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\title{Combinatorics of Polyhedra for $n$-Categories}
\author{Todd Trimble}%\footnote{(c) 1999 Todd H. Trimble.}}
\date{September 5, 1999, \textcopyright 1999 Todd H. Trimble }
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\newtheorem*{exampleOne}{Example 1}
\begin{document}
\maketitle
In these notes, we give some combinatorial techniques for constructing
various polyhedra which appear to be intimately connected with the
geometry of weak $n$-categories. These include
\begin{enumerate}[\qquad 1.\quad]
\item Associahedra (and ``monoidahedra"; see below);
\item Permutoassociahedra, and the cellular structure of
Fulton-MacPherson compactifications of moduli spaces
\item ``Functoriahedra" (related to the $A_n$ maps of Stasheff)
\end{enumerate}
Our basic techniques use derivations on operads and bar constructions.
In part A, we introduce derivations, which should be regarded as
``boundary operators" on (set-valued) operads satisfying a Leibniz rule.
Such derivations can be used to construct poset-valued operads; taking
nerves, one gets polyhedral operads. In this way we reconstruct the
associahedra and the Fulton-MacPherson compactifications.
\part{Derivations on operads}
Sections A.1. and A.2. preface the main definitions which begin in % TODO ref
section A.3: they establish notation and terminology and categorical
formalizations of fairly trivial facts. The reader may wish to skip
them or refer back to them as necessary.
\section{}
Initially, we work with set-valued non-permutative operads $M$, which may
be viewed as models of a multisorted algebraic theory with operations
$\sub_k: M_{p} \times M_{q} \to M_{p+q-1}$. (This is the appropriate presentation
when working with \emph{non-unital} operads.) The sorts in this case are
parametrized by natural numbers p, so that we have for each operad $M$
an underlying sequence of sets. Such a sequence may be regarded as a
functor $N \to \Set$ on the discrete category $\NN$ of natural numbers.
When working with permutative operads, it is often convenient to work
with functors on the category of finite sets and bijections, rather
than with functors on the permutation category (which is a skeleton).
(Logically it makes no difference, but some things are clearer and
easier to describe in the former setting.) There are analogues of the
$\sub_{k}$ operations, parametrized by pairs $(S, E)$ where $S$ is a finite
set and $E$ is a subset. Let $S/E$ denote the (pointed) set in which all
of the points of $E$ are identified with a basepoint; if $E$ is empty then
$S/E$ is the disjoint sum of $S$ with a basepoint. Then there is a
substitution map
\begin{align}
\sub_{S, E}: M(S/E) \times M(E) \to M(S).
\end{align}
Throughout these notes, we work in a setting which can be adapted to
both non-permutative and permutative operads.
\section{}
The functor category $[\NN, \Set]$ is a Boolean topos. In this case, this
means that the power object of a sequence is the sequence of power sets.
We let $P: [\NN, \Set] \to [\NN, \Set]$ denote the covariant power object
functor. Thus if $f: X \to Y$ is a morphism ($=$ sequence of functions),
then $Pf: PX \to PY$ sends a sequence of subsets $S_n$ to the sequence of
images $f_{n}(S_n)$.
The functor $P$ carries a structure of monoidal monad. The monoidal
structure map $PA \times PB \to P(A \times B)$ sends a pair of subsets to their
product. The monad unit $u_X: X \to PX$ sends a sequence of elements to
the corresponding sequence of singletons, and the multiplication
$m_X: PPX \to PX$ sends a sequence of subset-collections to the
corresponding sequence of unions. (It is easily checked that $u$ and
$m$ are monoidal natural transformations, so we do get a monoidal monad.)
The algebras of $P$ are sup-lattice objects, \textit{i.e.}, sequences of sup-lattices.
Using the monoidal monad structure on $P$, one can produce a closed
symmetric monoidal structure on $\PAlg$. There is an underlying functor
from $\PAlg$ into the category of (idempotent) commutative monoids in
$[\NN, \Set]$.
The morphisms $A \to B$ in the Kleisli category of $P$ are maps $A \to PB$
in $[\NN, \Set]$, i.e., relations between $A$ and $B$. Thus a binary relation
on A is a Kleisli morphism $A \to A$, and the hom-object $\Kl(A, A)$ of
binary relations is a monoid $\PAlg(PA, PA)$ in the category of sup-lattices.
The reflexive transitive closure of a binary relation $R$ on $A$ may be
defined as the geometric series $1 + R + R^2 + \dots$ in this sup-lattice
monoid. Of course, this is an incarnation of the free monoid construction,
although in this case, $R^\ast = 1 + R + R^2 + \dots$ gives an idempotent monad $T$.
A poset may be defined as an object $A$ together with a $T$-algebra in
$\Kl(A, A)$. Here it just means a sequence of ordinary posets.
All of this section carries over to species, braided species, and
more generally permutation representations of arbitrary groupoids,
starting from the observation that each of these gives a Boolean topos.
In fact, the Boolean assumption isn't even needed.
\section{}
On $[\NN, \Set]$, we have a substitution tensor product whose monoids
are non-permutative operads. If $X$ is a sequence of sets, we let
$OX$ denote the sequence underlying the free operad on $X$.
Using the monoidal structure on the covariant power set functor $P$,
each operad $M$ induces an operad structure on $PM$. Thus we have for
example an operad $POX$. It is very easy to see that the elementhood
relation $e_{M} \rightarrowtail M \times PM$ defines a suboperad.
\begin{definition}
If $A$ is a set-valued operad, and $B$ an operad valued in
commutative monoids, then a derivation $d: A \to B$ is a
map in $[\NN, \Set]$ such that
$d(\sub_{k}(a, b)) = \sub_{k}(da, b) + \sub_{k}(a, db)$.
\end{definition}
For the remainder of these notes, we take $B$ to be of the form $PM$, and
the main examples of derivations take the form $M \to PM$. Such derivations
extend uniquely to derivations $PM \to PM$ which preserve arbitrary sups.
Given a derivation $d: M \to PM$, it is suggestive to regard the elements m
of $M$ as ``cells", and $d(m)$ as the collection of face-cells lying on the
boundary of $m$. The idea is to use $(M, d)$ as combinatorial data to describe
a polyhedral operad. (It seems that Clemens Berger has a notion of
``cellular operad", but I don't know his definition, or how it fits
with the notions given here.)
