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\title{Hermitian $K$-theory: On a theorem of Giffen}
\author{Marco Schlichting}
\date{\today}
\begin{document}
\bibliographystyle{alpha}
\address{Marco Schlichting, Universit\"at GHS Essen, Fb 6 Mathematik,
45117 Essen, Germany}
%\thanks{}
\email{marco.schlichting@uni-essen.de}
\subjclass{19D06, 19G38, 11E70}
\keywords{hermitian $K$-theory, Q-construction}
\begin{abstract}
Let $\E$ be an exact category with duality.
In \cite{CharLee} a category $\W(\E)$ was introduced and the authors
asserted that the loop space of the topological realization of $\W(\E)$ is
homotopy equivalent to Karoubi's $U$-theory space of $\E$ when $\E=\P(R)$, the
category of finitely generated projective modules over a ring $R$ with
an involution if $2$ is invertible in $R$.
Unfortunately, their proof contains a mistake.
We present a different proof which avoids their argument.
\end{abstract}
\vspace{5ex}
\maketitle
\section*{\sc Introduction}
%\vskip\aftersection
\vskip\beforetruc
Let $R$ be a ring with unit, and let $R \to R:a \mapsto \overline{a}$
be an involution, \ie $\overline{a+b}=\overline{a}+\overline{b}$,
$\overline{ab}=\overline{b}\overline{a}$, $\overline{1}=1$ and $\overline{\overline{a}}=a$ (see
\cite[I 1.1.]{knus}).
Let $\eps \in R$ be a central element with $\eps \overline{\eps}=1$.
In analogy to algebraic $K$-theory, Karoubi \cite{karoubi:period}
defined the hermitian
$K$-theory groups of $R$ as the homotopy groups of a space
$_{\eps}K^h(R) \simeq{_{\eps}K_0^h(R)} \times B{_{\eps}O(R)}^+$.
Here ${_{\eps}K_0^h(R)}$ is the Grothendieck group of the abelian
monoid of isometry classes of finitely generated projective
right $R$-modules equipped with a non-degenerate $\eps$-symmetric form.
There is a hyperbolic functor from algebraic $K$-theory to hermitian
$K$-theory $h:K(R) \to
{_{\eps}K^h(R)}$ whose homotopy fiber is denoted by $_{\eps}U(R)$.
There is also a forgetful functor $f: {_{\eps}K^h(R)} \to K(R)$ whose
homotopy fiber is denoted by $_{\eps}V(R)$.
Karoubi's fundamental theorem in hermitian
$K$-theory states that there is a natural homotopy equivalence
$\Omega{_{\eps}U(R)}
\simeq {_{-\eps}V(R)}$ whenever $2$ is invertible in $R$ (see
\cite{Karoubi:thmfond}).
Giffen attempted
to reinterpret the fundamental theorem in a categorical framework.
To this end, he introduced a category $_{\eps}\W(R)$ which has been
proposed independently by Karoubi (unpublished).
We refer the reader to section \ref{constructions} for a description.
In \cite[3.1 Theorem, 3.7 Corollary]{CharLee}, Charney and Lee claimed
to have proved the
existence of a homotopy fibration
$$\xymatrix{ (1) & K(R) \ar[r]^{h} & _{\eps}K^h(R) \ar[r] &
_{\eps}\W(R)}$$
if $2$ is invertible in $R$.
This would identify, up to homotopy, the loop space of $_{\eps}\W(R)$ with $_{\eps}U(R)$.
Unfortunately, their proof contains an error:
the functor $\sigma^*$ of \cite[p.~177]{CharLee} doesn't act as an inner
automorphism as claimed.
We don't know whether the mistake can be fixed within their framework.
In this paper we present a different proof which avoids their argument.
\vskip\beforetruc
The main reason for being interested in the construction of $\W$ is
that it can be applied to any exact category with duality, whereas the definition
of Karoubi's
$U$-theory presupposes a definition of hermitian $K$-theory
which is not yet available for exact categories, in general.
