# Variations on the nerve theorem

Daniel A. Ramras  Department of Mathematical Sciences, Indiana University-
Purdue University Indianapolis, 402 N. Blackford, LD 270, Indianapolis, IN
46202, USA<EMAIL_ADDRESS>
###### Abstract.

Given a locally finite cover of a simplicial complex by subcomplexes,
Björner’s version of the Nerve Theorem provides conditions under which the
homotopy groups of the nerve agree with those of the original complex through
a range of dimensions. We extend this result to covers of CW complexes by
subcomplexes and to open covers of arbitrary topological spaces, without local
finiteness restrictions. Moreover, we show that under somewhat weaker
hypotheses, the same conclusion holds when one utilizes the _completed_ nerve
introduced by Fernández–Minian. Additionally, we prove homological versions of
these results, extending work of Mirzaii and van der Kallen in the simplicial
setting. Our main tool is the Čech complex associated to a cover, as analyzed
in work of Dugger and Isaksen. As applications, we prove a generalized
crosscut theorem for posets and some variations on Quillen’s Poset Fiber
Theorem.

###### 2020 Mathematics Subject Classification:

55P10, 55U10 (primary), 06A06 (secondary)

The author was partially supported by the Simons Foundation (grant #579789).

##  1\. Introduction

Given a cover $\mathcal{U}$ of a space $X$, the Borsuk nerve of $\mathcal{U}$
is the simplicial complex $\mathcal{N}(\mathcal{U})$ consisting of all finite,
non-empty subsets $\mathcal{F}\subseteq\mathcal{U}$ satisfying
$\bigcap\mathcal{F}\neq\emptyset$. Assume either that $X$ is a simplicial
complex and $\mathcal{U}$ is a collection of subcomplexes whose union is $X$,
or that $X$ is paracompact and each $U\in\mathcal{U}$ is open. The classical
Nerve Theorems state that if each non-empty finite intersection of sets from
$\mathcal{U}$ is contractible, then $X$ is homotopy equivalent to (the
geometric realization of) $\mathcal{N}(\mathcal{U})$; proofs of these
statements can be found in [5, 12] for simplicial complexes and [14] for
paracompact spaces. Ideas surrounding the Nerve Theorem play an important role
in several areas of pure and applied mathematics. Nerve theorems have been
used extensively in the study of homological stability [17, 23], in
topological combinatorics [3, 4, 5], and in topological data analysis (see [2]
and the many references therein).

There are a number of generalizations and variations on these results in the
literature. In one direction, Bauer et al. [2] recently showed that if
$\mathcal{U}$ is an open cover of an arbitrary topological space $X$, and each
non-empty finite intersection from $\mathcal{U}$ is weakly contractible, then
$X$ is weakly equivalent to $\mathcal{N}(\mathcal{U})$. In [11], Fernández and
Minian showed that when $X$ is a finite simplicial complex, a modified nerve
construction $\hat{\mathcal{N}}(\mathcal{U})$ captures the homotopy type of
$X$ when the finite intersections from $\mathcal{U}$ are disjoint unions of
contractible pieces. (When each such intersection is path connected,
$\hat{\mathcal{N}}(\mathcal{U})=\mathcal{N}(\mathcal{U})$.) In another
direction, Björner [6] and Nagórko [25] showed that in certain contexts, if
the intersection of every set of $k$ elements from $\mathcal{U}$ is
$(n-k+1)$–connected, then the homotopy groups of $X$ and
$\mathcal{N}(\mathcal{U})$ agree through dimension $n$. Björner’s result
applies to locally finite covers of simplicial complexes by subcomplexes,
while Nagorko’s applies to open covers of locally $n$–connected, separable
metric spaces. A homological version of Björner’s result, in which one deduces
a homology equivalence through dimension $n$ from analogous acyclicity
hypotheses on intersections, was proven earlier by Mirzaii and van der Kallen
[23] for covers of simplicial complexes by subcomplexes, without local
finiteness restrictions.

We provide generalizations of all of the above results, which apply both to
covers of CW complexes by subcomplexes and to open covers of locally path
connected topological spaces. In fact, we work more generally with what we
call _locally open_ covers (see Definition 2.8), which include both arbitrary
open covers of locally path connected spaces and open covers $\mathcal{U}$ of
arbitrary spaces in which each finite intersection from $\mathcal{U}$ is path
connected. The following is a slightly simplified version of the main result;
details are found in Theorem 2.13.

######  Theorem 1.1.

Let $\mathcal{V}$ be either a locally open cover of a topological space $X$,
or a cover of a CW complex $X$ by subcomplexes. Assume that for some
$n\geqslant 0$ and each set $\mathcal{F}\subseteq\mathcal{V}$ with cardinality
$0<|\mathcal{F}|\leqslant n$, each path component of the intersection
$\bigcap\mathcal{F}$ is $(n-|\mathcal{F}|+1)$–connected. Then there are
isomorphisms $\pi_{k}(X)\cong\pi_{k}|\hat{\mathcal{N}}(\mathcal{V})|$ for
$k\leqslant n$ and a surjection
$\pi_{n+1}(X)\twoheadrightarrow\pi_{n+1}|\hat{\mathcal{N}}(\mathcal{V})|$.

The maps in Theorem 1.1 are natural with respect to an appropriate notion of
morphisms between covered spaces.

The key observation behind Theorem 1.1 is that Björner’s connectivity
hypotheses match exactly with the hypotheses identified by Ebert and Randal-
Williams [10, Lemma 2.4] for showing that a map of simplicial spaces induces
an $(n+1)$–connected map between (thick) realizations:

######  Lemma 1.2 (Ebert–Randal-Williams).

Let $f_{*}\colon\thinspace X_{*}\to Y_{*}$ be a map of simplicial spaces. If,
for each $k\geqslant 0$, the map $f_{k}\colon\thinspace X_{k}\to Y_{k}$ is
$(n-k)$–connected, then $||f||\colon\thinspace||X_{*}||\to||Y_{*}||$ is
$n$–connected.

The proof of Theorem 1.1 for locally open covers proceeds by identifying
appropriate simplicial models for $X$ and $\hat{\mathcal{N}}(\mathcal{U})$,
using facts about the Čech complex due to Dugger and Isaksen [9]. We then
deduce the corresponding result for CW covers using Hatcher’s construction of
open neighborhoods for subcomplexes [14, Appendix] – this technique is exactly
the method Hatcher uses to show that basic results like the Seifert–Van Kampen
Theorem and the Mayer–Vietoris sequence apply to covers of CW complexes by
subcomplexes.

In Section 9, the strategy behind the proof of Theorem 1.1 is adapted to prove
a (co)homological version (Theorem 9.6), extending the homological nerve
theorem in [23]. The key point is that Lemma 1.2 admits an analogue in
(co)homology, obtained from an analysis of the spectral sequences associated
to the realization of a simplicial space (see Proposition 9.4).

It should be noted that we are not generalizing all aspects of the work on
nerves discussed above:

  * •

Björner shows that a particularly simple map from $X$ to its nerve induces an
isomorphism on homotopy in a range of dimensions (see Question 7.4).

  * •

Fernández and Minian obtain results about _simple_ homotopy type.

  * •

Nagórko allows more general dimension-theoretical hypotheses on the cover.

  * •

Bauer et al. obtain a functoriality statement using the functor carrying a
cover to the geometric realization of the nerve as a _simplicial complex_ ,
whereas we use the geometric realization of the nerve as a _poset_ , which has
many more simplices. While the latter is homeomorphic to the former via
barycentric subdivision, this homeomorphism is not natural for non-injective
simplicial maps, and in general morphisms of covers need not induce injective
maps between the nerves.

  * •

Mirzaii and van der Kallen deduce their result from a more general theorem on
covers of posets by highly connected subposets, which was further extended by
Looijenga and van der Kallen [17].

In Section 8, we present another variation on the Nerve Theorem, of a somewhat
different nature. Consider a CW complex $X$ and a cover $\mathcal{V}$ of $X$
consisting of subcomplexes. We show that in some circumstances where the nerve
of $\mathcal{V}$ _does not_ model the homotopy type of $X$, one may still
obtain a combinatorial model for $X$ from the intersections of the sets in
$\mathcal{V}$. This is done by considering intersections of all sets
containing a given cell of $X$. When $\mathcal{V}$ is locally finite, these
intersections form a subposet (though not a subcomplex) of
$\mathcal{N}(\mathcal{V})$, and hence we think of this as a _partial nerve_
construction.

In the final sections, we present some applications to topological
combinatorics. In Section 10 we generalize results on crosscuts and cutsets in
posets due to Folkman [13], Björner [5, 6] and Ottina [26]. In Section 11, we
apply our results to give several variations on Quillen’s Poset Fiber Theorem
[28]. The Poset Fiber Theorem may be viewed as a method for extracting
information about an order-preserving map $f\colon\thinspace P\to Q$ from the
cover of $Q$ by cones over (or under) its elements. Using this viewpoint,
together with our results on Čech complexes of CW covers, we prove the
following result (see Proposition 11.5).

######  Proposition 1.3.

Let $f\colon\thinspace P\to Q$ be a map of posets with $Q$ finite-dimensional,
and assume every finite set of minimal elements in $Q$ has a join. If
$f^{-1}(Q_{\geqslant m_{1}\vee\cdots\vee m_{k}})$ is $(n-k+1)$–connected for
every collection $m_{1},\ldots,m_{k}$ of minimal elements in $Q$, then $f$ is
$(n+1)$–connected.

Another way to obtain covers of posets is by considering _chains_ instead of
cones. This leads to another variation on Quillen’s theorem (see Proposition
11.12).

######  Proposition 1.4.

Let $f\colon\thinspace P\to Q$ be a map of posets, with $Q$ finite
dimensional. If $f^{-1}(m_{1}\cap\cdots\cap m_{k})$ is $(n-k+1)$–connected for
every set of maximal chains $m_{1},\ldots,m_{k}\subseteq Q$, then $f$ is
$(n+1)$–connected.

Each of these results has a homological analogue, in which vanishing of
homotopy groups is replaced by vanishing of homology in both the hypotheses
and the conclusions.