Many examples occurring in nature are derivations of the form $OX \to POX$.
Notice that derivations of this form correspond bijectively to maps
$X \to POX$ (much as linear maps $V \to A$ from a vector space $V$ into an
algebra $A$ correspond bijectively to derivations $TV \to A$ on the tensor
algebra).
\begin{example}
Let $X$ be empty in degrees $0$ and $1$, and terminal in degrees
$2$ and higher. Then the elements of $OX$ correspond bijectively
to (combinatorial) planar rooted trees, in which every vertex
adjacent to fewer than two incoming edges must be a leaf.
For $n>1$, the unit map $X \to OX$ sends the element in degree $n$
to the ``$n$-sprout" (no nodes except leaves and root). Let $[n]$
denote the n-sprout. Define $X \to POX$ by sending the element
in $X_n$ to the set of all trees in $(OX)_n$ of the form
$\sub_{k}([p], [q])$. Extend this uniquely to a derivation
$d: OX \to POX$.
\end{example}
This example is connected with Stasheff's associahedra, as we explain
in a moment. First, some generalities.
Each derivation $d: M \to PM$ may be regarded as a binary relation on $M$.
Consider its reflexive transitive closure $d^\ast$, \textit{i.e.}, the poset generated
by this binary relation, viewed as a map $M \to PM$, or as a sup-lattice
map $PM \to PM$.
\begin{proposition}
$d^\ast: PM \to PM$ is an operad map.
\end{proposition}
\begin{proof}
We must show that $d^\ast(\sub_{k}(x, y)) = \sub_{k}(d^\ast(x), d^\ast(y))$.
Being sup-lattice morphisms, the $\sub_{k}$ operations distribute
over arbitrary sums. Then we calculate, as with ordinary
derivations (except that we may drop binomial coefficients,
on account of the idempotency of $+$ in $PM$),
\begin{align}
d^{n}(\sub_{k}(x, y)) = \sum_{p+q=n} \sub_{k}(d^{p}(x), d^{q}(y))
\end{align}
and since $d^\ast = 1 + d + d^2 + \dots$, the desired equation
follows easily.
\end{proof}
\begin{remark}
There is a well-known theorem that if $A$ is a Banach algebra
and if $d: A \to A$ is a bounded linear derivation, then
$\exp(d): A \to A$ is a Banach algebra map. The proposition is
analogous: algebra multiplication is replaced by operad
substitution maps, the completeness property of the Banach
algebra by the sup-lattice property of $PM$, and $\exp(d)$
is replaced by the geometric series $d^\ast = 1 + d + d^2 + \dots $
\end{remark}
Since $P$ is a monoidal monad, the unit $u_M: M \to PM$ is an operad morphism,
and we may compose this with $d^\ast: PM \to PM$ to get an operad map $M \to PM$,
again denoted by $d^\ast: M \to PM$. Now, if we pull back (take the inverse
image of) the elementhood relation $e_{M} \to M \times PM$ along the map
$1 \times d^\ast: M \times M \to M \times PM$, we get a suboperad
\begin{align}
R \to M \times M,
\end{align}
since this is obtained as a pullback in the category of operads. In
other words, for each derivation $d: M \to PM$, we get a poset $(M, R)$
in the category of operads, which is the same as an operad in the
category of posets.
\begin{definition}
Given a derivation $d: M \to PM$, we denote by $M_d$ the
posetal operad whose characteristic map is $d^\ast: M \to PM$.
\end{definition}
\begin{exampleOne}[continued]
Given planar trees $t$, $t'$, let us write $t' \to t$
if $t'$ belongs to $d^\ast(t)$. Thus $\to$ is a poset relation. From
before, we have that $t'$ is contained in $d([n])$, where $[n]$ is
the $n$-sprout, if $t' = \sub_{k}([p], [q])$, \textit{i.e.}, if $t'$ has exactly
one internal edge, and $[n]$ is obtained from $t'$ by contracting that
internal edge. In general, $t' \to t$ if $t$ is obtained by
contracting a finite number of internal edges of $t'$. Since
contraction of all internal edges results in an $n$-sprout, we
see that the $n$-th component of the posetal operad $(OX)_d$
has $[n]$ as terminal object, so that its nerve is contractible.
With a little work, one can show that the realization of the
nerve of each component $(OX)_{d}(n)$ is homeomorphic to an $n$-disk.
Indeed, the nerve gives a natural triangulation of Stasheff's
associahedron in dimension $n$.
\end{exampleOne}
The remainder of part A is devoted to the construction of a polyhedral %TODO ref
operad which corresponds to a natural triangulation of the Fulton-MacPherson
compactification of moduli spaces (for $R^k$). In this case, we need to work
with permutative operads. We begin by defining a cell structure for the
moduli space.
For finite sets $S$ of cardinality greater than $1$, we let $F_k[S]$ denote the
spatial species of injective functions $S \to R^k$. For $S = \set{1, \dots, n}$, this
is usually denoted $F_k[n]$: the space of configurations of $n$ points in $R^k$.
We partition $F_k[n]$ into convex cells by using a kind of lexicographic order
(in which the $j$-th coordinate takes priority over the $i$-th coordinate if
$j > i$ [not $j < i$]). For example, $(x, y) < (x', y')$ if $y < y'$ or $y=y'$ and $x < x'$.
The points $p_1, \dots, p_n$ of a configuration may be rearranged in ascending
lexicographic order: $p_{f_1} < \dots < p_{f_n}$ for some permutation $f$. (Note: if
the species $F_k$ is defined on finite sets $S$, then we get an induced linear
order on $S$, not a permutation.) Let us define the type of the configuration
to be a tuple $(f; k_1, \dots, k_{n-1})$ where $f$ is the permutation and $k_j$ is
the index of the last coordinate where $p_{fj}$ and $p_{f(j+1)}$ disagree. One
often uses a ``bar notation" for denoting such types, by placing $k_j$ bars
between $f(j)$ and $f(j+1)$. For example, working with configurations in the
plane, $2||3|1$ is the type of a configuration of three points where points 1
and $3$ lie on the same horizontal line lying above point $2$, and point $3$ is to
the left of point $1$. It is immediate that the region of configurations of
common type form a convex cell. Letting $T(k, n)$ denote the set of types, we
have a typing function $F_k[n] \to T(k, n)$ whose convex fibers partition the
configuration space.