We feel that the category $\W$ should play a similar role in hermitian $K$-theory as
Quillen's $Q$-construction does in algebraic $K$-theory.
The author's interest in writing this article stems from the attempt
to generalize hermitian
$K$-theory of rings to exact categories or even to chain complexes as
in the spirit of Thomason and Trobaugh \cite{TT}.
A localization theorem for hermitian $K$-theory whose proof depends on
the result of this article will be published in
\cite{Jandme1}.
This can be considered as a first step to the targeted generalization.
The statement of \cite[3.1 Theorem]{CharLee} differs from
$(1)$ insofar as the authors of \cite{CharLee} don't assume $2$ to
be invertible in $R$.
The methods of our proof do not yield this more general version.
\vskip\beforetruc
The plan of the article is as follows.
In the first section we recollect some facts about symmetric monoidal
categories and the bar construction all of which are well known.
The second section introduces a
hermitian analog of Waldhausen's $S_{\cdot}$-construction whose
definition emerged from discussions I had with Jens Hornbostel.
We also recall the definition of Giffen's category $\W$.
In the third section we prove the main result which is contained in
theorem \ref{thm:giffen} and proposition \ref{prop:giffen}.
The appendix recalls definitions and basic facts about exact
categories with duality and shows that, up to equivalence of such
categories, we can always ``identify an object with its double dual''.
This simplifies notations and proofs.
We use the word ``space'' interchangeably for ``topological space of
the homotopy type of a CW-complex'', ``simplicial set'', or
``category'', the homotopy categories whereof they are objects are
known to be equivalent.
\vskip\beforetruc
\noindent
{\it Acknowledgments.} I'd like to thank Daniel Grayson and Jens
Hornbostel for the fruitful
discussions I had with them and Rick Jardine for pointing out the
reference \cite{moerdijk}.
\section{\sc Symmetric monoidal categories}
\label{symcatetc}
\pph
\label{bar}
Let $M$ be a monoid in the category of spaces which acts on a
space $X$.
There is a simplicial space ${ \Bar}_{\cdot}(M,X)$,
called bar construction, whose space of $n$-simplices is
$ \Bar_n(M,X)= M^n \times X$ (the simplicial space is $B(*,M,X)$ in
\cite[section 7]{may:fibrations} and $EM \times_M X$ in \cite[IV
5.3]{GoerssJardine}).
The face maps are given by the multiplication in $M$ and by the action of
$M$ on $X$.
We write $\Bar_{\cdot}(M)$ for $\Bar_{\cdot}(*,M)$.
Let $M$ be a monoid in the category of simplicial sets
acting on a simplicial set $X$.
We say that it acts invertibly up to homotopy if for every vertex $v
\in M_0$ the map $X \to X$ induced by restricting the action of $M$ to the
vertex $v$ is a homotopy equivalence.
A homology version of the following lemma can also be found in
\cite[Theorem IV 5.15]{GoerssJardine}.
\lem
\label{Bar fibration}
{\it
Let $M$ be a monoid in the category of simplicial sets
acting on a simplicial set $X$
invertibly up to homotopy, then the sequence
$$ X \rightarrow \Bar_{\cdot}(M,X) \rightarrow \Bar_{\cdot}(M)$$ is
a homotopy fibration.
Here the first map is ``inclusion of zero simplices'', and the second map is
induced by the $M$-equivariant map $X \rightarrow *$.
}
\prf
This is \cite[2.1 Theorem]{moerdijk} with $\Bbb{C}=M$,
$N(\Bbb{C})=\Bar_{\cdot}(M)$, $N(X_{\Bbb{C}})=\Bar_{\cdot}(M,X)$ and
$X(C)=\gamma^*(X(C))=X$.
\Qed
\rem
By taking nerves and diagonals when appropriate, lemma
\ref{Bar fibration} remains true when we replace simplicial sets by
categories or bisimplicial sets.