The paper is organized as follows. In Section 2 we review the completed nerve,
discuss its basic properties, and formulate the main result, Theorem 2.13.
Section 3 introduces the Čech complex, which is a central tool in the proof.
We prove Theorem 2.13 for locally open covers in Section 4 and for CW covers
in Section 6, after establishing some tools for working with CW covers in
Section 5. In Section 7, we discuss the case of simplicial covers of
simplicial complexes somewhat further, giving a more direct proof of Theorem
2.13 for locally finite simplicial covers satisfying somewhat stricter
hypotheses. Partial nerves are introduced and studied in Section 8.
Homological versions of the main result, and of the results from Section 8,
are proved in Section 9. Crosscuts are discussed in Section 10, and fiber
theorems in Section 11.

Acknowledgments: The author thanks Bernardo Villarreal and Omar Antolín-
Camarena for helpful discussions.

##  2\. Nerve constructions

We begin by setting out some basic conventions what will simplify the notation
involved in working with the Borsuk nerve and the completed version introduced
in [11].

It will be convenient to use unary notation for unions and intersections; for
instance, given a set $S$ (whose elements are again sets), we define

$\bigcap S:=\\{x\colon\thinspace\exists A\in S,\,x\in A\\}$

and similarly for $\bigcup S$. We use the same symbol to denote a function
$f\colon\thinspace S\to T$ and the induced function $2^{S}\to 2^{T}$ on power
sets; the latter map is “simplicial” in the sense that it does not increase
cardinality and distributes over unions.

A simplicial complex is a set $K$ such that each element (simplex) $\sigma\in
K$ is a non-empty finite set, and $\emptyset\neq\sigma\subseteq\tau\in K$
implies $\sigma\in K$. Morphisms in the category
$\underline{\mathrm{SimpComp}}$ of simplicial complexes (_simplicial maps_)
are functions $f\colon\thinspace K\to L$ that preserve unions and do not
increase cardinality: $f(\sigma\cup\tau)=f(\sigma)\cup f(\tau)$ and
$|f(\sigma)|\leqslant|\sigma|$ (equivalently, simplicial maps are those
functions sending singleton sets to singleton sets and satisfying
$f(\sigma)=\bigcup_{v\in\sigma}f(\\{v\\})$). Note that if a simplicial map
$f\colon\thinspace K\to L$ is bijective, then it is an isomorphism of
simplicial complexes (that is, $f^{-1}$ is automatically simplicial).

Morphisms in the category $\underline{\mathrm{Poset}}$ of posets are order-
preserving functions. We have functors

$\underline{\mathrm{SimpComp}}\longrightarrow\underline{\mathrm{Poset}}\stackrel{{\scriptstyle\Delta}}{{\longrightarrow}}\underline{\mathrm{SimpComp}}.$

The first functor simply attaches to a simplicial complex its tautological
ordering via set-theoretic inclusion (its _face poset_). We will use the same
symbol to denote both a simplicial complex its associated poset, and we leave
this functor unnamed. The functor $\Delta$ sends a poset $P$ to its order
complex, which is the simplicial complex $\Delta P$ consisting of finite, non-
empty chains (totally ordered subsets) of $P$; if $f\colon\thinspace P\to Q$
is order-preserving, then $f\colon\thinspace 2^{P}\to 2^{Q}$ send chains to
chains and, as mentioned above, is automatically simplicial, and $\Delta
f\colon\thinspace\Delta P\to\Delta Q$ is just the restriction of $f$ to
chains.

There is a functor from $\underline{\mathrm{Poset}}$ to the category
$\underline{\mathrm{Cat}}$ of small categories, given by viewing a partially
ordered set $P$ as a category in which there is a unique morphism from $p$ to
$q$ when $p\leqslant q$ and no morphisms from $p$ to $q$ otherwise. Again we
leave this functor unnamed.

We need to specify functorial a model for geometric realization of posets.
This functor will pass through the category $\underline{\mathrm{SSet}}$ of
simplicial sets, which we view as functors
$\underline{\Delta}^{\mathrm{op}}\to\underline{\mathrm{Set}}$. Here
$\underline{\Delta}\subseteq\underline{\mathrm{Poset}}$ is the full
subcategory on the objects $[n]=\\{0,\ldots,n\\}$, $n=0,1,\ldots$. Geometric
realization defines a functor
$\underline{\mathrm{SSet}}\to\underline{\mathrm{Top}}$, $X_{*}\mapsto|X_{*}|$;
see [10] for a discussion of geometric realization and its generalization to
(semi-)simplicial spaces, which will be needed later in this article. We have
a functor
$N_{*}\colon\thinspace\underline{\mathrm{Poset}}\to\underline{\mathrm{SSet}}$
(also called the nerve, unfortunately) sending $P$ to the simplicial set whose
$n$–simplices are sequences $p_{0}\leqslant\cdots\leqslant p_{n}$ in
$P^{n+1}$, with face and degeneracy maps given by deletion and repetition,
respectively. (This is simply the nerve of the category associated to $P$.) We
will write $|P|$ for the geometric realization of the simplicial set $N_{*}P$;
so $|P|$ has a canonical CW structure with an $n$–cell for each non-degenerate
simplex $p_{0}<\cdots<p_{n}$. Sending a sequence $p_{0}<\cdots<p_{n}$ to the
set $\\{p_{0},\ldots,p_{n}\\}$ gives a bijection between the non-degenerate
simplices in $N_{*}P$ and the elements of $\Delta P$, and this induces a
homeomorphism from the geometric realization of the simplicial complex $\Delta
P$ to $|P|$. Throughout the paper, all topological concepts applied to a poset
$P$ really refer to topological properties of $|P|$.

###  2.1. The completed nerve

When considering a cover $\mathcal{V}$ of a space $X$ in isolation, it is
natural to view $\mathcal{V}$ as a subset of the power set $2^{X}$. However,
we will see in Section 11 that allowing indexed covers, in which the same
subset of $X$ can appear multiple times, affords better functoriality. For
this reason, we will view covers as functions from an index set to $2^{X}$.

######  Definition 2.1.

Let $X$ be a set and let $\mathcal{V}\colon\thinspace I\to 2^{X}$ be a
function. Given a subset $J\subseteq I$, we will use the shorthand notation
$\bigcap J$ to denote the intersection $\bigcap_{j\in J}\mathcal{V}(j)$.

The $($Borsuk$)$ nerve of $\mathcal{V}$ is the simplicial complex

$\mathcal{N}(\mathcal{V}):=\\{\mathcal{F}\subseteq I\colon\thinspace
0<|\mathcal{F}|<\infty\textrm{ and }\bigcap\mathcal{F}\neq\emptyset\\}.$

This level of generality is convenient, since we will consider the case where
$\mathcal{V}$ is a collection of subsets of some topological space, and the
case where $\mathcal{V}$ is a collection of subcomplexes of an (abstract)
simplicial complex.

######  Definition 2.2.

Consider a topological space $X$ and a function $\mathcal{V}\colon\thinspace
I\to 2^{X}$. The completed nerve of $\mathcal{V}$ is the poset
$\hat{\mathcal{N}}(\mathcal{V})$ whose elements are pairs $(\mathcal{F},C)$
with $\mathcal{F}$ a finite, non-empty subset of $I$ and $C$ a path component
of $\bigcap\mathcal{F}$, with order relation given by

$(\mathcal{F},C)\leqslant(\mathcal{F}^{\prime},C^{\prime})\iff\mathcal{F}\subseteq\mathcal{F}^{\prime}\textrm{
and }C^{\prime}\subseteq C.$

Note that when $\mathcal{F}\subseteq\mathcal{F}^{\prime}$, the requirement
$C^{\prime}\subseteq C$ just means that $C$ is the path component of
$\bigcap\mathcal{F}$ containing the points in
$C\subseteq\bigcap\mathcal{F}^{\prime}\subseteq\bigcap\mathcal{F}$.

Fernández and Minian [11, Corollary 3.10] showed that when $X$ is a finite
simplicial complex, and $\mathcal{V}$ is a cover of $X$ by subcomplexes whose
intersections are either empty or a disjoint union of contractible subspaces,
the completed nerve is simple homotopy equivalent to $X$; we give another
proof of this fact in Proposition 7.3.

There are a number of ways to view $\hat{\mathcal{N}}(\mathcal{V})$. The
canonical CW structure on the geometric realization
$|\hat{\mathcal{N}}(\mathcal{V})|$ has a cell for each chain in the poset
$\hat{\mathcal{N}}(\mathcal{V})$, but Fernández and Minian [11, Section 3]
observed that $|\hat{\mathcal{N}}(\mathcal{V})|$ has a (regular) CW structure
whose cells are in bijection with $\hat{\mathcal{N}}(\mathcal{V})$ itself
(with $(\mathcal{F},C)$ corresponding to a cell of dimension
$|\mathcal{F}|+1$). We now give another formulation of this fact.

######  Definition 2.3.

Let $\mathcal{V}\colon\thinspace I\to 2^{X}$ be a collection of subspaces of
the space $X$. Choose a total ordering $\leqslant$ on $I$. We define
$\hat{\mathcal{N}}_{*}(\mathcal{V})$ to be the simplicial set with
$k$–simplices

$\hat{\mathcal{N}}_{k}(\mathcal{V})=\\{((i_{0},\ldots,i_{k}),C)\colon\thinspace
i_{0}\leqslant i_{1}\leqslant\cdots\leqslant i_{k}\in I\textrm{ and
}C\in\pi_{0}(\mathcal{V}(i_{0})\cap\cdots\cap\mathcal{V}(i_{k}))\\}$

and with the simplicial structure map associated to
$\phi\colon\thinspace[m]\to[n]$ given by

$(\\{i_{0},\ldots,i_{n}\\},C)\mapsto(\\{i_{\phi(0)},\ldots,i_{\phi(m)},[C]),$

where $[C]$ is the path component of $C$ in
$\bigcap_{j}\mathcal{V}(i_{\phi(j)})$.

This simplicial set is _almost_ a simplicial complex, except that there may be
multiple simplices sharing the same boundary; for instance, if $i,j\in I$ and
$\mathcal{V}(i)$, $\mathcal{V}(j)$ are path connected but
$\mathcal{V}(i)\cap\mathcal{V}(j)$ is not, then $i$ and $j$ correspond to
vertices in $\hat{\mathcal{N}}_{0}(\mathcal{V})$, and each component
$C\in\pi_{0}(\mathcal{V}(i)\cap\mathcal{V}(j))$ corresponds to a (distinct)
edge in $\hat{\mathcal{N}}_{1}(\mathcal{V})$ connecting $i$ and $j$.

