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\title{Subdivisions of small categories}
\author{Notes for REU by J.P. May}

\date{Summer, 2008}

\begin{document}

\maketitle

Let $\sA$ be a (small) category.  For example, monoids (sets with associative and unital products) 
can be identified with categories with a single object.  Analogously, 
posets can be identified with those categories $\sA$ with at most one arrow between any two objects 
by defining $x \leq y$ if there is an arrow $x\rtarr y$ between the objects $x$ and $y$ of $\sA$.

In particular, we write $[n]$ for the poset $\{0 <1 \cdots < n\}$.  If we think of $[n]$
as a finite space, then a continuous map $f\colon [m]\rtarr [n]$ is a monotonic, or non-decreasing, function, $f(i)\leq f(j)$ if $i\leq j$.  We can equally well regard such a function 
as a functor $[m]\rtarr [n]$.   We are perilously close to defining the fundamental notion
of a simplicial object in a possibly large category $\sC$, so let's do so.   

Let $\DE$ denote the 
category of posets $[n]$ and monotonic maps between them.  This category is generated
by certain canonical monotonic maps.  We have the ``face map'' $\de_i\colon [n] \rtarr [n+1]$
which is the monomorphism that misses $i$.  That is $\de_i(j) = j$ if $j<i$ and $\de_i(j)= j+1$ 
if $j\geq i$.  We also have the surjection $\si_i\colon [n+1]\rtarr [n]$ that hits $i$ twice.
That is, $\si_i(j) = j$ if $j\leq i$ and $\si_i(j) = j-1$ if $j > i$.   Every morphism in $\DE$
is a composite of these morphisms, and they satisfy certain easily determined identities.  A
simplicial object in $\sC$ is a {\em contravariant} functor $\DE \rtarr \sC$.  In detail,
it is a sequence of objects $C_n\in \sC$ together with face maps $d_i\colon C_n\rtarr C_{n-1}$
and degeneracy maps $s_i\colon C_n\rtarr C_{n+1}$, $0\leq i\leq n$, such that 
$$d_i\com d_j=d_{j-1}\com d_i \ \ \ \text{if}\ \ i<j $$
$$d_i\com s_j=\begin{cases}
s_{j-1}\com d_i  \ \  \ \ \ \text{if}\ \  i<j \\
\id \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if} \ \ i=j\ \ \text{or}\ \ i=j+1\\
s_j\com d_{i-1} \ \ \ \ \ \text{if} \ \ i>j+1.
\end{cases}$$
$$s_i\com s_j=s_{j+1}\com s_i \ \ \text{if}\ \ i\leq j.$$

When $\sC = Set$ is the category of sets, we obtain $sSet$, the category of simplicial sets.  
For example, if
$S$ is a topological space, we obtain the simplicial set $SX$ such that $S_nX$ is the
set of all continuous maps from the standard topological $n$-simplex $\DE_n$ to $X$.
Explicitly, 
\[ \DE_n= \{(t_0,\ldots\!,t_n)|0\leq t_i\leq 1,\ \textstyle{\sum}t_i=1\}
\subset \bR^{n+1}. \]
We have the ``face maps'' 
$$\de_i: \DE_{n-1}\rtarr \DE_n,\ \ 0\leq i\leq n,$$
specified by
$$\de_i(t_0,\ldots\!,t_{n-1})=(t_0,\ldots\!,t_{i-1},0,t_i,\ldots\!,t_{n-1})$$
and ``degeneracy maps''\index{degeneracy map}
$$\si_i: \DE_{n+1}\rtarr \DE_{n},\ \ 0\leq i\leq n,$$
specified by
$$\si_i(t_0,\ldots\!,t_{n+1})=(t_0,\ldots\!,t_{i-1},t_i+t_{i+1},\ldots\!, t_{n+1}).$$
Precomposing with these maps, we obtain the maps $d_i$ and $s_i$ that make $SX$ into
a simplicial set.  It has long been known that we can use simplicial
sets pretty much interchangeably with topological spaces when studying homotopy theory.