Let $M_k[n]$ (``moduli space") denote the orbit space of $F_k[n]$ under the
action of translations and positive dilatations on $R^k$. Observe that the
typing function factors through $M_k[n]$, so that we have an induced typing
function $M_k[n] \to T(k, n)$.
Keeping $k$ fixed, we let $T_n = T(k, n)$ for $n>1$, and take $T_0$ and $T_1$ to be
empty: this gives a species $T$. We will define a boundary $d: T \to POT$ whose
induced derivation $d: OT \to POT$ provides data for a posetal operad, whose
nerve components naturally triangularize the Fulton-MacPherson (or really
``Axelrod-Singer" for the real case) compactifications. First, a few
preliminaries.
First, it is convenient to view a type in terms of a collection of mutually
disjoint transitive relations $|_{1}$, $|_{2}, \dots, |_{k}$ on the set of
indices $\set{1, \dots, n}$, whose join (in the poset of transitive relations)
is a linear order $f(1) < \dots < f(n)$. Namely, define these $|_{j}$ to be
the minimal relations such that
\begin{enumerate}[\qquad (1)\quad]
\item $f(i) |_{j} f(i+1)$ if there are $j$ bars between $f(i)$ and $f(i+1)$;
\item $p |_{i} q$ and $q |_{j} r$ implies $p |_{\max(i, j)} r$.
\end{enumerate}
(The reader should consider what this means geometrically.)
Instances of $p |_{i} q$ shall be called ``consequences" of the type.
The maximum $i$ where $p |_{i} q$ occurs is called the ``degree" of the type.
By (2), and the fact that the transitive join of all the relations $|_{i} $
is a linear order on $\set{1, \dots, n}$, we see that the transitive join of $|_{i} $
for $i = 1$ up to $j$ must be a disjoint sum of linear orders. Each of these
orders will be called a $j$-connected component, or $j$-component for short.
We may also speak of the $j$-component of a given element in $\set{1, \dots, n}$.
In what follows, we will need certain kinds of ``shufflings". Given two
disjoint $j$-components $A$ and $B$ in a type, a $j$-shuffle of them is a shuffle
on the underlying linear orders, in such a way that $a$ in $A$ is interposed
between $b$ and $b'$ in $B$ only when $b |_{j} b'$, and $b$ in $B$ is interposed between
$a$ and $a'$ in $A$ only when $a |_{j} a'$. Together with all the $|_{i}$ relations on
$A$ and on $B$, one adjoins new relations $b |_{j} a$, $a |_{j} b'$ in the former
case, and $a |_{j} b$, $b |_{j} a'$ in the latter. As a special case, we have
the notion of ``intercalating" one $j$-component within another, where one
places the entire component B between two successive elements $a$, $c$ of $A$.
Or, if $a < b < c$ are successive elements in $A$, we get the same thing by
replacing $b$ by $B$, and we speak then of intercalating $B$ at $b$.
Now let us give rules for describing the boundary operator $d: T \to POT$.
\begin{enumerate}[\qquad Rule (1)]
\item
If $t$ and $t'$ are two types, let us say $t'$ is in $d(t)$ if
$t'$ can be obtained from t as follows: if $t$ has the linear order
$f(1) < \dots < f(n)$, then there exists $i$ such that $f(i) |_{j} f(i+1) $
in $t$, and $t'$ is obtained by removing this pair and the set of
consequences derived using this pair, followed by $(j-1)$-shuffling
the $(j-1)$-components of $f(i)$ and $f(i+1)$, and deriving a new set
of consequences from the relations which result.
\\
\item
The remaining elements in $d(t)$ are of the form $\sub_{E}(t', t'')$,
where E is a subset of $S = \set{1, \dots, n}$, $t$ belongs to $X[S] = X_n$,
$t'$ to $X[S/E]$, and $t''$ to $X[E]$ (see section A.1. for the $\sub_E$
operations). Such an element belongs to $d(t)$ if the following
conditions hold. Let $b$ be the basepoint in $S/E$, and suppose $t'$
has degree $d$. Then the type $t''$ on $E$ should be the same as that
obtained by restriction of $t$-consequences on $S$ to the subset $E$,
and there should exist a list of sets $C_1, \dots, C_d$, nested in
ascending order, where each $C_j$ is a $j$-component in $t''$ which,
when intercalated at $b$ in the $j$-component of $b$ in $t'$, yields
only correct consequences of $t$.
\end{enumerate}
The ideas which govern this description are as follows. First, each type
$t$ represents a convex cell in moduli space, and the boundary of $t$ (in
the moduli space) should be the set of types whose cells are codimension
$0$ parts of the boundary of the cell for $t$. If $p$ represents a configuration
moving through the closure of the $t$-cell, with the points of $p$ ordered
lexicographically, then $p$ undergoes a type transition when the relation
between two points $p(i)$ and $p(i+1)$ changes at the $j$-th coordinate, from
$p(i)_j < p(i+1)_j$ to $p(i)_j = p(i+1)_j$. (We are assuming $p(i)$ and $p(i+1)$
agree in coordinates with indices greater than $j$.) After the transition,
the largest index where $p(i)$ and $p(i+1)$ disagree is at most $j-1$. And in
order for $p$ after the transition to be interior to the codimension $0$
stratum of the boundary, that largest index must be equal to $j-1$. Now we
take into account the possible ways in which the $(j-1)$-th coordinates
differ, while still maintaining the order of points in $p$, and we are led
to the first part of the description of the boundary operator.
