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\begin{document}
\begin{frontmatter}
\title{Direct limits and fixed point sets%
\thanksref{foot}}
\thanks[foot]{2000 Mathematics Subject Classifications.
Primary: 18A30.
% L<,L>
Secondary: 06A06, 18B05, 18B35, 18B45, 20A99, 20M30.
% posets Set P_cat G_cat gps... M-sets
Preprint versions:
\ /\protect\linebreak[0]%
{$\!\sim$}gbergman/papers/%
dirlimfix.\{tex,dvi,ps\},~
~arXiv:math.CT/0306127.}
\author{George M. Bergman}
\address{University of California,
Berkeley, CA 94720-3840,
USA\\
{\rm gbergman@math.berkeley.edu}
}
\begin{keyword}
action of a group or monoid on a set;
set-valued functor on a category; commutativity of limits with
direct limits (filtered colimits); partially ordered set.
\end{keyword}
\begin{abstract}
For which groups $G$ is it true that whenever one forms a direct
limit of left $G$-sets, $\coLm_{i\in I}\,X_i,$ the set of its fixed
points, $(\coLm_I\,X_i)^G,$ can be obtained as the direct limit
$\coLm_I(X_i^G)$ of the fixed point sets of the given $G$-sets?
An easy argument shows that this is the case if and only if $G$ is
finitely generated.
If we replace ``group $G$'' by ``monoid $M$'', the answer is the
less familiar condition that the improper left congruence on $M$
be finitely generated; equivalently, that $M$ be finitely generated
under multiplication and ``right division''.
Replacing our group or monoid with a small category $\fb E,$
the concept of a set on which $G$ or $M$ acts with that
of a functor $\fb E\rightarrow \fb{Set},$ and the
fixed point set of an action with the limit of a functor,
a criterion of a similar nature is proved.
Specialized criteria are obtained in the cases where $\fb E$
has only finitely many objects and where $\fb E$ is a
(generally infinite) partially ordered set.
If one allows the codomain category $\fb{Set}$ to be replaced with other
categories, and/or allows direct limits to be replaced with other
classes of colimits, one enters a vast area open
to further investigation.
\end{abstract}
\end{frontmatter}
% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\section{Introduction.}\label{outline}
Although the next three sections, concerning fixed point sets of
group and monoid actions, require no familiarity
with category theory, I will (with apologies to the non-categorical
reader) frame this introduction in category-theoretic terms.
It is a familiar observation that ``left universal constructions
respect left universal constructions and right universal constructions
respect right universal constructions'' \cite[\S\S7.7-7.8]{245}.
Thus, when one takes a limit of limits, or a colimit of
colimits (in a context where the relevant limits or colimits
all exist), one can reverse the order of the two limit operations,
or of the two colimit operations, without changing the result.
In contrast, left and right universal constructions do not in general
respect one another.
(For instance, the free group on a direct product set $X\,{\times}\,Y$
is not isomorphic to the direct product of the free group on $X$
and the free group on~$Y).$
But there are classes of cases where, anomalously,
certain limits commute with certain colimits.
For instance, given directed systems of sets $(X_i)_I$ and
$(Y_i)_I$ indexed by the same partially ordered set $I,$
one finds that $\coLm (X_i\,{\times}\,Y_i)\cong (\coLm X_i)\times
(\coLm Y_i).$
Indeed, the fact that we can construct a direct limit of algebras
by putting an algebra structure on the direct limit of their
underlying sets is a consequence of this fact, given that
algebra operations on $X$ are set maps $X{\times}\cdots{\times}X\to X.$
This note investigates the question of which small categories
$\fb E$ have the property that limits of functors from $\fb E$ to
$\fb{Set}$ always commute with direct limits, that is, with
colimits over directed partially ordered sets.
It has been observed
\cite[Thm.~IX.2.1, p.211]{CW}, \cite[Thm.~4.73, p.72]{MK} that
this happens if $\fb E$ is a finite category, i.e., has only
finitely many objects and finitely many morphisms.