\pph
\label{pph1}
Recall that a symmetric monoidal category is a category $\C$ equipped
with a functor $\oplus: \C \times \C \to \C$, a unit object $0$ and
for all objects $A,B,C$ with natural isomorphisms $\alpha: A \oplus (B \oplus C)
\stackrel{\sim}{\to} (A \oplus B) \oplus C$, $\sigma: A \oplus B
\stackrel{\sim}{\to} B \oplus A$ and $\eta: 0 \oplus A
\stackrel{\sim}{\to} A$ making certain diagrams commutative (see
\cite[VII 7, XI]{ML}).
A symmetric monoidal category is called strict if $\alpha$ and $\eta$
are identity morphisms.
A morphism in the category of small symmetric monoidal categories is a
functor $f: \C \to \D$ together with natural isomorphisms $\tau:
f(A) \oplus f(B) \stackrel{\sim}{\to} f(A \oplus B)$, $\eps: 0 \to
f(0)$ making certain diagrams commute.
A morphism in the category of small strict symmetric monoidal
categories is a morphism between symmetric monoidal categories such
that $\tau$ and $\eps$ are the identity maps.
For more details see for instance \cite[VII 7, XI]{ML}, \cite[section
1]{thomason:colim}.
Every symmetric monoidal category is equivalent to a strict symmetric
monoidal category (see \cite[4.2]{May:perm}, \cite[XI 3]{ML}).
Considering a small strict symmetric monoidal category as a monoid in
the category of small categories,
the Bar construction of \ref{bar} is then a functor from strict symmetric
monoidal categories to simplicial symmetric monoidal categories.
If $\C$ is a category, then we write $i\C$ for the subcategory which has the
same objects as
$\C$ and whose morphisms are the isomorphisms in $\C$.
In the case of $\C$ symmetric monoidal, a category $(i\C)^{-1}\C$ has been
constructed in \cite[p.~219]{quillen/grayson} in a functorial way.
There is an inclusion of categories $\C \to (i\C)^{-1}\C$ which is a
group completion provided that all translations $\oplus A: i\C \to
i\C$ are faithful \cite[2), p.~220]{quillen/grayson}.
In this case we write $\C^+$ for $(i\C)^{-1}\C$.
Remark that the functor $\C \mapsto \C^+$ sends strict symmetric
monoidal categories to
strict ones.
We need the following well known lemma.
\lem
\label{level group completion}
{\it
Let $\C_*$ be a simplicial strict symmetric monoidal category such
that translations in $i\C_n$ are faithful for $n \in \N$ then the map
$$|\C_*| \rightarrow |\C_*^+|$$
is a group completion.
In particular, if the monoid $\pi_0(|\C_*|)$ is a group, then the above map is a
homotopy equivalence.
}
\prf
The two topological monoids $|\C_*|$ and $|\C_*^+|$ are both
H-spaces which meet the hypothesis of the ``group
completion theorem'', \ie for which the natural transformation of
functors $id \rightarrow \Omega
|p \mapsto \Bar_{p}(id)|$ is a group completion \cite[section 15]{may:fibrations}.
It therefore suffices to show that $|\Bar_{*}(|\C_*|)| \rightarrow
|\Bar_{*}(|\C_*^+|)|$ is a homotopy equivalence.
Realizing in a different order, we see that the this map is
homeomorphic to $|q \mapsto |\Bar_{*}(\C_q)|| \rightarrow |q
\mapsto |\Bar_{*}(\C_q^+)|| $.
But the last map is degree wise a homotopy equivalence because
$\C_q \rightarrow \C_q^+$ being a group completion of symmetric
monoidal categories
implies that $|\Bar_{*}(\C_q)| \rightarrow
|\Bar_{*}(\C_q^+)|$ is a homotopy equivalence.
\Qed
\section{\sc The $\r_{\cdot}$-construction and Giffen's category
$\W$}
\label{constructions}
\dfn
\label{dfn:excatwdual}
A {\it category with duality} is a triple $(\C,*,\eta)$ with
\begin{enumerate}
\item $\C$ a category,
\item $*:\C \to \C^{op}$ a functor and
\item $\eta: id_{\C} => **$ a natural equivalence such that
\item for all objects $A$ of $\C$ we have $1_{A^*}= \eta_A^* \circ
\eta_{A^*}$.