As for the second part, the idea is that we are peering into a first-order
infinitesimal neighborhood of a singular configuration, blowing it up,
and examining what types of moving non-singular configurations can approach
the inverse image or blow-up of the singular configuration. Now a singular
configuration, as a function from a finite set $S$ into $R^k$, factors through
a quotient $S/\sim$ which maps injectively into $R^k$; in order to get the
highest dimensional stratum (of the boundary of a $t$-cell in the
Fulton-MacPherson compactification), we restrict to equivalence relations
on $S$ which merely squash a single subset $E$ down to a point. Then we have
a non-singular configuration on $S/E$, so that the singularity is concentrated
on $E$; after blowing up, the points of $E$ are then distinguished according
to tangent directions, thus giving non-singular configurations on $E$
in the blow-up. The resulting pairing of a type on $S/E$ with a type on $E$ may
be viewed as an instance of a formal $\sub_{E}$ operation in the operad freely
generated from the species of types, and the description of how boundaries
of types intersect these applications of $\sub_{E}$ merely reflect properties
of configuration-types as some of their points become infinitely close.
As this example of a polyhedral operad is somewhat complicated, it may be
well to calculate the explicit structure of a few cells. In what follows,
$\ast$ denotes the basepoint $E/E$ in a quotient $S/E$ of $S$. Hence $\sub(\ast||3, 1|2)$
indicates the expression obtained by substituting $1|2$ for $\ast$ in $\ast||3$.
Substituted expressions are indicated using a bracket notation, \textit{e.g.},
$[1|2]||3$.
\begin{example}
The boundary of $1||2$ is $\set{1|2, 2|1}$, by rule (1), and $1|2$, $2|1$
each have empty boundaries. So the picture is
\begin{align}
\diag{8}{5}
{
{1|2}
\ar@{-}[r]^{1||2}
&
{1|2}
}
\end{align}
and one should think of a braiding $c_{x, y}: xy \to yx$.
\end{example}
\begin{example}
The boundary of $1||[2|3] = \sub(1||\ast, 2|3)$ is calculated by the
Leibniz rule. We get
$d(1||[2|3]) = \sub(d(1||\ast), 2|3) + \sub(1||\ast, d(2|3)) $
where the second summand is $0$ since $2|3$ has empty boundary.
From the preceding example, the first summand is
$\sub(1|\ast, 2|3) + \sub(\ast|1, 2|3) = \set{ 1|[2|3], [2|3]|1 } $
so the picture becomes
\begin{align}
\diag{8}{5}
{
{1|[2|3]}
\ar@{-}[r]^{1||[2|3]}
&
{[2|3]|1}
}
\end{align}
which is reminiscent of $c_{x, yz}: x(yz) \to (yz)x$.
\end{example}
\begin{example}
To calculate the boundary of $1|2|3$, we use rule (2). The
only consequences are $1|2$, $2|3$, and $1|3$; and the only ones
which occur as intercalated relations are $1|2$ and $2|3$.
Hence the boundary is $\set{ [1|2]|3, 1|[2|3] }$. The picture is
\begin{align}
\diag{8}{5}
{
{[1|2]|3}
\ar@{-}[r]^{1|2|3}
&
{1|[2|3]}
}
\end{align}
reminiscent of an associativity $a_{xyz}: (xy)z \to x(yz)$.
\end{example}
\begin{example}
It follows easily from Leibniz that $d(1|[2||3]) = \set{ 1|[2|3], 1|[3|2] }$.
\end{example}
\begin{example}
For $1|2||3$, we first apply rule (1), where we remove the
relation $2||3$. This leaves only the consequence $1|2$, so we
perform all possible $1$-shufflings of $3$ with $1|2$. This produces
$1|2|3$, $1|3|2$, $3|1|2$ as part of the boundary.
We can also apply rule (2), which is trickier. Let us start by
writing down the complete set of consequences of $1|2||3$: they
are $1|2$, $2||3$, $1||3$. Now, types involve two or more entries,
so the substituted expression must be one of these consequences.
Now $[1|2]||3$ is clearly a boundary cell: we have
$C_1 = C_2 = 1|2$, and intercalating $1|2$ for $\ast$ in $\ast||3$ gives
the original type expression; therefore this is OK. $1|[2||3]$
is also a boundary cell, which we verify by taking $C_1 = 2$ as
the $1$-component: intercalating $2$ in $1|\ast$ gives the correct
consequence $1|2$. Finally, $[1||3]|2$ is a boundary: by taking
$C_1 = 1$, intercalation in $\ast|2$ gives a correct consequence. It
is easy to check that no other boundary cells are possible.
Of course, these boundary cells also have non-trivial boundaries,
which are calculated essentially as in our prior examples. The
reader might like to verify that the resulting poset has a nerve
where the star of $1|2||3$ looks like
\begin{align}
\diag{4}{6}
{
&
{[1|2]|3}
\ar@{-}[rr]^{[1|2]||3}
\ar@{-}[dl]_{1|2|3}
&&
{3|[1|2]}
\ar@{-}[dr]^{3|1|2}
\\
{1|[2|3]}
\ar@{-}[dr]_{1|[2||3]}
&&
{1|2||3}
&&
{[3|1]|2}
\ar@{-}[dl]^{[1||3]|2}
\\
&
{1|[3|2]}
\ar@{-}[rr]_{1|3|2}
&&
{[1|3]|2}
}
\end{align}
% [1|2]||3
% [1|2]|3 ---------- 3|[1|2]
% / \
% 1|2|3/ \3|1|2
% / \
% 1|[2|3] 1|2||3 [3|1]|2
% \ /
%1|[2||3]\ /[1||3]|2
% \ /
% 1|[3|2] ----------- [1|3]|2
% 1|3|2
which of course is an instance of a braiding hexagon.
\end{example}
\begin{example}
For $[1||2]||3$, the boundary is calculated using Leibniz:
it is easy to check that this yields
$\set{ [1|2]||3, [2|1]||3, [1||2]|3, 3|[1||2] }$. These cells
also have non-trivial boundaries, and the resulting picture
looks like an instance of naturality for the braiding.
\end{example}
\begin{example}
Here is our granddaddy example. We calculate the structure
of $1||2||3$. Using rule (1), we can remove $1||2$ and $1$-shuffle
the $1$-components of $1$ and $2$ (which are just $1$ and $2$ again),
or remove $2||3$ and $1$-shuffle the $1$-components of $2$ and $3$
(which are $2$ and $3$ again). This yields the cells
$1|2||3$, $2|1||3$, $1||2|3$, $1||3|2$.
