
$\underline{\mathcal{C}}:\mathrm{Top}_{*}\to\mathrm{Top}_{*}$ is defined by

(5.21) $\underline{\mathcal{C}}(X):=\coprod_{j\geq
0}\,\left(\mathcal{C}(j)\times_{S_{j}}X^{j}\right)/\sim$

where the equivalence relation is of the form

$(c,x_{1},\ldots,x_{i-1},\ast,x_{i+1},\ldots,x_{j})\sim(\sigma_{i}(c),x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{j})$

for certain natural maps $\sigma_{i}:\,\mathcal{C}(j)\to\mathcal{C}(j-1)$ (see
[48, Construction 2.4]). A $\underline{\mathcal{C}}$-algebra is then defined
to be a space $A\in\mathrm{Top}_{*}$ with an action map
$\xi:\underline{\mathcal{C}}(A)\to A$ satisfying natural associativity and
unitality conditions. Opposite to the notion of a
$\underline{\mathcal{C}}$-algebra is that of a
$\underline{\mathcal{C}}$-functor, which a functor $F$ on $\mathrm{Top}_{*}$
equipped a morphism $F\circ\underline{\mathcal{C}}\to F$ defining a right