\end{enumerate}
The duality of $(\C,*,\eta)$ is called {\it strict}
if $**=id_{\A}$ and if
$\eta_A=1_A$ for all objects $A$ of $\C$.
A map $f:(\A,*) \to (\B,\sharp)$ between categories with strict
duality is a functor $f:\A \to \B$ such that $f^{op} \circ *=\sharp \circ f$.
A category with duality $(\E,*,\eta)$ is called {\it exact} if $\E$ is an exact
category (see \cite{quillen:higher},
\cite[Appendix A]{keller:chain}) and if $*: \E \to \E^{op}$ is an
exact functor.
Since every (exact) category with duality is equivalent to a
(exact) category with strict duality (see \ref{lem:dualstrict}) we
will assume our (exact) categories with
duality to have a strict duality.
\teil{The $\r_{\cdot}$-construction}
\label{R_dot construction}
Let $\Delta$ be the category of finite ordered sets and monotonic maps as
morphisms.
As usual, denote by $[n]$ the ordered set $\{0<... \sharp \circ f$ a natural
equivalence such that
\item for all objects $A$ of $\A$ the following diagram commutes
$$\xymatrix{ f(A) \ar[r]^{f(\alpha_A)} \ar[d]_{\beta_{f(A)}} &
f(A^{**}) \ar[d]^{\tau_{A^*}} \\
f(A)^{\sharp \sharp} \ar[r]_{\tau_A^{\sharp}} &
f(A^*)^{\sharp}.}
$$
\end{enumerate}
Composition is defined by
$(g, \sigma) \circ (f, \tau) = (g \circ f, \rho)$ with
$\rho_A=\sigma_{f(A)} \circ g(\tau_A)$.
A {\it map of additive (exact) categories with duality} $(f,
\tau)$ is a map of
categories with duality with $f$ an additive (exact) functor.
A map of \spredus is called {\it strict map} if $f^{op} \circ * =
\sharp \circ f$ and if $\tau=id$.
\rem
Composition of maps between (additive, exact)
categories with duality is associative.
Composition of strict maps is strict.
\dfn
\label{dfn:trafo_excatwdual}
Given two morphisms $(f_i, \tau_i):(\A,*,\alpha) \to
(\B,\sharp,\beta)$, $i=0,1$, between categories with duality,
a natural transformation $t:(f_0, \tau_0) => (f_1, \tau_1)$ is a
natural transformation of functors $t:f_0 => f_1$ such that for all
objects $A$ of $\A$ the following diagram commutes
$$\xymatrix{f_0(A^*) \ar[r]^{\tau_{0,A}} \ar[d]_{t_{A^*}} &
f_0(A)^{\sharp}\\
f_1(A^*) \ar[r]^{\tau_{1,A}} &
f_1(A)^{\sharp}. \ar[u]_{t_A^{\sharp}}}
$$
We call $t$ a natural equivalence if for all objects $A$ of $\A$ the
map $t_A$ is an isomorphism.
\pph
The above notions define a category $Dualcat$ of small \predus and maps
of \predus.
There is also a category $sDualcat$ of small \spredus and strict
maps of \predus.
There is an inclusion of categories $\iota: sDualcat \to Dualcat$.
\dfn
\label{dfn:dualeq}
A map $(f, \tau):(\A,*,\alpha) \to (\B,\sharp,\beta)$ is called {\it
an equivalence of \predus} if there is a map of \predus
$(g,\sigma): (\B,\sharp,\beta) \to (\A,*,\alpha)$ such that
$(f, \tau)\circ (g,\sigma)$ and $(g,\sigma) \circ (f, \tau)$ are
equivalent to $id_{(\B,\sharp,\beta)}$ and $id_{(\A,*,\alpha)}$
respectively.
\lem
\label{lem:dualeq}
{\it Let $(f, \tau):(\A,*,\alpha) \to (\B,\sharp,\beta)$ be a map of
\predus.