Using rule (2), we first write the consequences $1||2$, $2||3$,
$1||3$. Now $1||3$ is impossible as a substituted expression,
since it equals its own $2$-component, and intercalating it
in $2||\ast$ or $\ast||2$ leads to a consequences false in $1||2||3$.
The only correct possibilities are $[1||2]||3$ and $1||[2||3]$.
We have already done the essential calculations for obtaining
the structure of these boundary cells. To save space, we will
not exhibit the full-blown $3$-cell structure of $1||2||3$: it
turns out to be a $3$-cell which mediates between the two
ways of giving a commutative diagram proof of the Yang-Baxter
identity in a braided monoidal category. In other words, it
is a structure cell which obtains in a braided monoidal
$2$-category.
\end{example}
\part{Bar constructions}
In this section, we initially work over a base topos $\Set$; all of our
constructions apply in a context where we work over a Grothendieck
topos $\mathscr{E}$ as base. Later in this section, we take $\mathscr{E}$ to be the topos
of Joyal species.
Classically, bar constructions have been used to build classifying
bundles, free resolutions for group cohomology, and similar constructs.
A common characteristic between these constructions is the production of
an acyclic algebraic structure (\textit{e.g.}, a contractible $G$-space $EG$, or
a free resolution of a $G$-module). We begin by formalizing this
characteristic in terms of a universal property for bar constructions.
\section{Acyclic structures}
We follow the algebraists' convention, taking the simplicial category
to mean the category of finite ordinals (including the empty ordinal)
and order-preserving maps. It is well-known that $\Delta$ is initial
amongst strict monoidal categories equipped with a monoid, which in
$\Delta$ is $1$. This induces a monad $1$ $+$ $-$ on $\Delta$, called the \emph{translation
monad}. By composition, this in turn induces a pullback comonad $P$ on simplicial
sets; by the Kan construction, it has a left adjoint which is a monad $C$
on simplicial sets, called the \emph{cone monad}.
To explain this terminology, we recall the topologists' convention, where
$\Delta$ is restricted to the full subcategory $\Delta_+$ of non-empty ordinals.
If $\SSet$ denotes the category of simplicial sets under the algebraists'
convention, and $\SpSet$ that under the topologists' convention, then
by restriction we get a functor $\SSet \to \SpSet$, which has a left
adjoint. The left adjoint augments a $S_{+}$ set $X$ by its set of path
components $\pi_{0}(X)$. Starting with $X$, we can apply the left augmentation,
followed by the cone monad, followed by restriction $\SSet \to \SpSet$:
this gives a monad which, passing to geometric realization, is the
mapping cone of $X \to \pi_{0}(X)$. The right adjoint to this monad carries
a comonad structure; the category of algebras over the monad is equivalent
to the category of coalgebras over the comonad. This category of
algebras/coalgebras could be called the ``\emph{acyclic topos}": the algebras $X$
are acyclic as simplicial sets. More to the point, the algebra structure
$CX \to X$ witnesses this acyclicity by providing a representative basepoint
for each path component of $X$, together with a well-behaved simplicial
homotopy which contracts each component down to its basepoint.
\begin{definition}
An \emph{acyclic structure} is an algebra over $C$ (or coalgebra over $P$).
An \emph{$S$-acyclic structure} is one augmented over a set $S$.
\end{definition}
A morphism between acyclic structures is just a $C$-algebra map; a morphism
between $S$-acyclic structures is one whose component at $S$ is the identity.
It doesn't matter whether the monad $C$ is taken under the algebraists' or
topologists' convention: the category of $C$-algebras in $\SSet$ is equivalent to
the category of $C$-algebras in $\SpSet$, since given a $C$-algebra in $\SSet$,
\begin{align}
\diag{8}{5}
{
\dots
\ar@{..>}@<6pt>[r]
\ar@{..>}[r]
\ar@{..>}@<-6pt>[r]
&
X_{1}
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
X_{0}
\ar[r]
&
X_{-1}
},
\end{align}
it follows from acyclicity that this portion of the simplicial structure
is a split coequalizer, so that the augmentation map is the usual
augmentation to its set of path components.
In the sequel, it will be useful to regard an acyclic structure,
as a functor $X: \Delta^{op} \to \Set$, as a right coalgebra $X\,T \to X$
over the translation comonad $T: \Delta^{op} \to \Delta^{op}$.
\section{Abstract bar constructions}
Next, let us recall the formalism which leads to bar constructions.
Let $U: A \to E$ be a monadic functor over a category $E$ (in most of
our applications, $E$ will be a topos). Let $F: E \to A$ be the left
adjoint, so that we have a monad $M: E \to E$ and a comonad $C: A \to A$.
The comonad may be regarded as a comonoid in the endofunctor category
$[A, A]$ (under the monoidal product given by endofunctor composition).
Since $\Delta^{op}$ is initial amongst strict monoidal categories
equipped with a comonoid, there exists a unique monoidal functor
\begin{align}
\Delta^{op} \to [A, A]
\end{align}
sending the comonoid $1$ to the comonoid $C$. Now if $X$ is an object of
$A$ (\textit{i.e.}, an $M$-algebra), we have an evaluation functor $ev_{X}: [A, A] \to A$.
Thus we have the following composition which leads to a simplicial
$E$-object:
\begin{align}
\diag{8}{5}
{
\Delta^{op}
\ar[r]
&
[A, A]
\ar[r]^{ev_{X}}
&
A
\ar[r]^{U}
&
E
}
\end{align}
and this defines what we mean by the bar construction, denoted $B(M, M, X)$.
More explicitly, this is an augmented simplicial object of the form
\begin{align}
\diag{8}{5}
{
\dots
\ar@{..>}@<6pt>[r]
\ar@{..>}[r]
\ar@{..>}@<-6pt>[r]
&
MMX
\ar@<3pt>[r]^{\mu_X}
\ar@<-3pt>[r]_{M\,\xi}
&
MX
\ar[r]^{\xi}
&
X
},
\end{align}
and this simplicial object admits an acyclic structure where the contracting
homotopy is built out of components of the unit $u$ of the monad $M$:
\begin{align}
\diag{8}{5}
{
{M^{n}\,X}
\ar[r]^{u_{M^{n}X}}
&
{M^{n+1}\,X}
}.