######  Proposition 2.4.

Let $\mathcal{V}\colon\thinspace I\to 2^{X}$ be a collection of subspaces of
the space $X$, and choose a total ordering on $I$. Then the non-degenerate
$k$–simplices in $\hat{\mathcal{N}}_{*}(\mathcal{V})$ are those of the form
$((i_{0},\ldots,i_{k}),C)$ with the $i_{j}$ distinct, and there is a
homeomorphism
$|\hat{\mathcal{N}}(\mathcal{V})|\cong|\hat{\mathcal{N}}_{*}(\mathcal{V})|$.
Moreover, for each non-degenerate $k$–simplex $\eta$ in
$\hat{\mathcal{N}}_{*}(\mathcal{V})$, the smallest sub-simplicial set of
$\hat{\mathcal{N}}_{*}(\mathcal{V})$ containing $\eta$ is isomorphic to the
standard $k$–simplex.

###### Proof.

Recall that the canonical CW structure on a simplicial set has one $k$–cell
for each non-degenerate $k$–simplex [22]. The face (respectively, degeneracy)
maps act on $((i_{0},\ldots,i_{k}),C)\in\hat{\mathcal{N}}_{k}(\mathcal{V})$ by
deleting (repeating) elements from the list $(i_{0},\ldots,i_{k})$ (and
sending $C\in\pi_{0}\left(\bigcap_{j}V(i_{j})\right)$ to its path component in
the intersection of the resulting list), so the non-degenerate simplices in
$\hat{\mathcal{N}}_{k}(\mathcal{V})$ are just those without repetitions. It
follows that faces of $((i_{0},\ldots,i_{k}),C)$ are in bijection with the
non-empty subsets of $\\{i_{0},\ldots,i_{k}\\}$, and this gives the desired
isomorphism with the $k$–simplex. It now follows from the proof of [18,
Theorem 1.7, p. 80] that $|\hat{\mathcal{N}}_{*}(\mathcal{V})|$ is
homeomorphic to the geometric realization of its poset of closed cells,
ordered by inclusion, which is precisely $\hat{\mathcal{N}}(\mathcal{V})$. ∎

Given a cover $\mathcal{V}\colon\thinspace I\to 2^{X}$ of a space $X$, it is
helpful to view the completed nerve of $\mathcal{V}$ in terms of the
Grothendieck construction. There is a tautological functor
$\tau\colon\thinspace\mathcal{N}(\mathcal{V})^{\mathrm{op}}\to\underline{\mathrm{Top}}$
sending $\mathcal{F}$ to $\bigcap\mathcal{F}$ and sending morphisms to
inclusion the maps between these intersections. Composing $\tau$ with the path
component functor $\underline{\mathrm{Top}}\to\underline{\mathrm{Set}}$ gives
a functor

(1)
$\pi_{0}\colon\thinspace\mathcal{N}(\mathcal{V})^{\mathrm{op}}\to\underline{\mathrm{Set}}.$

In general, the Grothendieck construction on a functor
$f\colon\thinspace\mathcal{C}\to\underline{\mathrm{Set}}$ is another category,
which we denote $\int_{C}f$. If the domain category is (the category
associated to) a poset $P$, then $\int_{P}f$ is also poset, whose elements are
pairs $(p,x)$ with $p\in P$ and $x\in f(p)$, and $(p,x)\leqslant(q,y)$ if and
only if $p\leqslant q$ and the function

$f(p\leqslant q)\colon\thinspace f(p)\to f(q)$

carries $x$ to $y$. Applying this construction to the functor (1) yields the
following result.

######  Lemma 2.5.

There is an isomorphism of posets

$\hat{\mathcal{N}}(\mathcal{V})^{\mathrm{op}}\cong\int_{\mathcal{N}(\mathcal{V})^{\mathrm{op}}}\pi_{0}.$

######  Remark 2.6.

Thomason’s theorem [33] now states that $|\hat{\mathcal{N}}(\mathcal{V})|$ is
the homotopy colimit of the functor
$\pi_{0}\colon\thinspace\mathcal{N}(\mathcal{V})^{\mathrm{op}}\to\underline{\mathrm{Set}}\hookrightarrow\underline{\mathrm{Top}}$.

We will now show that $\hat{\mathcal{N}}(\mathcal{V})$ is homotopy equivalent
to the (potentially) smaller poset consisting of all path components
$C\in\pi_{0}\left(\bigcap\mathcal{F}\right)$
($\mathcal{F}\in\mathcal{N}(\mathcal{V})$). Here, and subsequently, we
consider the set of path components $\pi_{0}(Z)$ of a space $Z$ to be a
collection of subsets of $Z$ (so for $C\in\pi_{0}Z$, we have $C\in 2^{Z}$).

######  Proposition 2.7.

Let $(X,\mathcal{V})$ be a partial cover of the space $X$. Define

$\overline{\mathcal{V}}=\bigcup_{\mathcal{F}\in\mathcal{N}(\mathcal{V})}\pi_{0}\left(\bigcap\mathcal{F}\right)\subseteq
2^{X},$

with the ordering inherited from $2^{X}$. Then the order-preserving map

$q\colon\thinspace\begin{array}[t]{ >{\displaystyle}l >{{}}c<{{}}
>{\displaystyle}l
}\hat{\mathcal{N}}(\mathcal{V})^{\mathrm{op}}&\longrightarrow&\overline{\mathcal{V}}\\\
(\mathcal{F},C)&\longmapsto&C\end{array}$

is a homotopy equivalence.

###### Proof.

This is a simple application of Quillen’s Poset Fiber Theorem [28]. The
Quillen fiber of $q$ above $C_{0}\in\overline{\mathcal{V}}$ is the poset

$F:=q^{-1}(\overline{\mathcal{V}}_{\geqslant
C_{0}})=\\{(\mathcal{F},C)\in\hat{\mathcal{N}}(\mathcal{V})\colon\thinspace
C_{0}\subseteq C\\}.$

Note that $(\mathcal{F},C)\in F$ if and only if $C_{0}\subseteq\mathcal{V}(i)$
for all $i\in\mathcal{F}$ and $C$ is the (unique) path component of
$\bigcap\mathcal{F}$ containing $C_{0}$. So $F$ is isomorphic to the face
poset of the full simplicial complex on the set $\\{i\in I\colon\thinspace
C_{0}\subseteq\mathcal{V}(i)\\}$, and is, in particular, contractible. ∎

###  2.2. Covered spaces and functoriality of the completed nerve

We need to define appropriate domain categories for the nerve constructions.
These are variants on the category of covered spaces in [2]. First we
introduce the type of covers to which our arguments apply.

######  Definition 2.8.

Let $X$ be a topological space. A collection $\mathcal{V}\colon\thinspace I\to
2^{X}$ is called a _locally open cover_ of $X$ if

  1. (1)

The interiors of the sets in $\mathcal{V}(I)$ form an open cover of $X$, and

  2. (2)

For each $\mathcal{F}\in\mathcal{N}(\mathcal{V})$, the intersection
$\bigcap\mathcal{F}$ is the topological disjoint union $($coproduct$)$ of its
path components.

Note that for open covers, Condition (2) is equivalent to requiring that each
path component of the open set $\bigcap\mathcal{F}$ is itself open, which
always holds if $X$ is locally path connected. Also, Condition (2) is vacuous
if each intersection $\bigcap\mathcal{F}$ is path connected, as is assumed in
the statement of the classical Nerve Theorem.

######  Definition 2.9.

The category $\underline{\mathrm{PCov}}$ of partial covers has objects
$(X,\mathcal{V})$, where $X$ is a space and $\mathcal{V}\colon\thinspace I\to
2^{X}$ is a collection of subsets of $2^{X}$. A morphism

$(X,\mathcal{V})\longrightarrow(Y,\mathcal{W}\colon\thinspace J\to 2^{X})$

in this category is a pair $(f,\phi)$, where $f\colon\thinspace X\to Y$ is a
map and $\phi\colon\thinspace I\to J$ is a function satisfying
$f(\mathcal{V}(i))\subseteq\mathcal{W}(\phi(i))$ for all $i\in I$. Composition
is given by $(f,\phi)\circ(g,\psi)=(f\circ g,\phi\circ\psi)$.

The categories $\underline{\mathrm{Cov}}$ and $\underline{\mathrm{LOCov}}$ are
the full subcategories of $\underline{\mathrm{PCov}}$ on the objects
$(X,\mathcal{V})$ for which $\mathcal{V}$ is, respectively, a cover $($meaning
that $\bigcup\mathcal{V}=X$$)$ or a locally open cover. Similarly,
$\underline{\mathrm{CWCov}}$ is the full subcategory of
$\underline{\mathrm{Cov}}$ on those $(X,\mathcal{V})$ such that $X$ admits a
CW structure with each $V\in\mathcal{V}$ a subcomplex.

It is straightforward to check that $\underline{\mathrm{PCov}}$ is a category.

If $\alpha=(f,\phi)\colon\thinspace(X,\mathcal{V}\colon\thinspace I\to
2^{X})\to(Y,\mathcal{W}\colon\thinspace J\to 2^{Y})$ is a morphism in
$\underline{\mathrm{PCov}}$ and $\phi$ is a bijection, then the induced map
$\phi^{-1}\colon\thinspace 2^{J}\to 2^{I}$ may or may not send elements of
$\mathcal{N}(\mathcal{W})\subseteq 2^{J}$ to elements of
$\mathcal{N}(V)\subseteq 2^{I}$.

######  Definition 2.10.

We say that a morphism

$(f,\phi)\colon\thinspace(X,\mathcal{V}\colon\thinspace I\to
2^{X})\longrightarrow(Y,\mathcal{W}\colon\thinspace J\to 2^{X})$

is an _equivalence_ if $\phi$ is a bijection and $\phi^{-1}\colon\thinspace
J\to I$ maps $\mathcal{N}(\mathcal{W})$ to $\mathcal{N}(V)$.