If $f:\A \to \B$ is an equivalence of categories then $(f, \tau)$ is
an equivalence of \predus.}
\prf
This is left to the reader (hint: use an inverse $g$ of $f$ with good
properties).
\Qed
\lem
\label{lem:dualstrict}
{\it
There is a strictifying functor $^s:Dualcat \to sDualcat:(\A,*,\alpha)
\mapsto (\A^s,*^s,\alpha^s)$ and a natural transformation
$(E,\eta):id => (\iota\circ ^s)$,
$(E,\eta)_{(\A,*,\alpha)}= (E^{\A},\eta^{\A}):
(\A,*,\alpha) \to (\A^s,*^s,\alpha^s)$ which is an equivalence of
\predus for any \predu
$(\A,*,\alpha)$.
In other words, any \predu is equivalent to a \spredu and maps
between \predus
can by strictified in a functorial way.
}
\prf
The objects of $\A^s$ are two copies of the objects of $\A$:
$Ob\A^s = Ob\A \times \{0,1\}$.
We will write $A_0$ for $(A,0)$ and $A_1$ for the object $(A,1)$.
Morphisms in $\A^s$ are defined as follows.
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\item Maps from $A_0$ to $B_0$ correspond bijectively to maps from
$A$ to $B$ in $\A$.
\item Maps from $A_1$ to $B_0$ correspond bijectively to maps from
$A^*$ to $B$ in $\A$.
\item Maps from $A_0$ to $B_1$ correspond bijectively to maps from
$A$ to $B^*$ in $\A$.
\item Maps from $A_1$ to $B_1$ correspond bijectively to maps from
$A^*$ to $B^*$ in $\A$.
\end{enumerate}
Composition is induced by the composition in $\A$.
The duality functor $*^s:\A^s \to {\A^s}^{op}$ is defined as follows.
On objects it is $A_i^{*^s}=A_{1-i}$.
\begin{enumerate}
\item Let $\ffi: A_0 \to B_0$, \ie $\ffi: A \to B$ then $\ffi^*:B^* \to
A^*$ and we define $\ffi^{*^s}:=\ffi^*:B_1 \to A_1$.
\item Let $\ffi: A_1 \to B_0$, \ie $\ffi: A^* \to B$ then $\ffi^*:B^* \to
A^{**}$ and we define $\ffi^{*^s}:=\alpha_A^{-1}\circ\ffi^*:B_1 \to A_0$.
\item Let $\ffi: A_0 \to B_1$, \ie $\ffi: A \to B^*$ then $\ffi^*:B^{**} \to
A^*$ and we define $\ffi^{*^s}:=\ffi^*\circ \alpha_B:B_0 \to A_1$.
\item Let $\ffi: A_1 \to B_1$, \ie $\ffi: A^* \to B^*$ then $\ffi^*:B^{**} \to
A^{**}$ and we define $\alpha_A^{-1}\circ \ffi^{*^s}\circ
\alpha_B:=\ffi^*:B_0 \to A_0$.
\end{enumerate}
Observe that the $\alpha^{-1}$'s always compose on the left and the
$\alpha$'s on the right in $a),..d)$.
This shows that $*^s:\A^s \to {\A^s}^{op}$ is a functor.
We calculate $*^s\circ*^s=id$.
This is obviously true on objects.
On morphisms we have:
\begin{enumerate}
\item Let $\ffi: A_0 \to B_0$ then
$\ffi^{*^s*^s}=(\ffi^*)^{*^s}=\alpha_B^{-1}\ffi^{**}\alpha_A=\ffi$
by \ref{dfn:excatwdual} (3).
\item Let $\ffi: A_1 \to B_0$ then
$\ffi^{*^s*^s}=(\alpha_A^{-1}\ffi^*)^{*^s}=\alpha_B^{-1}\ffi^{**}{\alpha_A^*}^{-1}=\ffi\alpha_{A^*}^{-1}{\alpha_A^*}^{-1}=\ffi$ by \ref{dfn:excatwdual} (3) and (4).