For a simplicial set $K$, we define a space $|K|$, called the ``geometric realization'' of $K$,
as follows. As a set
$$ |X|=\textstyle{\coprod}_{n\geq 0} (K_n\times \DE_n)/(\sim),$$
where the equivalence relation $\sim$ is generated by
$$(k,\de_iu)\sim(d_i(k),u) \ \ \text{for}\ \ k\in K_n \ \ \text{and}\ \ u\in \DE_{n-1}$$
and
$$(k,\si_iv)\sim(s_i(k),v) \ \ \text{for}\ \ K\in K_n \ \ \text{and}\ \ v\in \DE_{n+1}.$$
Topologize $|K|$ by giving 
$$|K|^n\equiv \textstyle{\coprod}_{0\leq n\leq q} (K_n\times \DE_n)/(\sim)$$ 
the quotient topology and then giving $|K|$ the topology of the union, so that a subset
is closed if it intersects each $|K|^n$ in a closed subset.  Write $|k,u|$ for points of
$|K|$.  Say that $(k,u)$ is nondegenerate if $k\in K_n$ is not of the form $s_i j$ for 
any $i$ and any $j\in K_{n-1}$ and if $u\in \DE_n$ is an interior point.  Every $(k,u)$ 
is equivalent to one and only one nondegerate point.

Define $\ga: |SX|\rtarr X$ by
$$\ga|f,u|=f(u)\ \ \text{for}\ \ f:\DE_n\rtarr X \ \ \text{and}\ \ u\in \DE_{n}.$$
It is a fact that $\ga$ is a
weak homotopy equivalence for every space $X$, although we shall not prove that.  There
is also a map $\io\colon K\rtarr S|K|$ of simplicial sets specified by $\io(k)(u) = |k,u|$
for $k\in K_n$ and $u\in\DE_n$.  Again, as we also shall not prove, $|\io|\colon |K|\rtarr |S|K||$ is a homotopy equivalence.

There is a neat relationship between $|-|$ and $S$.  They are left and right adjoint functors,
meaning that there is a bijection, natural in both variables between morphism sets:
\[ \text{Top}(|K|,X) \iso \text{sSet}(K,SX). \]
It is specified by letting $f$ correspond to $g$ if $f(|k,u|) = g(k)(u)$.

There is also a construction that assigns a simplicial {\em set} $K^s$ to a simplicial 
{\em complex} $K$.  The idea is to allow repeated elements in the sets of simplices.
It is easiest to define $\sK(X)^s$ for a poset $X$, and then the construction gives a
functor, but one can use any arbitrarily chosen ordering of the set of vertices to apply the 
definition more generally.  One lets the set $\sK(X)^s_n$ of $n$-simplices be the set 
of sequences $x_0\leq x_1\leq \cdots \leq x_n$.  Deleting $x_i$ gives $d_i$, and repeating
$x_i$ gives $s_i$.   The geometric realization gives the geometric realization $|\sK(X)^s|$.
When $X$ is finite this is homeomorphic to any choice of geometric realization as we defined
it earlier, but the definition $|\sK(X)|=|\sK(X)^s|$ works in general and gives a functor of $X$.

With this as background, we turn to the homotopy theory of small categories $\sA$.  We construct
a simplicial set $N\sA$ called the nerve of $\sA$.   Regarding $[n]$ as a category, we define
the set $N_n\sA$ of $n$-simplices to be the set of functors $[n]\rtarr \sA$.  Regarding a 
monotonic function $f\colon [m] \rtarr [n]$ as a functor, precomposition with $f$ gives us the
required contravariant functoriality on $\DE$.    The definition should look very similar to
the definition of the total singular functor $S$ from spaces to simplicial sets.  It gives us
a functor $N$ from $\sC\!at$, the category of small categories and funtors between them, to
simplicial sets.  We define $B\sA = |N\sA|$.  This is called the {\em classifying space}
of the category $\sA$.  When $G$ is a group regarded as a category with a single object,
$BG$ is called the classsifying space of the group $G$.   These are fundamentally important
constructions in topology and its applications.   