######  Proposition 2.11.

The nerve construction extends to a functor

$\mathcal{N}\colon\thinspace\underline{\mathrm{PCov}}\to\underline{\mathrm{SimpComp}},$

which sends a morphism
$\alpha=(f,\phi)\colon\thinspace(X,\mathcal{V})\to(Y,\mathcal{W})$ to the
simplicial map

${\mathcal{N}(\mathcal{V})}$${\mathcal{N}(\mathcal{W}).}$${\mathcal{F}}$${\phi(\mathcal{F})}$$\scriptstyle{\alpha_{*}}$

If $\alpha$ is an equivalence, then $\alpha_{*}$ is an isomorphism of
simplicial complexes.

The completed nerve construction extends to a functor

$\hat{\mathcal{N}}\colon\thinspace\underline{\mathrm{PCov}}\longrightarrow\underline{\mathrm{Poset}}$

which sends sends $(X,\mathcal{V})$ to $\hat{\mathcal{N}}(\mathcal{V})$ and
sends $\alpha=(f,\phi)\colon\thinspace(X,\mathcal{V})\to(Y,\mathcal{W})$ to
the order-preserving map

${\hat{\mathcal{N}}(\mathcal{V})}$${\hat{\mathcal{N}}(\mathcal{W})}$${(\mathcal{F},C)}$${(\phi(\mathcal{F}),[f(C)])}$$\scriptstyle{\hat{\alpha}_{*}}$

where $[f(C)]\in\pi_{0}\left(\bigcap\phi(\mathcal{F})\right)$ is the path
component of $f(C)$.

If $\alpha$ is an equivalence, and for each
$\mathcal{F}\in\mathcal{N}(\mathcal{V})$, the composite

(2) $\bigcap\mathcal{F}\stackrel{{\scriptstyle
f}}{{\longrightarrow}}f\left(\bigcap\mathcal{F}\right)\hookrightarrow\bigcap\phi(\mathcal{F})$

induces a bijection on path components, then $\hat{\alpha}_{*}$ is an order
isomorphism.

This can be proved by an elementary tracing of the definitions. Alternatively,
it follows from general functoriality properties of the Grothendieck
construction, as discussed in Ramras [29, Section 2] or, in more generality,
[16, Chapter 10]. We leave details to the reader.

###  2.3. The Completed Nerve Theorem

We now formulate our main theorem, relating the homotopy type of a completed
nerve to that of the underlying space. The proof will be divided into two
stages, treating locally open covers first in Section 4 and then CW covers in
Section 6.

######  Definition 2.12.

Recall that a topological space $Z$ is said to be $n$–connected $(n\geqslant
0)$ if it is non-empty and $\pi_{k}(Z,z)=0$ for all $z\in Z$ and all
$k\in\\{0,\ldots,n\\}$. We say that $Z$ is $(-1)$–connected if and only if it
is non-empty, and all spaces are considered to be $n$–connected for
$n\leqslant-2$.

We say that a map $f\colon\thinspace Z\to W$ is $n$–connected $(n\geqslant 0)$
if for each $z\in Z$, the induced map $\pi_{k}(Z,z)\to\pi_{k}(Z,f(z))$ is an
isomorphisms for $k\in\\{0,\ldots,n-1\\}$ and a surjection for $k=n$. All maps
are considered to be $n$–connected for all $n\leqslant-1$. We say that $f$ is
a weak equivalence if it is $n$–connected for all $n\geqslant 0$.

Note that we view $\pi_{0}(Z,z)$ as the set of all based homotopy classes of
maps $(S^{0},1)\to(Z,z)$ (where $S^{0}=\\{-1,1\\})$, so if $Z$ is
$n$–connected for some $n\geqslant 0$, then it is path connected, and if
$f\colon\thinspace X\to Y$ is $0$–connected, then the induced map

(3) $f_{*}\colon\thinspace\pi_{0}(X)\to\pi_{0}(Y)$

is surjective, while if $f$ is $n$–connected for some $n\geqslant 1$ then (3)
is bijective.

######  Theorem 2.13.

Consider an object $(X,\mathcal{V})$ in either $\underline{\mathrm{LOCov}}$ or
$\underline{\mathrm{CWCov}}$. Assume that for some $n\geqslant 0$ and each set
$\mathcal{F}\in\mathcal{N}(\mathcal{V})$ with cardinality
$0<|\mathcal{F}|\leqslant n$, every $C\in\pi_{0}(\bigcap\mathcal{F})$ is
$(n-|\mathcal{F}|+1)$–connected. Then there is a natural zig-zag of the form

(4)
$X\stackrel{{\scriptstyle\simeq}}{{\longleftarrow}}|\check{\mathcal{C}}(\mathcal{V})|\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}|\check{\mathcal{C}}^{\delta}(\mathcal{V})|\stackrel{{\scriptstyle\simeq}}{{\longleftarrow}}|N_{*}\mathrm{Simp}\,(\check{\mathcal{C}}^{\delta}(\mathcal{V}))|\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}|N_{*}\hat{\mathcal{N}}(\mathcal{V})|\cong|\hat{\mathcal{N}}(\mathcal{V})|$

connecting $X$ to the completed nerve of $\mathcal{V}$, with $\pi$ an
$(n+1)$–connected map and all other maps weak equivalences. In particular,
when $X$ is path connected, so is $|\hat{\mathcal{N}}(\mathcal{V})|$, and
there are isomorphisms
$\pi_{k}(X)\cong\pi_{k}|\hat{\mathcal{N}}(\mathcal{V})|$ for $k\leqslant n$
and a surjection
$\pi_{n+1}(X)\twoheadrightarrow\pi_{n+1}|\hat{\mathcal{N}}(\mathcal{V})|$.

Note that when $n=0$, the conditions on $\mathcal{N}(\mathcal{V})$ are
vacuous.

The spaces appearing in (4), which are variations on the Čech complex
construction, will be defined in Sections 3 and 4. Naturality of the zig-zag
means that each of the maps in (4) is part of a natural transformation of
functors $\underline{\mathrm{PCov}}\to\underline{\mathrm{Top}}$.

We will see in Section 9 that the strategy behind the proof of Theorem 2.13
also yields a (co)homological version.

######  Remark 2.14.

As noted in the Introduction, even when all intersections $\bigcap\mathcal{F}$
are path connected, so that $\hat{\mathcal{N}}(\mathcal{V})=\mathcal{N}(V)$,
it is important that we view the assignment
$(X,\mathcal{V})\mapsto|\mathcal{N}(\mathcal{V})|$ as the composite functor

$\underline{\mathrm{LOCov}}\xrightarrow{\mathcal{N}(\cdot)}\underline{\mathrm{SimpComp}}\longrightarrow\underline{\mathrm{Poset}}\stackrel{{\scriptstyle|\cdot|}}{{\longrightarrow}}\underline{\mathrm{Top}},$

rather than forming the geometric realization of $\mathcal{N}(\mathcal{V})$ as
a simplicial complex.

Before introducing Čech complexes and proving the theorem, we wish to explain
what Theorem 2.13 tells us about homotopy in low dimensions.

######  Corollary 2.15.

For every object $(X,\mathcal{V}\colon\thinspace I\to 2^{X})$ in either
$\underline{\mathrm{LOCov}}$ or $\underline{\mathrm{CWCov}}$, there is a
bijection $\pi_{0}X\cong\pi_{0}|\hat{\mathcal{N}}(\mathcal{V})|$, given by
sending the the component of a vertex
$(\mathcal{F},C)\in\hat{\mathcal{N}}(\mathcal{V})$ to the component of $X$
containing $C$. Furthermore, there is a surjection

$\pi_{1}(X,x)\twoheadrightarrow\pi_{1}(|\hat{\mathcal{N}}(\mathcal{V})|,(\\{i\\},C))$

whenever $i\in I$ and $x\in C\in\pi_{0}\mathcal{V}(i)$.

If, in addition, for every $i\in\mathcal{V}$ we have
$\pi_{1}(\mathcal{V}(i),x)=0$ for all $x\in V$, then there are isomorphisms
$\pi_{1}(X,x)\cong\pi_{1}(|\hat{\mathcal{N}}(\mathcal{V})|,(\\{i\\},C))$
$($with $x$, $V$, and $C$ as before$)$ as well as corresponding surjections on
$\pi_{2}$.

###### Proof.

We phrase the proof for locally open covers; the proof for CW covers is
similar, but simpler. Note that every locally open cover satisfies the
hypotheses of Theorem 2.13 with $n=0$, and for $n=1$ the only requirement is
that the the path components of each $V\in\mathcal{V}$ must be simply
connected. Hence Theorem 2.13 tells us that the zig-zag (4) induces a
bijection on $\pi_{0}$ and has the claimed behavior on $\pi_{1}$ and
$\pi_{2}$.

It remains only to check that the effect of (4) on path components is as
described in the Corollary. When $X$ is path connected there is nothing to
check (since we know
$\pi_{0}(\hat{\mathcal{N}}(\mathcal{V}))\cong\pi_{0}(X)$). In general, let $C$
be the path component of $x\in X$. First, we define

$\mathcal{V}\cap C\colon\thinspace I\to 2^{C}$

by

$(\mathcal{V}\cap C)(i)=\mathcal{V}(i)\cap C.$

We claim that $\mathcal{V}\cap C$ is a locally open cover of $C$. First, since
the interiors of the sets in $\mathcal{V}(I)$ cover $X$, for each $x\in C$ we
have $x\in U\subseteq\mathcal{V}(i)$ for some open set $U\subseteq X$ and some
$i\in I$, and now $x\in U\cap C\subseteq\mathcal{V}(i)\cap C$ shows that $x$
is in the interior of $\mathcal{V}(i)\cap C$. Next, we know that for each
$i_{1},\ldots,i_{k}\in I$, the intersection $\bigcap_{j}\mathcal{V}(i_{j})$ is
the coproduct of its path components, and we need to verify that the same
holds for $\bigcap_{j}(\mathcal{V}(i_{j})\cap
C)=(\bigcap_{j}\mathcal{V}(i_{j}))\cap C$. But for each path-connected set
$D\subseteq\bigcap_{j}\mathcal{V}(i_{j})$, we have either $D\cap C=\emptyset$
or $D\cap C=D$, so $\left(\bigcap_{j}\mathcal{V}(i_{j})\right)\cap C$ is the
union of a subset of $\pi_{0}\left(\bigcap_{j}\mathcal{V}(i_{j})\right)$,
which suffices.