\item Let $\ffi: A_0 \to B_1$ then
$\ffi^{*^s*^s}=(\ffi^*\alpha_B)^{*^s}=\alpha_B^*\ffi^{**}\alpha_A=\alpha_B^*\alpha_{B^*}\ffi=\ffi$ by \ref{dfn:excatwdual} (3) and (4).
\item Let $\ffi: A_1 \to B_1$ then
$\ffi^{*^s*^s}=(\alpha_A^{-1}\ffi^{*^s}\alpha_B)^{*^s}=\alpha_B^*\ffi^{**}{\alpha_A^*}^{-1}=\alpha_B^*\ffi^{**}{\alpha_A}^*=\alpha_B^*{\alpha_B}^*\ffi=\ffi$ by \ref{dfn:excatwdual} (3) and (4).
\end{enumerate}
It follows that $(\A^s,*^s,\alpha^s)$ is a \predu such that
$*^s\circ*^s=id$ and $\alpha^s=1$.
Hence $(\A^s,*^s,\alpha^s)$ is a \spredu.
We define a map of \predus $(E^{\A},\eta^{\A}):
(\A,*,\alpha) \to (\A^s,*^s,\alpha^s)$ in the following way.
The underlying functor $E^{\A}:\A \to \A^s$ sends $A$ to $A_0$.
It is the identity on morphisms.
By definition of the composition in $\A^s$, $E^{\A}$ is an additive functor.
We let $\eta^{\A}_{A}=1_{A^*}:A^*_0 \to A_1$.
One verifies that $\eta^{\A}:E\circ * => *^s \circ E$ is a natural
equivalence satisfying \ref{dfn:mapof_excatwdual} (3).
Since $E^{\A}:\A \to \A^s$ is an equivalence of categories,
$(E^{\A},\eta^{\A}): (\A,*,\alpha) \to (\A^s,*^s,\alpha^s)$ is an
equivalence of
\predus by the above lemma \ref{lem:dualeq}.
Given a map of \predus $(g,\sigma):(\A,*,\alpha) \to (\B,\sharp,\beta)$,
we construct a map $(g,\sigma)^s=(g^s,\sigma^s):(\A^s,*^s,\alpha^s) \to
(\B^s,\sharp^s,\beta^s)$ making the following diagram commute
$$(I) \ \ \ \ \ \ \ \xymatrix{
(\A,*,\alpha) \ar[r]^{(g,\sigma)}
\ar[d]_{(E^{\A},\eta^{\A})} &
(\B,\sharp,\beta) \ar[d]^{(E^{\B},\eta^{\B})}\\
(\A^s,*^s,\alpha^s) \ar[r]_{(g^s,\sigma^s)} &
(\B^s,\sharp^s,\beta^s).}
$$
On objects $g^s$ is $A_i \mapsto g(A)_i$.
On morphisms it is defined in the following way.
\begin{enumerate}
\item $\ffi: A_0 \to B_0$, \ie $\ffi: A \to B$ goes to $g(\ffi)$.
\item $\ffi: A_1 \to B_0$, \ie $\ffi: A^* \to B$ goes to
$g(\ffi)\circ\sigma_A^{-1}:g(A)_1 \to g(B)_0$.
\item $\ffi: A_0 \to B_1$, \ie $\ffi: A \to B^*$ goes to $\sigma_B
\circ g(\ffi):g(A)_0 \to g(B)_1$.
\item $\ffi: A_1 \to B_1$, \ie $\ffi: A^* \to B^*$ goes to
$\sigma_B\circ g(\ffi)\circ\sigma_A^{-1}$.
\end{enumerate}
As above $g^s$ is a functor.
We let $\sigma^s=1$.
It is obviously a natural equivalence satisfying
\ref{dfn:mapof_excatwdual} (3).
We leave it to the reader to verify the commutativity of (I) and of
the fact that strictifying is a functor, \ie
$[(g,\sigma)\circ(f,\tau)]^s=(g,\sigma)^s\circ(f,\tau)^s$.