More generally, it occurs whenever $\fb E$ has finitely many
objects and finitely {\em generated} morphism-set
(\cite[Prop.~7.9.3]{245} $=$ Corollary~\ref{CfgE} below).
The result of \S\ref{group} (first paragraph of
the above abstract) is equivalent to the statement that
if $\fb E$ is a one-object category whose morphisms form a {\em group},
this finite generation condition is necessary as well as sufficient.
In a general one-object category $\fb E,$ the morphisms form a
monoid $M.$
By the result noted above, finite generation of $M$ is
sufficient
for the construction of limits over $\fb E$ (i.e., fixed-point sets of
$\!M\!$-sets) to commute with that of direct limits, but in this case it
is not necessary.
In \S\ref{moncrit} we obtain two
criteria each of which is necessary as well as sufficient.
We find in \S\ref{cat} that one of these, finite generation
of the improper left congruence on $M,$ when reformulated as
finite presentability of the trivial $\!M\!$-set, generalizes to
arbitrary small categories $\fb E,$ while the other, finite
generation of $M$ under multiplication and ``right division'',
generalizes nicely to categories $\fb E$ with finitely many objects.
In \S\ref{poset} we examine the case where $\fb E$ is the category
$J_\fb{cat}$ induced by a partially ordered set $J,$ and
translate our general criterion into a condition on~$J.$
Half of the condition we get can be stated in familiar language: It says
that the set of minimal elements of $J$ is finite, and every
element lies above a minimal element.
(This is in fact necessary and sufficient for
the comparison maps associated with our limits and colimits to be
{\em injective} in all cases;
it is also {\em necessary} for them always to be surjective.)
The remaining condition appears to be new.
In language which we shall define, it says that the set of elements
of $J$ ``critical'' with respect to the minimal elements is
finite, and that these critical elements
``gather'' all minimal elements under every element of~$J.$
Note that the results of this paper only concern
limits and colimits of functors to $\fb{Set};$
the behavior of functors to other
categories can be strikingly different.
For instance \cite[Exercise~7.9.5]{245},
in $\fb{Set}^{\rm op},$ direct limits do not in general commute with
equalizers, though equalizers are limits over a certain finite category;
but they do commute with not necessarily finite small products;
so we have both negative and positive deviations from the behavior of
$\!\fb{Set}\!$-valued functors.
Clearly, it would be interesting to investigate more classes of cases
of commutativity between limits and colimits: for functors with
codomains other than $\fb{Set},$ and for
colimits over categories other than directed partially ordered sets.
If we fix one of the three variables -- the small category over which
we take limits, the small category over which we take colimits,
and the codomain category -- then we get a Galois connection
\mbox{\cite[\S5.5]{245}} on the other two, and can study the
resulting closure operators.
The exercises in \cite[\S{IX.2}]{CW} and the results and
exercises at the end of \cite[\S7.9]{245} give scattered results
along these lines, but for the most part,
the topic seems wide open for study!
I am indebted to Birge Huisgen-Zimmermann and Ken Goodearl for
organizing the gathering at which I first spoke about some of these
results, to Max Kelly, Arthur Ogus, and Boris Schein for references to
the literature, and to the referee for several helpful corrections
and suggestions.
The present note has various possible audiences, ranging from any
mathematician who uses direct limits, to the specialist in
semigroups or categories or partially ordered sets.
I hope the reader will be patient with my reviewing
details that may be familiar to him or her, and also with my following,
in \S\ref{moninit}, a somewhat leisurely path of motivation
to the results on monoids.
\section{Direct limits and group actions.}\label{group}
Recall that a partially ordered set $(I,\leq)$ is said
to be {\em directed} if for every pair of elements $i,j\in I,$
there exists $k\in I$ majorizing both,
i.e., satisfying $k\geq i$ and $k\geq j.$
A {\em directed system of sets} means a family of sets
$(X_i)_{i\in I}$ indexed by a nonempty directed partially
ordered set $I,$ and given with connecting maps
$\alpha_{i,j}\<{:}\ X_i\to X_j$ $(i\leq j)$ such that
each $\alpha_{i,i}$ is the identity map of $X_i,$ and whenever
$i\leq j\leq k,$ one has $\alpha_{i,k}=\alpha_{j,k}\,\alpha_{i,j}.$
(So a more complete notation for the directed
system is $(X_i,\alpha_{i,j})_{i,j\in I}.)$
In this situation one has the concept of the {\em direct limit} of
the given system.