\end{align}
(The ``co-associativity" axiom for the right $T$-coalgebra structure holds
by naturality of $u$.)
\begin{definition}
An $X$-acyclic $M$-algebra is a simplicial $M$-algebra whose
underlying simplicial $E$-object admits an $X$-acyclic structure.
\end{definition}
Notice that we do not require any compatibility between the $M$-algebra
structure and the acyclic structure: the acyclic structure map is not
required, for example, to be a map of simplicial $M$-algebras.
A morphism of $X$-acyclic $M$-algebras is just a map of simplicial $M$-algebras
whose underlying simplicial $E$-map is a morphism of $X$-acyclic structures.
The category of $X$-acyclic $M$-algebras may thus be rendered as a ($2$-)pullback
of
\begin{align}
\diag{8}{5}
{
&
X\mathrm{-Acyclic}(E)
\ar[d]^{\text{``forget"}}
\\
[\Delta^{op}, A]
\ar[r]_{[1, U]}
&
[\Delta^{op}, E]
}
\end{align}
where the vertical arrow is the obvious forgetful functor from the
category of $X$-acyclic structures in $E$.
The universal property of the bar construction is enunciated in the following
\begin{theorem}
$B(M, M, X)$ is initial in the category of $X$-acyclic $M$-algebras.
\end{theorem}
\begin{proof}
The method of proof closely parallels that of the acyclic
models theorem familiar from homological algebra. Let $Y$ be
an acyclic $M$-algebra augmented over $X$:
\begin{align}
\diag{8}{5}
{
\dots
\ar@{..>}@<6pt>[r]
\ar@{..>}[r]
\ar@{..>}@<-6pt>[r]
&
Y_{1}
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
Y_{0}
\ar[r]
&
X
},
\end{align}
so that the $T$-coalgebra structure or contracting homotopy has
the form $X \to Y_{0} \to Y_{1} \to \dots$ In order to get the desired
simplicial $M$-algebra map $B(M, M, X) \to Y$ which preserves the
contracting homotopies, we are forced to use the diagram
\begin{align}
\diag{8}{5}
{
X
\ar[r]^{u_{X}}
\ar[d]_{1}
&
MX
\ar[r]^{u_{MX}}
\ar[d]^{\phi_1}
&
MMX
\ar[r]^{u_{MMX}}
\ar[d]^{\phi_2}
&
MMMX
\ar@{..>}[r]
\ar[d]^{\phi_3}
&
\cdots
\\
X
\ar[r]_{h_0}
&
Y_0
\ar[r]_{h_1}
&
Y_1
\ar[r]_{h_2}
&
Y_2
\ar@{..>}[r]
&
\cdots
}
\end{align}
where $\phi_{n+1}$ is defined as the unique $M$-algebra map which
extends $(h_n)(\phi_{n})$ along $u_{M^{n}X}$. Thus uniqueness is clear;
what remains is to check that the $\phi_n$ form the components of
a simplicial map $\phi$. The proof of this is sketched in the
lemma and corollary which follow.
\end{proof}
Let $\Simp(E)$ denote the category of simplicial objects in $E$. The pullback
comonad $P$ may be regarded as a comonoid in the endofunctor category
$[\Simp(E), \Simp(E)]$, so that there is an induced monoidal functor
\begin{align}
\Delta^{op} \to [\Simp(E), \Simp(E)]
\end{align}
which we may compose with evaluation at a simplicial object $Y$. If
moreover Y carries an acyclic structure, then there is an induced acyclic
structure on this composite, regarded as a simplicial object in $\Simp(E)$:
\begin{align}
\diag{15}{8}
{
\Delta^{op}
\ar[r]
&
[\Simp(E), \Simp(E)]
\ar[r]^{ev_Y}
&
\Simp(E)
}
\end{align}
We denote this bisimplicial object by $B(Y, T, T)$, where $T$ is the aforesaid
translation comonad. Since $P\,Y = Y\,T$, it has the form
\begin{align}
\diag{8}{5}
{
\dots
\ar@{..>}@<6pt>[r]
\ar@{..>}[r]
\ar@{..>}@<-6pt>[r]
&
YTT
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
YT
\ar[r]
&
YTT
}.
\end{align}
Observe that applying evaluation $\Simp(E) \to E$ at the augmented component
(the component we had earlier indexed by the numeral $-1$), this yields $Y$ again:
\begin{align}
\diag{8}{5}
{
\dots
\ar@{..>}@<6pt>[r]
\ar@{..>}[r]
\ar@{..>}@<-6pt>[r]
&
Y_1
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
Y_0
\ar[r]
&
Y_{-1} = X
}.
\end{align}
\begin{lemma}
Let $Y$ be an acyclic $M$-algebra. Let $B(M, M, Y)$ be the bisimplicial
object formed as the bar construction on $Y$. Then there exists a
map of $Y$-acyclic $M$-algebras (valued in $\Simp(E)$)
\begin{align}
B(M, M, Y) \to B(Y, T, T)
\end{align}
whose value at the terminal object $0$ in $\Delta^{op}$ is the identity
map on $Y$.
\end{lemma}
Thinking of $B(M, M, Y) \to B(T, T, Y)$ as a transformation between functors
of the form $\Delta^{op} \to \Simp(E)$, we may post-compose by evaluation
$\Simp(E) \to E$ at the augmented component. This yields precisely
\begin{align}
B(M, M, X) \to Y
\end{align}
whence follows a corollary which completes the proof of the theorem:
\begin{corollary}
There exists an $X$-acyclic $M$-algebra map $B(M, M, X) \to Y$.
\end{corollary}
\begin{proof}[Proof of lemma]
We construct $B(M, M, Y) \to B(Y, T, T)$ inductively, beginning
with the identity on $Y$. The next component is of the form
$\theta_{1}: MY \to YT$, namely the composite
\begin{align}
\diag{10}{5}
{
MY
\ar[r]^{Mh}
&
MYT
\ar[r]^{\xi T}
&
YT
}
\end{align}
where $h$ is the right $T$-coalgebra structure and $\xi$ the
$M$-algebra structure. It is immediate that $\theta_{1}$
preserves the homotopy component (\textit{i.e.}, $h = \theta_{1}(u_Y))$,
and since $\theta_{1}$ is an $M$-algebra map, it quickly
follows that $(Y\,\ep)\,\theta_{1} = \xi$, where $\ep$ is the
counit of the comonad $T$.