\Qed
\rem
\label{Wdefext}
For $(\E,\eta,*)$ an (exact) category with (non necessarily strict) duality,
one can extend the definitions of this article to obtain the obvious
definitions for $\E_h$, $\r_{\cdot}\E$ and $\W(\E)$.
The category $\E_h$ for instance has objects isomorphisms $\phi:X \to
X^*$ such that $\phi=\phi^* \circ \eta_X$ and morphisms $a:\phi \to
\phi'$ such that $a: X \to X'$ with $a^* \circ \phi' \circ a = \phi$
(compare \ref{pph:C_h}), similarly for the $\W$ and
$\r_{\cdot}$-constructions.
The natural transformation $(E, \eta)$ induces homotopy
equivalences between the $\r_{\cdot}$ and $\W$-constructions of an
exact category with duality and of its strictification.
\rem
\label{permstrict}
For $(\A,*)$ an (pre-) additive category with strict duality, we can extend
the duality of $\A$ to the associated strict additive category $L\A$.
Recall that an object of the category $L\A$ is a finite string
$n(A_1,...,A_n)$ of objects of $\A$.
A morphism from $n(A_1, ..., A_n)$ to $m(B_1, ..., B_m)$ is a map
$(...((A_1 \oplus A_2) \oplus A_3)...\oplus A_n) \to
(...((B_1 \oplus B_2) \oplus B_3)...\oplus B_m)$.
We extend the duality on $\A$ by $n(A_1,...,A_n)^*=n(A_n^*,...,A_1^*)$.
The natural inclusion $\A \to L\A: A \mapsto 1(A)$ is a map of
categories with strict duality.
It is an equivalence if and only if $\A$ is additive.
\lem
\label{filtcolim}
{\it Every idempotent complete additive category with duality is
equivalent to a filtered
colimit of additive
categories with duality all of which are equivalent to $(\P(R),\eta,
*)$ for some ring with involution $(R,\bar{\ })$ and
$*=\operatorname{Hom}_R(\ \ ,R)$ as in \ref{ringwinv}.
}
\prf
According to lemma \ref{lem:dualstrict} and remark \ref{permstrict} we can
suppose our additive category with duality $(\A, \alpha, *)$ to be
strict and to have a
strict duality satisfying $(A\oplus B)^*=B^* \oplus A^*$.
Call a full subcategory with duality finitely
generated if there are finitely many objects in $\A$, called
generators, such that
it is the full subcategory of objects of $\A$ which are direct
factors of finite sums of the generators.
The set of all such subcategories becomes a filtered partially ordered
set $\mathcal{I}$ under inclusion.
Obviously $\A = \colim_{i \in \mathcal{I}}i$.
Taking the sum of the generators we see that every category in
$\mathcal{I}$ is actually generated by a single object.
It therefore suffices to prove the lemma for categories $\A$ as in
\ref{permstrict} generated by a single object, say $A$.
Let $R= End_{\A}(A \oplus A^*)$.
The ring $R$ is equipped with an involution
$\bar{\ }:\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \mapsto
\left(\begin{array}{cc}d^*&b^*\\c^*&a^*\end{array}\right)$.
The inclusion of the one object category with strict duality $(R,
\bar{\ })$ into $(\A,*)$ given by the identity on morphisms
respects the duality and extends to an equivalence between the
idempotent completion of $L(R,\bar{\ })$ and the one of
$L(\A,*)$.
The latter category with duality is equivalent to $(\A,*)$.
The former category is equivalent to
$(\P(R), \operatorname{Hom}_R(\ ,R))$.
The equivalence is induced by the functor $(id,\tau): (R,\bar{\ }) \to
(\P(R), \operatorname{Hom}_R(\ ,R))$ where the natural equivalence
$\tau_R:R \to \operatorname{Hom}_R(R ,R)$ sends $a$ to the map $r
\mapsto \bar{a}r$.
This functor extends to an equivalence between the idempotent
completions of the associated strict additive categories with
duality (see \ref{permstrict} and \ref{lem:dualeq}).
\Qed
\newpage
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\end{document}
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