It now follows from Theorem 2.13 that $|\hat{\mathcal{N}}(\mathcal{V}\cap C)|$
is path connected. The inclusion $C\xhookrightarrow{\iota_{C}}X$ induces a map
of locally open covers

$(\iota_{C},\mathrm{Id})\colon\thinspace(C,\mathcal{V}\cap
C)\to(X,\mathcal{V}),$

and naturality of the zig-zag (4) shows that on $\pi_{0}$, (4) maps $C$ to the
(unique) component in the image of $|\hat{\mathcal{N}}(\mathcal{V}\cap
C)|\to|\hat{\mathcal{N}}(C)|$, which contains all vertices (zero-simplices) of
the form $(\\{i\\},C)$. ∎

######  Remark 2.16.

We note that the bijection
$\pi_{0}X\cong\pi_{0}(\hat{\mathcal{N}}(\mathcal{V}))$ can also be proven
directly, and does not require any conditions on the path components of
intersections from $\mathcal{V}$. Briefly, the inverse of the function
$(\mathcal{F},C)\mapsto[C]$ is given by sending the path component $[x]$ of
$x\in X$ to $(\\{i\\},[x])$, where $x$ is in the interior of $\mathcal{V}(i)$;
the main step is to show that this inverse is well-defined. Given a path
$\gamma\colon\thinspace[0,1]\to X$ with $\gamma(0)=x$ and $\gamma(1)=y$, one
builds a path from $(\\{i\\},[x])$ to $(\\{j\\},[y])$ $($where $y$ lies in the
interior of $\mathcal{V}(j)$$)$ in the 1-skeleton of
$|\hat{\mathcal{N}}(\mathcal{V})|$ by covering $[0,1]$ with $($relatively$)$
open intervals mapping into the interiors of sets in $\mathcal{V}(I)$, and
choosing an appropriate finite subcover.

##  3\. The Čech complex

In this section we introduce the simplicial machinery needed to prove the main
results.

######  Definition 3.1.

Let $\mathcal{V}:I\to 2^{X}$ be a partial cover of the topological space $X$.
The Čech complex of $\mathcal{V}$ is the simplicial space
$\check{\mathcal{C}}_{*}(\mathcal{V})\colon\thinspace\underline{\Delta}^{\mathrm{op}}\to\underline{\mathrm{Top}}$
whose $n$th level is given by

(5) $\check{\mathcal{C}}_{n}(\mathcal{V}):=\\{((i_{0},\ldots,i_{n}),x)\in
I^{n+1}\times X\colon\thinspace x\in V_{i_{0}}\cap\cdots\cap V_{i_{n}}\\},$

where $I^{n}$ is given the discrete topology. To define the simplicial
structure on $\check{\mathcal{C}}(X)$, let $\phi\colon\thinspace[k]\to[n]$ be
an order-preserving map. Then $\phi$ defines a function
$\phi^{*}\colon\thinspace I^{n}\to I^{k}$, with

$\phi^{*}(i_{0},\ldots,i_{n})=(i_{\phi(0)},\ldots,i_{\phi(k)}),$

and the structure map associated to $\phi$ is given by
$(\phi^{*},\mathrm{Id}_{X})$. Tracing the definitions shows that this defines
a functor
$\check{\mathcal{C}}_{*}(\mathcal{V})\colon\thinspace\Delta^{\mathrm{op}}\to\underline{\mathrm{Top}}$.

Note that $\check{\mathcal{C}}_{n}(\mathcal{V})$ is the coproduct, over the
set $I^{n+1}$, of all the intersections $V_{i_{0}}\cap\cdots\cap V_{i_{n}}$.
It will be convenient, notationally, to write this coproduct in the form (5).

It is helpful to recognize the Čech complex as the (categorical) nerve of a
certain internal category in $\underline{\mathrm{Top}}$. For this we need a
simple lemma.

######  Lemma 3.2.

Let $\mathcal{V}\colon\thinspace I\to 2^{X}$ be a partial cover. Then for each
$n\geqslant 1$, there is a homeomorphism

$\check{\mathcal{C}}_{n}(\mathcal{V})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\lim\left(\check{\mathcal{C}}_{n-1}(\mathcal{V})\xrightarrow{d_{0}^{n-1}}\check{\mathcal{C}}_{0}(\mathcal{V})\stackrel{{\scriptstyle
d_{1}}}{{\longleftarrow}}\check{\mathcal{C}}_{1}(\mathcal{V})\right)=:\check{\mathcal{C}}_{n-1}(\mathcal{V})\times_{\check{\mathcal{C}}_{0}(\mathcal{V})}\check{\mathcal{C}}_{1}(\mathcal{V}),$

induced by the simplicial structure maps
$\check{\mathcal{C}}_{n}(\mathcal{V})\stackrel{{\scriptstyle
d_{n}}}{{\longrightarrow}}\check{\mathcal{C}}_{n-1}(\mathcal{V})$ and
$\check{\mathcal{C}}_{n}(\mathcal{V})\stackrel{{\scriptstyle
d_{0}^{n-1}}}{{\longrightarrow}}\check{\mathcal{C}}_{1}(\mathcal{V})$ in the
simplicial space $\check{\mathcal{C}}_{*}(\mathcal{V})$.

###### Proof.

It follows from the simplicial identities and the universal property of limits
that $d_{n}$ and $d_{0}^{n-1}$ induce a continuous map

$\alpha\colon\thinspace\check{\mathcal{C}}_{n}(\mathcal{V})\longrightarrow\check{\mathcal{C}}_{n-1}(\mathcal{V})\times_{\check{\mathcal{C}}_{0}(\mathcal{V})}\check{\mathcal{C}}_{1}(\mathcal{V}).$

Tracing the definitions, one sees that $\alpha$ is a (continuous) bijection,
so we just need to check that $\alpha$ is an open map. Each open set in
$\check{\mathcal{C}}_{n}(\mathcal{V})$ is a coproduct of sets of the form
$\\{(i_{0},\ldots,i_{n})\\}\times(U\cap V_{i_{0}}\cap\cdots\cap V_{i_{n}})$,
where $U\subseteq X$ is open and $i_{0},\ldots,i_{n}\in I$. So it suffices to
observe that $\alpha(\\{(i_{0},\ldots,i_{n})\\}\times(U\cap
V_{i_{0}}\cap\cdots\cap V_{i_{n}}))$ is equal to the intersection of
$\check{\mathcal{C}}_{n-1}(\mathcal{V})\times_{\check{\mathcal{C}}_{0}(\mathcal{V})}\check{\mathcal{C}}_{1}(\mathcal{V})$
with the set

$\left(\\{(i_{0},\ldots,i_{n-1})\\}\times(U\cap V_{i_{0}}\cap\cdots\cap
V_{i_{n-1}})\right)\times\left(\\{(i_{n-1},i_{n})\\}\times(U\cap
V_{i_{n-1}}\cap V_{i_{n}})\right),$

which is open in
$\check{\mathcal{C}}_{n-1}(\mathcal{V})\times\check{\mathcal{C}}_{1}(\mathcal{V})$.
∎

######  Proposition 3.3.

Let $\mathcal{V}\colon\thinspace I\to 2^{X}$ be a partial cover of the space
$X$. Then $\check{\mathcal{C}}_{*}(\mathcal{V})$ is isomorphic to the nerve of
an internal category $\underline{\mathcal{V}}$ in $\underline{\mathrm{Top}}$
with object space $\check{\mathcal{C}}_{0}(\mathcal{V})=\coprod_{i\in
I}\mathcal{V}(i)$ and morphism space
$\check{\mathcal{C}}_{1}(\mathcal{V})=\coprod_{(i,j)\in
I^{2}}\mathcal{V}(i)\cap\mathcal{V}(j)$, and with domain and range maps given
by the face operators $d_{1}$ and $d_{0}$ in
$\check{\mathcal{C}}_{*}(\mathcal{V})$ $($respectively$)$.

###### Proof.

We need to define a continuous composition operation for this category. By
definition of an internal category, the domain of this operation must be the
fiber product
$\check{\mathcal{C}}_{1}(\mathcal{V})\times_{\check{\mathcal{C}}_{0}(\mathcal{V})}\check{\mathcal{C}}_{1}(\mathcal{V})\cong\check{\mathcal{C}}_{2}(\mathcal{V})$
appearing in Lemma 3.2, so we can define composition by simply composing this
homeomorphism with the face map
$d_{1}\colon\thinspace\check{\mathcal{C}}_{2}(\mathcal{V})\to\check{\mathcal{C}}_{1}(\mathcal{V})$.
The nerve of the resulting category, by definition, is a simplicial space with
$n$th level given by an $n$–fold iterated fiber product of the form

$\check{\mathcal{C}}_{1}(\mathcal{V})\times_{\check{\mathcal{C}}_{0}(\mathcal{V})}\check{\mathcal{C}}_{1}(\mathcal{V})\times_{\check{\mathcal{C}}_{0}(\mathcal{V})}\cdots\times_{\check{\mathcal{C}}_{0}(\mathcal{V})}\check{\mathcal{C}}_{1}(\mathcal{V}),$

and general properties of limits provide natural homeomorphisms from these
iterated fiber products to the ones from Lemma 3.2. These maps combine to give
the desired isomorphism of simplicial spaces. ∎

We call the category $\underline{\mathcal{V}}$ in Proposition 3.3 the Čech
category of $\mathcal{V}$. We will need two general facts about internal
category theory in $\underline{\mathrm{Top}}$. Let $\mathcal{C}$ and
$\mathcal{D}$ be internal categories in $\underline{\mathrm{Top}}$. A functor
$\mathcal{C}\to\mathcal{D}$ is called _continuous_ if its defining functions
between object and morphism spaces are continuous, and a natural
transformation between functors $\mathcal{C}\to\mathcal{D}$ is called
continuous if its defining function from objects in $\mathcal{C}$ to morphisms
in $\mathcal{D}$ is continuous. It is immediate from the definitions that a
continuous functor $\mathcal{C}\to\mathcal{D}$ induces a map of simplicial
spaces $N_{*}\mathcal{C}\to N_{*}\mathcal{D}$ between the nerves, and hence a
continuous map between their geometric realizations. A natural transformation
induces a continuous functor $\mathcal{C}\times\\{0,1\\}\to\mathcal{D}$, where
$\\{0,1\\}=[1]$ is the poset with $0<1$, and the projection maps induce an
isomorphism of simplicial spaces
$N_{*}(\mathcal{C}\times\\{0,1\\})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}N_{*}(\mathcal{C})\times
N_{*}(\\{0,1\\})$. Moreover, since $|N_{*}(\\{0,1\\})|\cong[0,1]$ is compact,
the projections onto the factors induce a homeomorphism
$|N_{*}(\mathcal{C})\times
N_{*}(\\{0,1\\})|\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}|N_{*}(\mathcal{C})|\times[0,1]$.
In summary, we have established the following well-known fact. (See also the
proof of [9, Proof of Lemma 2.4].)