We leave to the reader to check that if we inductively
define $\theta_{n}: M^{n}Y \to YT^{n}$ as the composite
\begin{align}
\diag{10}{5}
{
M^{n}Y
\ar[r]^{M^{1}\theta_{n-1} }
&
MYT^{n-1}
\ar[r]^{\theta_{1}T^{n-1}}
&
YT^{n}
}
\end{align}
then it easily follows that
$(\theta_{n})(u_{M^{n-1}Y}) = (hT^{n-1})(\theta_{n-1})$, \textit{i.e.},
$\theta$ preserves the homotopies (preserves acyclic structure).
The fact that $\theta_{n}$ preserves face and degeneracy maps
follows by induction: since $\theta_{n}$ is an $M$-algebra map
by construction, it suffices to check that the relevant
diagrams which obtain by precomposing with a unit u commute,
but since $\theta$ preserves homotopies, one can exploit the
naturality of the homotopies to convert the diagrams into
ones where the inductive assumption applies. In short, the
argument is similar to the usual one for the acyclic models
theorem, and this completes our sketch of the proof.
\end{proof}
\section{Applications}
A classical application of bar constructions is the Milgram bar construction
of a classifying bundle (say of a discrete group $G$). As is well known, the
total space $EG$ is a contractible space on which $G$ acts freely. What appears
less well known is the following theorem.
Let us define a \emph{contractive space} to be a space (in a suitable topological
category, such as the category of compactly generated Hausdorff spaces)
which is an algebra over the cone monad. Here, the cone monad means
the mapping cone of the map $X \to 1$ into the one-point space, and this
is the monad whose algebras are pointed spaces equipped with a continuous
action by the unit interval $I$, the monoid whose multiplication is ``inf",
such that multiplication by $0$ sends every point to the basepoint. An
algebra structure may be viewed as a well-behaved homotopy which contracts
the space to a point.
\begin{theorem}
$EG$ is initial amongst $G$-spaces whose underlying space is equipped
with a contractive structure.
\end{theorem}
\begin{proof}
Let $R: \SSet \to \Top$ be geometric realization. $EG$ is formed as
$RB(G, G, 1)$, where the bar construction is applied to the monad
$G \times -$ on $\Set$. This is a $1$-acyclic space, \textit{i.e.}, a contractive space.
If $X$ is any other contractive $G$-space, we wish to demonstrate that
there is exactly one contractive $G$-map $EG \to X$.
If $S: \Top \to \SSet$ is the singularization functor right adjoint to
$R$, then a map $EG \to X$ gives rise to $B(G, G, 1) \to SX$. Since $G \times -$
as a functor $\Top \to \Top$ is cocontinuous, it is easy to see that
a $G$-map $EG \to X$ gives rise to a $G$-map $B(G, G, 1) \to S\,X$. Next, let
$C$ be the cone monad acting on the category of pointed spaces; then
$C$ is also cocontinuous (it has a right adjoint given by the path
space functor), and it follows as before that a $C$-map $EG \to X$ gives
rise to a $C$-map $B(G, G, 1) \to SX$. Indeed, contractive $G$-maps
$EG \to X$ are in bijective correspondence with $1$-acyclic $G$-maps
$B(G, G, 1) \to SX$ in $S$-set, and there is exactly one of these by
the theorem of the last section. The proof is complete.
\end{proof}
Now let $E$ be the topos of Joyal species, $[P, \Set]$, where $P$ denotes the
permutation category. If $Op$ denotes the category of permutative operads,
then the underlying functor $Op \to [P, \Set]$ is monadic. Let $O$ denote
the monad for this adjunction. If $M$ is an $O$-algebra (an operad), then
there is an associated bar construction $B(O, O, M)$. Needless to say,
it is acyclic.
\begin{example}[Associahedra revisited]
Let $t_+$ be empty in degree $0$, and
terminal in higher degrees. There is a unique operad structure
on $t_+$, as a suboperad of the terminal operad $t$.
Form the bar construction $B(O, O, t_{+})$: this carries a
simplicial $O$-algebra structure, i.e., a simplicial operad
structure. If we form the operad quotient in which every
unary operation is identified with the identity operation,
then the operad which results is a permutative version of the
associahedral operad given in part $A$.
\end{example}
\begin{example}
Now let $t$ be the terminal operad, and form the simplicial
operad $B(O, O, t)$. Again take the operad quotient in which
the operation in $t$ of degree $1$ is identified with the operad
identity. The result is a simplicial operad called the
``\emph{monoidahedral operad}", denoted $M$.
The principle is that we have included a generating operation
in degree $0$, so that we obtain not just higher associativity
laws as in the associahedral operad, but higher unit laws as
well. In contrast to the associahedral operad, the nerve
components here are infinite-dimensional (again, due to the
presence of a nullary operation).
\end{example}
This last example has interesting applications to Trimble's work on
Grothendieck's fundamental $n$-groupoids. Working in the category of
bipointed spaces and bipointed maps, let $I$ denote the unit interval,
bipointed by the pair $(0, 1)$. On the category of bipointed spaces, there
is a monoidal product given by ``wedges" $X \vee Y$, where the second point of
$X$ is identified with the first point of $Y$, and the wedge is bipointed by
the first point of $X$ followed by the second point of $Y$. Then the $n$-fold
wedge of $I$ is canonically identified with the interval $[0, n]$, bipointed
by the pair $(0, n)$.
Now suppose we adapt the monoidahedral operad to the non-permutative setting.
Let $\Bip(I, I^n)$ denote the space of bipointed maps from $I$ to its $n$-fold
wedge; this is the $n$-th component of a tautological non-permutative
operad structure. Notice that these spaces are convex, so that if we take
as basepoint in $\Bip(I, I^n)$ the map $I \to I^n$ given by multiplication by $n$,
then by convexity there is an induced contractive structure on each of these
spaces. Thus we get a contractive spatial operad $\Bip(I, I^\ast)$.