The nerve functor $N$ is accompanied by a functor $c\colon sSets \rtarr \sC\!at$.  It is 
left adjoint to $N$, meaning that
\[ \sC\!at(c K, \sA)\iso \text{sSet} (K, N\sA). \]
This means that it is conceptually sensible, but it does not have good homotopical properties.
For a simplicial set $K$, the objects of the category $cK$ are the vertices (= $0$-simplices) of
$K$.  To construct the morphisms, one starts by thinking of the $1$-simplices $y$ as maps
$d_1y\rtarr d_0y$.  One forms all words (formal composites) that make sense, that is, whose
targets and sources match up.  Then one imposes the relations on morphisms determined by
\[   s_0x = \text{id}_x \ \ \text{for} \ \ x\in K_0 \ \ \text{and}\ \ d_1z = d_0 z\com d_2z \ \ \text{for} \ \ z\in K_2. \] 
This makes good sense since if $K = N\sA$, then a $0$-simplex is an object $x$ of $\sA$,
a $1$-simplex $y$ is a map $d_1y\rtarr d_0y$ and $s_0x = \text{id}_x$, and a $2$-simplex $z$ 
is given by a pair of composable morphisms $d_2z$ and $d_0z$ together with their composite 
$d_1z$.  Therefore there is a natural map $cN\sA\rtarr \sA$ that is the identity on zero simplices and is induced by the identity on $1$-simplices.  In fact, it is an isomorphism of categories: it
is the identity on objects, and it presents the category in terms of generators given by the
morphism sets modulo relations determined by the category axioms. There is also a natural map 
$K\rtarr N cK$.  Applied to the boundary of an $n$-simplex $\DE[n]$, it is the inclusion 
$\pa \DE[n]\rtarr \DE[n]$ when $n>2$, as we shall see. For the adjunction, a functor 
$F\colon cK\rtarr \sA$ is constructed from a map of simplicial
sets $g\colon K\rtarr N\sA$ by letting $F$ be the unique functor that agrees with $g$
on objects (= $0$-simplices) and equivalence classes of morphisms (= $1$-simplices).  

Backing up, we define the standard simplicial $n$-simplex $\DE[n]$ to be the simplicial set 
whose $q$-simplices are the monotonic functions $\si\colon [q]\rtarr [n]$; precomposition with 
monotonic functions $\xi\colon [p]\rtarr [q]$ gives the required contravariant functoriality on 
$\DE$.  The nondegenerate $q$-simplices in $\DE[n]$ are the monomorphisms (= strictly monotonic functions) $[q]\rtarr [n]$, and there is one for each subset of $[n]$ of cardinality $q+1$.  We 
may identify the set of all non-degenerate simplices with the poset of non-empty subsets of 
the set $[n]$ of $n+1$ elements, ordered by inclusion.  In other words, $\DE[n] = (\sK([n])^s$ 
is the simplicial set determined by the simplicial complex $\sK([n])$.  A monotonic function 
$\al\colon [m]\rtarr [n]$ gives a map $\al\colon \DE[m]\rtarr \DE[n]$ of simplicial sets that
sends $\si\colon [q]\rtarr [m]$ to $\al\com\si$.  Thus $\DE[-]$ is a covariant functor from
$\DE$ to simplicial sets.

The $n$-skeleton $K^n$ of a simplicial set $K$ is the subsimplicial set generated by 
the $q$-simplices for all $q\leq n$.  Visibly, $cK$ depends only on the  $2$-skeleton
$K^2$.  Therefore the inclusion $K^2\rtarr K$ of simplicial sets induces an isomorphism of 
categories $cK^2\rtarr cK$ for any $K$.  For $\DE_n$ this implies that 
$\pa \DE_n\rtarr \DE_n$ induces an isomorphism on application of $c$ when $n>2$. 
What is truly amazing is that this extreme loss of information disappears after
subdividing twice.  This is something I am trying to better understand myself.

The reader will find it easy to believe that there is a subdivision functor on simplicial
sets that generalizes the subdivision functor $Sd$ on simplicial complexes in the sense 
that $(SdK)^s \iso Sd(K^s)$ for a simplicial complex $K$  This allows 
one to define a subdivision functor on categories by setting $Sd \sA = cSdN\sA$.  One can
iterate subdivision, forming functors $Sd^2$ on both simplicial sets and categories.  What
is mind blowing at first is that the iterated subdivision $Sd^2 \sA$ is actually a poset 
whose classifying space $BSd^2\sA$ is homotopy equivalent to $B\sA$.   I will explain at
least the construction in a slow way to try to make the idea transparent.