######  Lemma 3.4.

A continuous natural transformation between continuous functors induces a
homotopy between their geometric realizations.

##  4\. Completed nerves of locally open covers

This section consists of the proof of Theorem 2.13 in the case of a locally
open cover $(X,\mathcal{V}\colon\thinspace I\to 2^{X})$. We will work our way
from left to write along the zig-zag (4). The first step is essentially due to
Dugger and Isaksen, and we state it separately for later reference.

######  Proposition 4.1 (Dugger-Isaksen).

Let $X$ be a topological space and let $\mathcal{V}\colon\thinspace I\to
2^{X}$ be a cover of $X$ such the interiors of the sets in $\mathcal{V}(I)$
form an open cover of $X$. Then the natural map
$e\colon\thinspace|\check{\mathcal{C}}(\mathcal{V})|\to X$ is a weak
equivalence.

###### Proof.

We follow the methods in Dugger–Isaksen [9, Section 2]. Consider the one-
element cover $\\{X\\}$ of $X$. Its associated Čech complex is the constant
simplicial space $X_{*}$, with $X_{n}=X$ for all $n\geqslant 0$ and all
simplicial structure maps equal to the identity. The Čech category
$\underline{\\{X\\}}$ has $X$ as both its object space and its morphism space,
and all morphisms are identity morphisms. The inclusions $V\hookrightarrow X$,
$V\in\mathcal{V}$, induce a continuous functor
$\epsilon\colon\thinspace\underline{\mathcal{V}}\to\underline{\\{X\\}}$.

The geometric realization of $\underline{\\{X\\}}$ is naturally homeomorphic
to $X$, and we will show that $e=|\epsilon|$ is a weak equivalence. For open
covers, this is [9, Theorem 2.1], and the proof in the present setting is
nearly identical. We briefly sketch the argument. A general recognition
principle for weak equivalences due to May [19] (see also [34, Theorem 6.7.9])
reduces the problem to verifying that

$e\colon\thinspace
e^{-1}(\mathrm{int}(\mathcal{V}(i_{1}))\cap\cdots\cap\mathrm{int}(\mathcal{V}(i_{n})))\to\mathrm{int}(\mathcal{V}(i_{1}))\cap\cdots\cap\mathrm{int}(\mathcal{V}(i_{n}))$

is a weak equivalence for each set
$\\{i_{1},\ldots,i_{n}\\}\in\mathcal{N}(\mathcal{V})$. Dugger and Isaksen show
by a direct analysis that for every open set $U\subseteq X$, $e^{-1}(U)$ is
naturally homeomorphic to the geometric realization of the Čech complex for
the indexed cover $\mathcal{W}\colon\thinspace I\to 2^{U}$ of $U$ defined by
$i\mapsto\mathcal{V}(i)\cap U$. Note that if $U$ is a subset of
$\mathcal{V}(i)$ for some $i\in I$ (for instance, if
$U=\mathrm{int}(\mathcal{V}(i_{1}))\cap\cdots\cap\mathrm{int}(\mathcal{V}(i_{n}))$
for some $\\{i_{1},\ldots,i_{n}\\}\in\mathcal{N}(\mathcal{V})$), then this
cover has $U$ in its image.

Next, [9, Lemma 2.4] states that if $\mathcal{W}\colon\thinspace J\to 2^{Y}$
is an open cover of a space $Y$ satisfying $\mathcal{W}(j_{0})=Y$ for some
$j_{0}\in J$, then $e\colon\thinspace|\check{\mathcal{C}}(\mathcal{W})|\to Y$
is a homotopy equivalence. In fact no assumptions on the subspaces $W_{j}$,
$j\neq j_{0}$, are needed for the proof. For completeness, we give the
argument in the categorical framework discussed in Section 3 (this is
essentially the “slick” proof alluded to after the proof of [9, Lemma 2.4]).
There is a continuous functor
$s\colon\thinspace\underline{\\{Y\\}}\to\underline{\mathcal{W}}$, defined on
objects by mapping $y\in Y$ to $(\\{j_{0}\\},y)$; the behavior of $s$ on
morphisms is then forced, since $\underline{\\{Y\\}}$ has only identity
morphisms. Note that $\epsilon\circ s$ is the identity. By Lemma 3.4, it will
now suffice to show produce a continuous natural transformation $\eta$ from
$s\circ\epsilon$ to the identity. Such a natural transformation is just a map

$\displaystyle\check{\mathcal{C}}_{0}(\mathcal{W})=\\{(j,y)\in J\times
Y:y\in\mathcal{W}(j)\\}\hskip 144.54pt$
$\displaystyle\longrightarrow\check{\mathcal{C}}_{1}(\mathcal{W})=\\{((j,k),y)\in
J^{2}\times Y:y\in\mathcal{W}(j)\cap\mathcal{W}(k)\\},$

and we define $\eta(j,y)=((j_{0},j),y)$, which is a morphism from
$(j_{0},y)=s\circ\epsilon(j,y)$ to $(j,y)$ itself. It is immediate from the
definitions that $\eta$ is continuous and natural. This completes the proof
that $e\colon\thinspace|\check{\mathcal{C}}(\mathcal{V})|\to X$ is a weak
equivalence. ∎

Next, we will compare the geometric realization of
$\check{\mathcal{C}}(\mathcal{V})$ to that of the simplicial set
$\check{\mathcal{C}}^{\delta}(\mathcal{V})$ defined by composing the functor
$\check{\mathcal{C}}(\mathcal{V})\colon\thinspace\Delta^{\mathrm{op}}\to\underline{\mathrm{Top}}$
with the path component functor
$\pi_{0}\colon\thinspace\underline{\mathrm{Top}}\to\underline{\mathrm{Set}}$.
Viewing $\check{\mathcal{C}}^{\delta}(\mathcal{V})$ as a (level-wise discrete)
simplicial space, there is a natural map of simplicial spaces
$\check{\mathcal{C}}(\mathcal{V})\to\check{\mathcal{C}}^{\delta}(\mathcal{V})$
given by sending each point in $\check{\mathcal{C}}_{k}(\mathcal{V})$ to its
path component, and the map $\pi$ in (4) is the induced map on realizations.
Note that continuity of the projection
$\check{\mathcal{C}}_{k}(\mathcal{V})\to\pi_{0}(\check{\mathcal{C}}_{k}(\mathcal{V}))$
is equivalent to our hypothesis that the $k$–fold intersections from
$\mathcal{V}$ are coproducts of their path components, so $\pi$ is indeed
continuous. Our connectivity hypothesis guarantees that for each $k\geqslant
0$, the map
$\check{\mathcal{C}}_{k}(\mathcal{V})\to\check{\mathcal{C}}^{\delta}_{k}(\mathcal{V})$
is $(n-k+1)$–connected, so by Lemma 1.2 the map
$||\check{\mathcal{C}}(\mathcal{V})||\longrightarrow||\check{\mathcal{C}}^{\delta}(\mathcal{V})||$
is $(n+1)$–connected.111The argument proof of Lemma 1.2 in [10] is based on a
glueing lemma for $n$–connected maps, found in tom Dieck [34, Theorem 6.7.9].
The definition of $n$–connected map given there requires only a surjection on
$\pi_{0}$, but it is elementary to check that the results hold in full
generality under the definition of $n$–connected map given in Definition 2.12.
Moreover, the projection maps
$||\check{\mathcal{C}}(\mathcal{V})||\to|\check{\mathcal{C}}(\mathcal{V})|$
and
$||\check{\mathcal{C}}^{\delta}(\mathcal{V})||\to|\check{\mathcal{C}}^{\delta}(\mathcal{V})|$
are homotopy equivalences by [31, Proposition A.1]:
$\check{\mathcal{C}}(\mathcal{V})$ and
$\check{\mathcal{C}}^{\delta}(\mathcal{V})$ are good simplicial spaces in the
sense of [31] because each of their degeneracy maps has the form
$A\hookrightarrow A\coprod B$ for some spaces $A,B$, and hence is a closed
cofibration. Naturality of the projection maps now implies that
$\pi\colon\thinspace|\check{\mathcal{C}}(\mathcal{V})|\longrightarrow|\check{\mathcal{C}}^{\delta}(\mathcal{V})|$
is $(n+1)$–connected as well.

To compare $|\check{\mathcal{C}}^{\delta}(\mathcal{V})|$ and
$|\hat{\mathcal{N}}(\mathcal{V})|$, we will introduce one more intermediate
space. For each simplicial set $K_{*}$, there is a natural weak equivalence
from the nerve of its category of simplices $N_{*}(\mathrm{Simp}\,(K_{*}))$ to
$K_{*}$ [15, Theorem 18.9.3], so it will suffice to compare
$\mathrm{Simp}\,(\check{\mathcal{C}}^{\delta}(\mathcal{V}))$ and
$\hat{\mathcal{N}}(\mathcal{V})$. The objects of
$\mathrm{Simp}\,(\check{\mathcal{C}}^{\delta}(\mathcal{V}))$ are in bijection
with the disjoint union of the sets
$\check{\mathcal{C}}^{\delta}_{k}(\mathcal{V})$, and hence may be written as
pairs $((i_{0},\ldots,i_{k}),C)$, where $i_{j}\in I$ for each $j$ and
$C\in\pi_{0}(\mathcal{V}(i_{0})\cap\cdots\cap\mathcal{V}(i_{k}))$. Morphisms
in this category from $((i_{0},\ldots,i_{k}),C)$ to
$((j_{0},\ldots,j_{l}),C^{\prime})$ correspond to order-preserving maps
$\phi\colon\thinspace[k]\to[l]$ satisfying $i_{p}=j_{\phi(p)}$ and
$C^{\prime}\subseteq C$ (equivalently, $C^{\prime}$ is sent to $C$ by the map
$\pi_{0}(\mathcal{V}(j_{0})\cap\cdots\cap\mathcal{V}(j_{l}))\to\pi_{0}(\mathcal{V}(i_{0})\cap\ldots\cap\mathcal{V}(i_{k}))$
induced by the inclusion
$\mathcal{V}(j_{0})\cap\cdots\cap\mathcal{V}(j_{l})\hookrightarrow\mathcal{V}(j_{\phi(0)})\cap\cdots\cap\mathcal{V}(j_{\phi(k)})=\mathcal{V}(i_{0})\cap\cdots\cap\mathcal{V}(i_{k})$).