Then there is a $1$-acyclic simplicial operad $S(\Bip(I, I^\ast))$, obtained by
applying singularization to the aforesaid spatial operad. It follows that
there is a unique 1-acyclic operad map $M \to S(\Bip(I, I^\ast))$. This map is
used to identify higher associativity and higher unit laws present in
$\Bip(I, I^\ast)$.
Before giving the next few examples (which are relevant to weak $n$-functors
and their geometry), we need a new definition. If $X$ and $Y$ are species,
we use $X \cdot Y$ to denote their substitution product.
\begin{definition}
Let $A$ and $B$ be operads. An $A-B$ bimodule is a species $X$ equipped
with structure maps $A\cdot X \to X$ and $X \cdot B \to X$, compatible in the
usual sense.
\end{definition}
It is tempting to try to define bimodule composition, where if $X$ is an
$A-B$ bimodule and $Y$ is a $B-C$ bimodule, then the tensor product $XY$ is an
$A-C$ bimodule given by an evident coequalizer of the form
\begin{align}
\diag{8}{5}
{
X \cdot B \cdot Y
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
X \cdot Y
\ar@{->>}[r]
&
XY
}
\end{align}
The trouble is that bimodule composition fails to be associative, because
while $- \cdot Y$ preserves colimits, $X \cdot -$ does not. However, in practice many
coequalizers of the type shown above split, and we can refer to triple
products $XYZ$ without essential ambiguity if the coequalizers ending with
$XY$ and $YZ$ split.
If $M$ is an operad, then the free $M$-bimodule monad $F$ is given by the
assignment $X \mapsto M \cdot X \cdot M$, and if $X$ is itself a bimodule, we obtain a
bar construction $B(F, F, X)$. Some examples follow.
\begin{example}
Let $t_+$ be the permutative version of the operad of example 1, % TODO ref
viewed as a bimodule over itself. Then $B(F, F, t_{+})$ is a
contractive $t$-bimodule whose components are triangularized
\emph{cubes}.
\end{example}
Let us calculate this in detail. In the language of Joyal species and
their analytic functors, we have the linear fractional transformation
\begin{align}
t_{+}(X) = \frac{X}{1-X}.
\end{align}
Let $t_{+}^n$ denote the $n$-fold substitution power of $t_{+}$. Then the
bar construction
\begin{align}
&
\diag{8}{5}
{
\dots
\ar@{..>}@<6pt>[r]
\ar@{..>}[r]
\ar@{..>}@<-6pt>[r]
&
t_{+}^5
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
t_{+}^3
\ar[r]
&
t_{+}
},
\intertext{has the form}
&
\diag{8}{5}
{
\dots
\ar@{..>}@<6pt>[r]
\ar@{..>}[r]
\ar@{..>}@<-6pt>[r]
&
\dfrac{X}{1-5X}
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
\dfrac{X}{1-3X}
\ar[r]
&
\dfrac{X}{1-X}
},
\end{align}
Looking at coefficients, a structure of species $X/(1-kX)$ on $n$ points
is given by a ``combing" or linear order on $\set{1, \dots, n}$ together with
a function $\set{1, \dots, n} \to \set{1, \dots, k}$. We abbreviate the set of such
structures by $k^n$. Keeping the object $n$ of the permutation category $P$
fixed, we get a simplicial object
\begin{align}
\left(
\diag{6}{5}
{
5^n
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
3^n
\ar[r]
&
1
}\right) =
\left(
\diag{6}{5}
{
5
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
3
\ar[r]
&
1
}\right)^n
\end{align}
where the latter power denotes $n$-fold cartesian product in $S$-set.
Passing now to the topologists' convention (\textit{i.e.} truncating the
augmented object $1$), the claim is that
\begin{align}
\diag{8}{5}
{
\dots
\ar@{..>}@<9pt>[r]
\ar@{..>}@<3pt>[r]
\ar@{..>}@<-3pt>[r]
\ar@{..>}@<-9pt>[r]
&
7
\ar@<6pt>[r]
\ar[r]
\ar@<-6pt>[r]
&
5
\ar@<3pt>[r]
\ar@<-3pt>[r]
&
3
}
\end{align}
is the nerve of a once-subdivided $1$-cube : $\cdot - \cdot - \cdot$.
Certainly there are $3$ $0$-cells, and $5$ minus $3$ non-degenerate $1$-cells.
If $n_k$ denotes the number of non-degenerate $k$-cells, then we have
\begin{align}
&
1\cdot(n_0) = 3
\\
&
1\cdot(n_1) + 1\cdot(n_0) = 5
\\
&
1\cdot(n_2) + 2\cdot(n_1) + 1\cdot(n_0) = 7
\end{align}
etc. in Pascal triangle fashion, so that $n_k = 0$ for $k>1$. It is then
very easy to demonstrate that we thus in fact get a subdivided $1$-cube
for the component $n=1$. For higher $n$, we take powers of this $1$-cube, and
this leads to an $n$-cube, suitably triangularized.
\begin{example}
Let $A$ be the associahedral operad of example 1, and consider % TODO ref
$t_{+}$ as an $A-A$ bimodule. Then the bar construction $B(F, F, t_{+})$
gives a canonical triangulation of the polyhedra which
parametrize the operations of Stasheff's $A_n$ maps (see
Homotopy Associativity of H-Spaces I, II).
\end{example}
This example deserves further comments. There is a strong analogy between
the cellular structure of the $A_n$ maps, and the cellular structure of the
data for bihomomorphisms, trihomomorphisms, etc., except that the $A_n$
structures and $A_n$ maps take account only of higher associativities and
their weak preservations, but do not take account of units.
To take account of units, the geometry of $A_n$ maps should be replaced by
the geometry of the bar construction $B(F, F, t)$, where the terminal operad
$t$ is regarded as a bimodule over the monoidahedral operad $M$ of example 2. %TODO ref
The polyhedra which result are again infinite-dimensional. I call these
polyhedra ``\emph{functoriahedra}"; the claim/conjecture is that they carry all
the cellular structure one desires of weak $n$-functors.
\vskip 5cm
\textcopyright 1999 Todd H. Trimble
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\end{document}