This is constructed by forming the disjoint union $\bigsqcup_IX_i,$
and dividing out by the least equivalence relation $\sim$
such that $x\sim \alpha_{i,j}(x)$
whenever $x\in X_i$ and $i\leq j.$
Denoting the resulting set
$\coLm_I\,X_i,$ and writing $[x]$ for the equivalence class
therein of $x\in\bigsqcup_IX_i,$ we get, for each $j\in I,$ a map
$\alpha_{j,\infty}\<{:}\ X_j\to \coLm_I\,X_i$
taking $x\in X_j$ to $[x].$
The characterization of $\coLm_I\,X_i$ that we will use here
is that it is a set given with maps
$\alpha_{j,\infty}\<{:}\ X_j\to \coLm_I\,X_i$ for
each $j\in I,$ such that every
element of $\coLm_I\,X_i$ is of the form $\alpha_{j,\infty}(x)$
for some $j\in I,$ $x\in X_j,$ and such that
\begin{xlist}\item\label{eqiff}
$\alpha_{i,\infty}(x)\,=\,\alpha_{j,\infty}(y)\,$ if and only if there
exists $k\geq i,\,j$ such that $\,\alpha_{i,k}(x)\,=\,\alpha_{j,k}(y).$
\end{xlist}
Property~(\ref{eqiff}) is easily deduced from the above
construction of $\coLm_I\,X_i,$ using the directedness of $I.$
Note that it includes the relations
\begin{xlist}\item\label{ijinf}
$\alpha_{i,\infty}(x)\,=\,\alpha_{j,\infty}(\alpha_{i,j}(x))\quad
(i\leq j\in I,~x\in X_i)$
\end{xlist}
corresponding to the generators of the equivalence relation
in that construction.
If $G$ is a group, then a directed system of left $\!G\!$-sets
means a directed system $(X_i,\alpha_{i,j})_{i,j\in I}$ of sets, such
that each $X_i$ is given with a left action of $G,$ and each of the
connecting maps $\alpha_{i,j}$ is a morphism of $\!G\!$-sets (a
$\!G\!$-equivariant map).
Henceforth we will generally omit the qualifier ``left''.
Given such a directed system, it is easy to verify
that $\coLm_I\,X_i$ admits a unique $\!G\!$-action making
the maps $\alpha_{i,\infty}$ morphisms of $\!G\!$-sets, i.e., such that
\begin{xlist}\item\label{g&ioo}
$g\,\alpha_{i,\infty}(x)\,=\,\alpha_{i,\infty}(gx)\quad
(g\in G,\ i\in I,\ x\in X_i).$
\end{xlist}
For any $\!G\!$-set $X,$ let us write
\begin{xlist}\item[]
$X^G\,=\,\{x\in X~|~(\<\forall g\in G)\,\ gx=x\}$
\end{xlist}
for the fixed-point set of the action.
If $(X_i,\alpha_{i,j})_{i,j\in I}$ is a directed
system of $\!G\!$-sets, we see that each map $\alpha_{i,j}$ carries the
fixed set $X^G_i$ into $X^G_j.$
Writing $\beta_{i,j}$ for the restriction of $\alpha_{i,j}$ to a
map $X^G_i\to X^G_j,$ we thus get a directed system of sets
$(X^G_i,\beta_{i,j}),$ and we can form its direct
limit $\coLm_I\,X^G_i.$
It is now straightforward to verify that one has a map
\begin{xlist}\item\label{iotaG}
$\iota:\ \coLm_I\,X^G_i\longrightarrow\,(\coLm_I\,X_i)^G,$ \ \ defined
by \ \ %
$\iota(\beta_{i,\infty}(x))\,=\,\alpha_{i,\infty}(x)$ $(x\in X^G_i).$
\end{xlist}
\begin{thm}\label{TGfixed}
If $G$ is a group, $I$ a directed partially ordered
set, and\linebreak[4] $(X_i,\alpha_{i,j})_{i,j\in I}$ a directed
system of $\!G\!$-sets, then the set-map $\iota$ of~{\rm(\ref{iotaG})}
is one-to-one.