For a set $C$ and a simplicial set $L$, one can form a new simplicial set $C\times L$ by
letting $(C\times L)_q = C\times L_q$, and similarly letting the faces and degeneracies
be induced by those of $L$.  A simplicial set $K$ can be reconstructed from the disjoint
union over $n$ of the simplicial sets $K_n\times \DE[n]$ for $n\geq 0$ by taking equivalence
classes under the equivalence relation generated by
\begin{equation}\label{ten}  (\al^*(k), \si) \simeq (k,\al_*(\si)) \end{equation}
for $k\in K_n$, $\si\in \DE[m]_q$, and $\al\colon [m]\rtarr [n]$.  Here $\al^*(k)\in K_m$ is given by
the fact that $K$ is a contravariant functor from $\DE$ to sets and $\al_*(\si)\in \DE[n]_q$ is given by the fact that $\DE[-]$ is a covariant functor from $\DE$ to simplicial sets.  The simplicial 
structure is induced from the simplicial structure on the $\DE[n]$. The point is that an arbitrary pair $(k,\ta)$ 
in $K_n\times \DE[n]_q$ is equivalent to the pair $(\ta(k),\io_q)$ in $K_q \times \DE[q]_q$, where 
$\io_q\colon [q]\rtarr [q]$ is the identity map viewed as a canonical $q$-simplex in $\DE[q]$, and
$\ta\colon [q]\rtarr [n]$ is viewed as a morphism of $\DE$, so that $\ta = \ta_*(\io_q)$.  Identifying equivalence classes of $q$-simplices with elements of $K_q$ in this faction, we find that the faces and degeneracies agree.  Indeed, for $\xi\colon [p]\rtarr [q]$, $\xi\com \io_p = \io_q \com \xi$ and 
\[   (k, \xi^*(\io_q)) = (k,\xi_*(\io_p))\simeq (\xi^*(k),\io_p). \]

We define $Sd \DE[n] = (\sK'[n])^s$.  That is, we take the simplicial set associated to the
barycentric subdivision of the simplicial complex $\sK[n]$, where we again regard $[n]$ as a
poset.  Just like $\DE[-]$, this gives a covariant functor $Sd\DE[-]$ from $\DE$ to simplicial
sets.  We use it in exactly the same way as above to construct $Sd K$ as a quotient of the 
disjoint union of the $\sK[n]\times Sd\DE[n]$ by the equivalence relation of the same form as (\ref{ten}).  This is obviously sensible!  We then define $Sd\sA = cSd N\sA$. 

---------------------- (Notes below)

It is classical that we have a canonical equivalence $Sd K\rtarr K$.  Applying this with 
$K = N\sA$ gives a canonical (realization) equivalence $Sd N\sA\rtarr N\sA$.  Checking that 
the functor $c$, preserves such equivalences (if it does) and using that $c\com N\iso \text{Id}$,
this gives a canonical equivalence $Sd\sA\rtarr \sA$.  We claim that a little more explicitness
shows that $SdSd\sA$ is a poset.  That is, want to reinterpret directly in terms of categories.
(I haven't had time yet).

Historical fact: Don W. Anderson claimed in passing that the last claim is easily 
verified, in his 1978 paper referred to in del Hoya's miserable attempt at a paper.

More accurately and conceptually, $Sd K = K\otimes_{\DE} Sd\DE[-]$, as we have defined it above,
can be rewritten as $K\otimes_{\DE}\DE(-,-)$.  Here the category $\DE$ is giving a cosimplicial simplicial set, namely the represented bifunctor $\DE(-,-)$.  We have the simplicial sets 
$\DE[n]([q]) = \DE([q],[n])$, one for each fixed $n$.  Fixing $q$
and letting $n$ vary, we have covariant functors.  In defining the tensor
product we use the covariant functoriality, but then the simplicial structure on the
result is induced by the contravariant functoriality.  In categorical situations like this
there is an evident associativity isomorphism of simplicial sets
\[ (K\otimes_{\DE} L)\otimes_{\DE}M \iso K\otimes_{\DE} (L\otimes_{\DE}M) \]
where $K$ is simplicial and $L$ and $M$ are cosimplicial simplicial.  Inductively, this implies
that
\[ Sd^n K \iso K\otimes_{\DE} Sd^n\DE[-]. \]
This gives a good hold on these functors, since $Sd^n\DE[-] = (\sK^{(n)}\DE[-])^s$ is just the
classical iterated barycentric subdivision, regarded as a simplicial set.

Can we say anything useful about $c(N\sA\otimes_{\DE} L)$ in general?

Can we simplify the proof (almost impossible to dig out of the literature)
that $Sd^2K$ comes from a classical simplicial complex?


\end{document}