We claim that there is a functor

$q\colon\thinspace\mathrm{Simp}\,(\check{\mathcal{C}}^{\delta}(V))\to\hat{\mathcal{N}}(\mathcal{V})$

sending $((i_{0},\ldots,i_{k}),C)$ to $(\\{i_{0},\ldots,i_{k}\\},C)$.
Comparing the above description of morphisms in
$\mathrm{Simp}\,(\check{\mathcal{C}}^{\delta}(V))$ with our description of the
order relation in $\hat{\mathcal{N}}(\mathcal{V})$ shows that if there is a
morphism $((i_{0},\ldots,i_{k}),C)\to((j_{0},\ldots,j_{l}),D)$ in
$\mathrm{Simp}\,(\check{\mathcal{C}}^{\delta}(V))$, then
$(\\{i_{0},\ldots,i_{k}\\},C)\leqslant(\\{j_{0},\ldots,j_{l}\\},D)$ in
$\hat{\mathcal{N}}(\mathcal{V})$, and since all diagrams in the poset
$\hat{\mathcal{N}}(\mathcal{V})$ commute it is automatic that $q$ respects
composition. We will apply Quillen’s Fiber Theorem [27, Theorem A] to show
that $q$ is a weak equivalence. Since all diagrams in
$\hat{\mathcal{N}}(\mathcal{V})$ commute, the Quillen fibers of $q$ are all
full subcategories of $\mathrm{Simp}\,(\check{\mathcal{C}}^{\delta}(V))$.
Hence it will suffice to show that for each
$(\mathcal{F},C)\in\hat{\mathcal{N}}(\mathcal{V})$, the full subcategory of
$\mathrm{Simp}\,(\check{\mathcal{C}}^{\delta}(V))$ on the set of objects

$\displaystyle\\{((j_{0},\ldots,j_{l}),D)\in\coprod_{k}\check{\mathcal{C}}_{k}^{\delta}(\mathcal{V})\colon\thinspace(\\{j_{0},\ldots,j_{l}\\},D)\leqslant(\mathcal{F},C)\\}$
$\displaystyle=\\{((j_{0},\ldots,j_{l}),D)\in\coprod_{k}\check{\mathcal{C}}_{k}^{\delta}(\mathcal{V})\colon\thinspace\\{j_{p}\\}_{p}\subseteq\mathcal{F}\textrm{
and }C\subseteq D\\}.$

is contractible. Note that for each list
$(\mathcal{W}(j_{0}),\ldots,\mathcal{W}(j_{l}))$ with
$\\{j_{p}\\}_{p}\subseteq\mathcal{F}$, there is a unique path component
$D\in\pi_{0}(\mathcal{W}(j_{0})\cap\cdots\cap\mathcal{W}(j_{l}))$ containing
$C$, and so objects in the fiber are in bijection with lists of elements from
the set $\mathcal{F}$. This gives an isomorphism from the above fiber category
to the simplex category of the nerve of the indiscrete category with object
set $\mathcal{F}$. Since the indiscrete category on a set $S$ (that is, the
category with object set $S$ and each morphism set a singleton) is equivalent
to the trivial category with one object and one morphism, its nerve is
contractible, and it follows from [15, Theorem 18.9.3] that the simplex
category of its nerve is contractible as well.

This completes the proof of Theorem 2.13 in the case of locally open covers.

##  5\. CW neighborhoods

Consider a CW cover $(X,\mathcal{V}\colon\thinspace I\to 2^{X})$. We will use
the open neighborhoods of subcomplexes constructed in [14, Appendix] to
enlarge the sets $K\in\mathcal{V}(I)$ into an open cover
$\mathcal{U}(\mathcal{V})\colon\thinspace I\to 2^{X}$, which comes with a
natural morphism of covers $(X,\mathcal{V})\to(X,\mathcal{U}(\mathcal{V}))$
that induces an isomorphism on nerves and a homotopy equivalence on all
intersections. In this section, we establish the essential properties of this
construction.

Let $X$ be a CW complex, with skeleta $X^{(n)}$. We will fix a choice of
characteristic map $D^{n}\to X$ for each $n$–cell of $X$, where $D^{n}$ is the
closed unit disk in $\use@mathgroup\M@U\symAMSb{R}^{n+1}$. If
$\phi\colon\thinspace D^{n}\to X$ is the characteristic map of an $n$-cell,
then we call $\phi(D^{n}\setminus\partial D^{n})$ an _open cell_ of $X$. When
$n=0$, we set $\partial D^{0}=\emptyset$, so each $0$–cell of $X$ is an open
cell.

Recall that each point $x\in X$ lies in a unique open cell, whose
characteristic map we denote by $\phi_{x}$; if the domain of $\phi_{x}$ is
$D^{n}$ then we say $x$ has dimension $n$. If $x=\phi_{x}(0)$, then we say
that $x$ is a center-point of $X$. We will define a function from $X$ to its
set of center-points, $x\mapsto\underline{x}$, by successively pushing $x$
radially to the boundary of its open cell until we reach a center-point. (This
function will be discontinuous unless $X$ is zero-dimensional.) Formally, if
$x$ is the center-point of an open cell $e$, then we define $\underline{x}=x$;
so in particular if $x\in X^{(0)}$ we have $\underline{x}=x$. If $x$ is _not_
a center-point, then $x=\phi_{x}(z)$ for some $z\neq 0$, and we set
$\underline{x}=\underline{\phi_{x}(z/|z|)}$. Note that if $x$ has dimension
$n$, then $\phi_{x}(z/|z|)$ has dimension strictly less than $n$, so this
provides an inductive definition of $\underline{x}$ over the skeleta of $X$,
and also shows that if $x\in X^{(n)}$, then $\underline{x}\in X^{(n)}$ as
well.

######  Definition 5.1.

For each subset $A\subseteq X$, we define

$U(A)=U(A,X)=\\{x\in X\colon\thinspace\underline{x}\in A\\}.$

We note that the set $U(A)$ is the neighborhood of $A$ constructed in [14,
Appendix] for the parameter $\epsilon=1$.

The following observation will be used to describe the topology on the
neighborhoods $U(A)$.

######  Lemma 5.2.

Let $Z$ be a topological space. Say $Y\subseteq Z$ is closed and $W\subseteq
Y$ is $($relatively$)$ open in $Y$. If $L\subseteq Z$ satisfies $Y\cap
L\subseteq W$ and $Y\cup L=Z$, then $W\cup L$ is open in $Z$.

###### Proof.

Write $W=V\cap Y$, with $V$ open in $Z$. Then

$W\cup L=(V\cap Y)\cup L=(V\cup L)\cap(Y\cup L)=(V\cup L)\cap Z=V\cup L,$

and therefore

$Z\setminus(W\cup L)=(Y\cup L)\setminus(V\cup L)=Y\setminus V,$

which is closed in $Z$ since $Y$ and $V^{c}$ are closed in $Z$. ∎

Here are the properties of these neighborhoods that we will need.

######  Proposition 5.3.

Let $X$ be a CW complex. The neighborhoods constructed in Definition 5.1
satisfy the following properties:

  1. (1)

$U$ is order-preserving: If $A\subseteq B$, then $U(A)\subseteq U(B)$.

  2. (2)

$U$ distributes over unions and intersections: for every $\mathcal{A}\subseteq
2^{X}$, we have

    1. (a)

$\bigcup_{A\in\mathcal{A}}U(A)=U\left(\bigcup\mathcal{A}\right)$

    2. (b)

$\bigcap_{A\in\mathcal{A}}U(A)=U\left(\bigcap\mathcal{A}\right)$.

  3. (3)

If $K$ is a subcomplex of $X$, then

    1. (a)

$K\subseteq U(K)$;

    2. (b)

An open cell of $X$ is contained in $U(K)$ if and only if it is contained in
$K$;

    3. (c)

$U(K)$ is open in $X$, as are its path components;

    4. (d)

$U(K)$ deformation retracts to $K$ $($strongly$)$.

###### Proof.

(1) & (2): Since $U(A)$ is defined as the inverse image of $A$ under a certain
function, it is immediate that $U$ is order-preserving and distributes over
unions and intersections.

(3a) & (b): Say $K\subseteq X$ is a subcomplex. Then a simple induction over
skeleta establishes (3a). To prove (3b), observe that if an open cell $e$ is
contained in $U(K)$, then its center-point $c$ lies in $U(K)$, so
$\underline{c}\in K$. But $\underline{c}=c$, so $c\in K$, and since $K$ is a
subcomplex this means $e\subseteq K$. The reverse implication in (3b) is
immediate from (3a).

(3c): It follows from the definitions that the path components of $U(K)$ are
all of the form $U(K^{\prime})$ with $K^{\prime}\in\pi_{0}K$. Hence it
suffices to show that each $U(K)$ is open. (Alternatively, since CW complexes
are locally contractible, they are locally path connected, so the path
components of an open set in $X$ are always open themselves). We must show
that the inverse image of $U(K)$ under the characteristic map of a cell in $X$
is always open. We will prove this by induction on dimension; note that for
0-cells the statement is vacuously true. Now assume the statement for all
cells of dimension less than $n$, and say $\phi\colon\thinspace D^{n}\to X$ is
a characteristic map with $\phi(z)\in U(K)$ for some $z\in D^{n}$. If
$\phi(0)\in U(K)$, then $\phi(0)=\underline{\phi(0)}\in K$, and since $K$ is a
subcomplex this means the entire cell $\phi(D^{n})$ lies in $K$, and hence in
$U(K)$ (by (3a)). If $\phi(0)\notin U(K)$, then $z\neq 0$ and hence
$\underline{\phi(z)}=\underline{\phi(z/|z|)}\in K$. This implies
$\phi(z/|z|)\in U(K)\cap X^{(n-1)}$. The induction hypothesis implies that
$U(K)\cap X^{(n-1)}$ is open in $X^{(n-1)}$, so its inverse image under
$w\mapsto\phi(w/|w|)$ is an open set $W\subseteq D^{n}\setminus\\{0\\}$
containing $z$. For each $w\in W$, we have
$\underline{\phi(w)}=\underline{\phi(w/|w|)}$, and since $\phi(w/|w|)\in U(K)$
we conclude $\phi(w)\in U(K)$ as well. So $W$ is an open neighborhood of $z$
in $\phi^{-1}(U(K))$.