Moreover, for any group $G,$ the following conditions are equivalent:
\begin{xlist}
\item\label{bijG}
For every directed partially ordered set $I$ and
directed system\linebreak[4] $(X_i,\alpha_{i,j})_{i,j\in I}$ of
$\!G\!$-sets, the set-map $\iota$ of~{\rm(\ref{iotaG})} is bijective.
\end{xlist}
\begin{xlist}
\item\label{Gfg}
$G$ is finitely generated.
\end{xlist}
\end{thm}\begin{pf}
The assertion of the first
sentence follows from~(\ref{eqiff}) and the fact
that the maps $\beta_{i,j}$ are restrictions of the $\alpha_{i,j}.$
To see that~(\ref{Gfg}) implies~(\ref{bijG}),
let $\{g_1,\dots,g_n\}$ be a finite generating set for $G,$ and
consider any element of $(\coLm_I\,X_i)^G,$ which we may write
$\alpha_{i,\infty}(x)$ for some $i\in I$ and $x\in X_i.$
The element $x\in X_i$ may not itself be fixed
under $G,$ but by assumption, for every $g\in G$ we have
$g\,\alpha_{i,\infty}(x)=\alpha_{i,\infty}(x),$ in other words,
$\alpha_{i,\infty}(gx)=\alpha_{i,\infty}(x).$
By~(\ref{eqiff}) this means that for each $g\in G$ there
exists $k(g)\geq i$ in $I$ such that
$\alpha_{i,k(g)}(gx)=\alpha_{i,k(g)}(x).$
Since $I$ is directed, we can find a common upper bound
$k$ for $k(g_1),\dots,k(g_n),$ and we see from the
$\!G\!$-equivariance of the maps $\alpha_{k(g_j),k}$ that
$\alpha_{i,k}(x)$ will be invariant under all
of $\{g_1,\dots,g_n\},$ hence will belong to $X^G_k.$
The element $\beta_{k,\infty}(\alpha_{i,k}(x))$ is thus an element
of $\coLm_I\,X^G_i,$ and~(\ref{ijinf}) shows that it
is mapped by~(\ref{iotaG}) to the given
element $\alpha_{i,\infty}(x)\in (\coLm_I\,X_i)^G,$ as required.
Conversely, if $G$ is a non-finitely-generated group, let
$I$ be the set of finitely generated subgroups of $G,$ partially
ordered by inclusion; this is clearly a directed partially ordered set.
For each $H\in I,$ let $X_H$ be the transitive $\!G\!$-set $G/H,$
and define connecting maps by $\alpha_{H_1,H_2}(gH_1) = gH_2$
for $H_1\leq H_2;$ this gives a directed system.
Since each $H\in I$ is a proper subgroup of $G,$ each of the
$\!G\!$-sets $X_H$ satisfies $(X_H)^G = \varnothing,$
so $\coLm_I\,(X_H)^G=\varnothing.$
On the other hand, any two elements $g_1H_1\in X_{H_1}$
and $g_2H_2\in X_{H_2}$ have the same image in
$X_{H_3}$ for any $H_3$ containing $H_1,\; H_2,$ and
$g_1^{-1}g_2\,,$ so $\coLm_I\,X_H$ is the one-point $\!G\!$-set.
Thus $(\coLm_I\,X_H)^G\neq\varnothing,$ and~(\ref{bijG}) fails.\qed
\end{pf}
{\em Digression.}
One may ask whether~(\ref{bijG}) is equivalent to the corresponding
statement with $I$ restricted to be the
set $\Nset$ of natural numbers with the usual ordering~${\leq\<},$
this being the kind of direct limit one generally first learns about.
If we call this weakened condition~(\ref{bijG}$\!_{\Nset}\!$),
I claim the proof of Theorem~\ref{TGfixed} may be adapted to
show that~(\ref{bijG}$\!_{\Nset}\!$) is equivalent to
\begin{xlist}
\item[\ \ (\ref{Gfg}$\!_{\Nset}\!$)]
Every chain $H_0\leq H_1\leq\dots$ of subgroups of $G$
indexed by ${\Nset}$ and having union $G$ is eventually constant.
\end{xlist}
Indeed, suppose $G$ is a group for which~(\ref{bijG}$\!_{\Nset}\!$)
fails, so that we have a directed system $(X_i)_{i\in\Nset}$ and
an element $\alpha_{j,\infty}(x)\in(\coLm_{\Nset}\,X_i)^G$
which is not in the image of $\iota.$
Then no $\alpha_{j,k}(x)$ lies in $X^G_k,$ and letting
$H_i$ be the isotropy subgroup of $\alpha_{j,j+i}(x)$ for each $i,$
it is easy to see that these subgroups give
a counterexample to~(\ref{Gfg}$\!_{\Nset}\!$).
Conversely, if we have a counterexample
to~(\ref{Gfg}$\!_{\Nset}\!$), then setting $X_i=G/H_i$
gives a counterexample to~(\ref{bijG}$\!_{\Nset}\!$).
But are there any groups that satisfy~(\ref{Gfg}$\!_{\Nset}\!$)
and not~(\ref{Gfg})?
Clearly~(\ref{Gfg}$\!_{\Nset}\!$)
cannot hold in any countable non-finitely-generated group.
It will also fail in any group which admits a homomorphism onto
a group in which it fails, from which one can show that it fails
in any non-finitely-generated abelian
group \cite[paragraph following Question~8]{Sym_Omega:1}.
However, examples are known of uncountable nonabelian groups
that satisfy~(\ref{Gfg}$\!_{\Nset}\!$):
Infinite direct powers of nonabelian simple
groups \cite{SK+JT}, full permutation groups on
infinite sets \cite{MN,Sym_Omega:1}, and others
\cite{MD+RG,MD+WCH,Sh,STcof}.
(Groups satisfying~(\ref{Gfg}$\!_{\Nset}\!$)
but not~(\ref{Gfg}) are said to be of ``uncountable cofinality''.
The same condition on modules has been studied under a surprising
variety of names \cite[p.895, top paragraph]{EKN}.)
\section{Monoid actions -- initial observations.}\label{moninit}
If we replace the group $G$ of the preceding section with a general
monoid $M,$ a large part of the discussion goes over unchanged.
Given a directed system $(X_i,\alpha_{i,j})_{i,j\in I}$ of left
$\!M\!$-sets, we get an $\!M\!$-set structure on $\coLm_I\,X_i,$
and there is a natural map
\begin{xlist}
\item\label{iotaM}
$\iota:\ \coLm_I\,X^M_i\longrightarrow\,(\coLm_I\,X_i)^M$\quad given
by $\iota(\beta_{i,\infty}(x))=\alpha_{i,\infty}(x)$ $(x\in X^M_i),$
\end{xlist}
which is always one-to-one; and again we may ask for which $M$ it
is true that
\begin{xlist}\item\label{bijM}
For every directed partially ordered set $I$ and
directed system $(X_i,\alpha_{i,j})_{i,j\in I}$ of
$\!M\!$-sets, the set-map $\iota$ of~(\ref{iotaM}) is bijective.
\end{xlist}
The argument used in the proof of Theorem~\ref{TGfixed},
(\ref{Gfg})\!$\implies$\!(\ref{bijG}),
shows that a {\em sufficient} condition is
\begin{xlist}\item\label{Mfg}
$M$ is finitely generated.
\end{xlist}
%
Attempting to reproduce the converse argument, we can say, as before,
that if $M$ is not finitely generated its finitely generated
submonoids $N$ form a directed partially ordered
set; however, there is no concept of
factor-$\!M\!$-set $M/N,$ as would be needed to continue the argument.
And in fact, there exist non-finitely-generated monoids for
which~(\ref{bijM}) holds.
For instance, let $M$ be the multiplicative monoid of any
field $F;$ note that $0\in M.$
Given an element $\alpha_{j,\infty}(x)\in (\coLm_I\,X_i)^M,$
we have $\alpha_{j,\infty}(x)= 0\,\alpha_{j,\infty}(x)=
\alpha_{j,\infty}(0x),$ hence there exists
$k\in I$ such that $\alpha_{j,k}(x)= \alpha_{j,k}(0x).$
We now observe that for every $u\in M$ we have
\begin{xlist}\item[]
$u\,\alpha_{j,k}(x)\ =\ u\,\alpha_{j,k}(0x)\ =\ %
\alpha_{j,k}((u\,0)x)\ =\ \alpha_{j,k}(0x)\ =\ \alpha_{j,k}(x),$
\end{xlist}
so $\alpha_{j,k}(x)\in X^M_k,$ so the arbitrary element
$\alpha_{j,\infty}(x)\in (\coLm_I\,X_i)^M$
is in the image of~(\ref{iotaM}).
Recalling that an element $z$ of a monoid $M$ is called
a {\em right zero} element if $uz=z$ for all $u\in M,$
we see that the above argument shows that a sufficient condition
for~(\ref{bijM}) to hold, clearly independent of~(\ref{Mfg}), is
\begin{xlist}
\item\label{rtz}
$M$ has at least one right zero element.
\end{xlist}
With a little thought, one can come up with a common generalization
of~(\ref{Mfg}) and~(\ref{rtz}).
Recall that a {\em left ideal} of a monoid means a subset $L$
closed under left multiplication by all elements of $M.$
Combining the ideas of the two preceding arguments, one can show
that~(\ref{bijM}) holds if
\begin{xlist}
\item\label{fgI}
$M$ has a nonempty left ideal $L$ which is finitely generated
as a semigroup.
\end{xlist}
But we can generalize this still further.
We don't need left multiplication by {\em every} element of $M$ to send
{\em every} element of $L$ into $L.$
We claim it suffices to assume that
\begin{xlist}\item[]
$M$ has a finitely generated subsemigroup $S$ such
that $\{a\in M~|\linebreak[0] aS\cap S\neq\varnothing\}$ generates $M.$
\end{xlist}
Indeed, assuming the above holds, and given as before a directed system
$(X_i)_{i\in I}$ of $\!M\!$-sets and an element
$\alpha_{j,\infty}(x)\in (\coLm_I\,X_i)^M,$ choose
$k\geq j$ such that for all elements $g$ of a finite
generating set for $S,$
we have $g\,\alpha_{j,k}(x)=\alpha_{j,k}(x);$
thus $\alpha_{j,k}(x)$ is invariant under the action of $S.$
Writing $\alpha_{j,k}(x)=y,$ note that
for any $a\in M$ such that $aS\cap S\neq\varnothing,$
if we take $s,t\in S$ such that $as=t,$ and apply the two sides of
this equation to $y,$ we get $ay=y,$ showing that $y$ is fixed
under the action of each such element $a.$
Since such elements generate $M,$ we can conclude
that $y\in X^M_k,$ from which~(\ref{bijM}) follows as before.
In the condition just considered, nothing is lost if we replace
the semigroup
$S$ by the monoid $S\cup\{1\}.$
(The same was not true of~(\ref{fgI}), where the property of being
an ideal would have been lost.)
So let us formulate that condition in the more natural form
\begin{xlist}\item\label{fgM0}
$M$ has a finitely generated submonoid $M_0$ such
that $\{a\in M~|\linebreak[2] aM_0\cap M_0\neq\varnothing\}$
generates $M.$
\end{xlist}
To see that this is strictly weaker than~(\ref{fgI}), consider
the monoid presented by infinitely many generators
$x_n$ $(n\in\Nset)$ and $y,$
and the relations saying that all the elements
$x_n\