(3d): In order to check continuity of the deformation retraction from $U(K)$
onto $K$ defined below, we will need some observations regarding the topology
on $U(K)$. Define $U^{(n)}(K)=(U(K)\cap X^{(n)})\cup K$. We claim that

(6) $U(K)\cap X^{(n)}=U(K^{(n)},X^{(n)}),$

where on the right we are taking the neighborhood with respect to the same
characteristic maps as we used for $X$. The containment $U(K)\cap
X^{(n)}\subseteq U(K^{(n)},X^{(n)})$ follows from the fact that the function
$x\mapsto\underline{x}$ maps $X^{(n)}$ to itself; the reverse containment is
immediate.

We claim that $U^{(n)}(K)$ is an open subset of the subcomplex $X^{(n)}\cup
K$. Equation (6) implies that $U^{(n)}(K)=U(K^{(n)},X^{(n)})\cup K$. We can
now appeal to Lemma 5.2, with $Z:=X^{(n)}\cup K$, $Y:=X^{(n)}$,
$W=U(K^{(n)},X^{(n)})$ (which is open in $Y$ by (3c)) and $L:=K$; note that
$Y\cap L=K^{(n)}\subseteq U(K^{(n)},X^{(n)})=W$ by (3a). Now the lemma tells
us that $W\cup L=U^{(n)}(K)$ is open in $Z=X^{(n)}\cup K$, as claimed.

Since $U^{(n)}(K)$ is open in $X^{(n)}\cup K$, it has the quotient topology
inherited from the characteristic maps for this subcomplex. Furthermore, since
$U(K)$ is open in $X$, it has the quotient topology inherited from _all_ the
characteristic maps for $X$, and since $U(K)=\bigcup_{n}U^{(n)}(K)$, it
follows that the inclusions $U^{(n)}(K)\hookrightarrow U(K)$ also induce a
quotient map

$\coprod_{n}U^{(n)}(K)\longrightarrow U(K).$

To define our deformation retraction, we begin by defining homotopies

$H^{(n)}\colon\thinspace U^{(n)}(K)\times[1/2^{n},1/2^{n-1}]\to U^{(n)}(K)$

($n=1,2,\ldots$) of $U^{(n)}(K)$ onto $U^{(n-1)}(K)$ with
$H^{(n)}_{1/2^{n}}=\mathrm{Id}_{U^{(n)}(K)}$ and

$r^{(n)}:=H^{(n)}_{1/2^{n-1}}\colon\thinspace U^{(n)}(K)\longrightarrow
U^{(n-1)}(K)$

a retraction (in fact, $H^{(n)}$ will be a strong deformation retraction onto
$U^{(n-1)}(K)$). We define $H^{(n)}(x,t)=x$ for each point $x\in U^{(n-1)}(K)$
and each $t\in[1/2^{n},1/2^{n-1}]$. For $x\in U^{(n)}(K)\setminus
U^{(n-1)}(K)$, we write $x=\phi_{x}(z)$ and define

$H^{(n)}(x,t)=\phi_{x}((2-2^{n}t)z+(2^{n}t-1)z/|z|);$

note that since $x\notin U^{(n-1)}(K)$, we know $x\notin K$, so $z\neq 0$.
Since $U^{(n)}(K)$ is the quotient of its inverse images under the attaching
maps in $X^{(n)}\cup K$ and $I$ is locally compact, $U^{(n)}(K)\times I$ is
the quotient of these inverse images crossed with $I$. Therefore to prove
continuity of $H^{(n)}$, it suffices to check that if $\psi$ is the
characteristic map of a cell in $X^{(n)}\cup K$, then
$H^{(n)}\circ(\psi,\mathrm{Id})$ is continuous, which is immediate from the
construction.

The desired deformation retraction of $U(K)$ onto $K$ is given by

$H|_{U^{(n)}(K)\times[1/2^{k},1/2^{k-1}]}(x,t)=H^{(k+1)}(r^{(k+1)}\circ\cdots\circ
r^{(n)}(x),t)$

for $n\geqslant 1$, $1\leqslant k\leqslant n$ (where, for $k=n$, the
expression $r^{(k+1)}\circ\cdots\circ r^{(n)}(x)$ is interpreted to simply
mean $x$) and $H(x,t)=x$ if $x\in U^{(n)}(K)$ and $t\leqslant 1/2^{n}$. This
function is well-defined on all of $U(K)\times I$ and is continuous on each
set $U^{(n)}(K)\times I$; since $U(K)\times I$ is the quotient of these
subspaces, it follows that $H$ itself is continuous. This completes the proof
of Property (3d). ∎

######  Corollary 5.4.

Let $(X,\mathcal{V}\colon\thinspace I\to 2^{X})$ be a CW cover. Then
$\mathcal{U}(\mathcal{V})\colon\thinspace I\to 2^{X}$,
$\mathcal{U}(\mathcal{V})(i)=U(\mathcal{V}(i))$, is a locally open cover, and
the map of covers

$(\mathrm{Id}_{X},\mathrm{Id}_{I})\colon\thinspace(X,\mathcal{V})\to(X,\mathcal{U}(\mathcal{V}))$

induces an order isomorphism

$\hat{\mathcal{N}}(\mathcal{V})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\hat{\mathcal{N}}(\mathcal{U}(\mathcal{V})).$

###### Proof.

We will use the various properties of $U$ established in Proposition 5.3.
Properties (2b) and (3c) imply that each finite intersection of spaces in
$\mathcal{U}(\mathcal{V})(I)$ is the disjoint union of its path components, so
$\mathcal{U}(\mathcal{V})$ is a locally open cover.

To show that this map induces an isomorphism between completed nerves, it
suffices, by Proposition 2.11, to show that it is an equivalence of covers and
induces bijections on path components for each
$\mathcal{F}\in\mathcal{N}(\mathcal{V})$. If
$\\{U(\mathcal{V}(i_{1})),\ldots,U(\mathcal{V}(i_{n}))\\}\in\mathcal{N}(\mathcal{U}(\mathcal{V}))$,
then by Property (2b) we have

$U(\mathcal{V}(i_{1}))\cap\cdots\cap
U(\mathcal{V}(i_{n}))=U(\mathcal{V}(i_{1})\cap\cdots\cap\mathcal{V}(i_{n})),$

and since this neighborhood is non-empty,
$\mathcal{V}(i_{1})\cap\cdots\cap\mathcal{V}(i_{n})$ must be non-empty as
well. So $(\mathrm{Id}_{X},\mathrm{Id}_{I})$ is an equivalence. It remains to
check that the maps (2) from Proposition 2.11 induce bijections on path
components. These maps are in fact homotopy equivalences, by Properties (2b)
and (3d). ∎

##  6\. Completed nerves of CW covers

In this section, we combine the results of the previous two sections to prove
Theorem 2.13 in the case of CW covers.

Let $(X,\mathcal{V})$ be a CW cover, and let
$\mathcal{U}=\mathcal{U}(\mathcal{V})$ be the cover constructed in Section 5.
By Corollary 5.4, $\mathcal{U}$ is a locally open cover of $X$, and we have a
map of covers $(X,\mathcal{V})\to(X,\mathcal{U})$. This map induces a
commutative diagram

${X}$${{|\check{\mathcal{C}}(\mathcal{V})|}}$${{|\check{\mathcal{C}}^{\delta}(\mathcal{V})|}}$${{|N_{*}\mathrm{Simp}\,\check{\mathcal{C}}^{\delta}(\mathcal{V})|}}$${{|N_{*}(\hat{\mathcal{N}}\mathcal{V})|}}$${{|\hat{\mathcal{N}}(\mathcal{V})|}}$${X}$${{|\check{\mathcal{C}}(\mathcal{U})|}}$${{|\check{\mathcal{C}}^{\delta}(\mathcal{U})|}}$${{|N_{*}\mathrm{Simp}\,\check{\mathcal{C}}^{\delta}(\mathcal{U})|}}$${{|N_{*}(\hat{\mathcal{N}}\mathcal{U})|}}$${{|\hat{\mathcal{N}}(\mathcal{U})|.}}$$\scriptstyle{\mathrm{Id}_{X}}$$\scriptstyle{q^{\mathcal{V}}}$$\scriptstyle{\cong}$$\scriptstyle{q^{\mathcal{U}}}$$\scriptstyle{\cong}$

We will show that the vertical map

$|\check{\mathcal{C}}(\mathcal{V})|\longrightarrow|\check{\mathcal{C}}(\mathcal{U})|$

is a weak equivalence, while the other vertical maps are isomorphisms. We
consider the other vertical maps first. Corollary 5.4 states that
$\hat{\mathcal{N}}(\mathcal{V})\to\hat{\mathcal{N}}(\mathcal{U})$ is an
isomorphism, and by naturality the same is true for the map
$N_{*}(\hat{\mathcal{N}}\mathcal{V})\to N_{*}(\hat{\mathcal{N}}\mathcal{U})$.
A similar argument shows that the map
$\check{\mathcal{C}}^{\delta}(\mathcal{V})\to\check{\mathcal{C}}^{\delta}(\mathcal{U})$
is an isomorphism, and by naturality, the induced map between the nerves of
the simplex categories is an isomorphism as well. The map
$\check{\mathcal{C}}(\mathcal{V})\to\check{\mathcal{C}}(\mathcal{U})$ is a
level-wise weak equivalence by Properties (2b) and (3d) from Proposition 5.3.
Since these simplicial spaces are proper, it follows that the induced map on
geometric realizations is a weak equivalence [31, Proposition A.1].

Commutativity of the above diagram implies that $q^{\mathcal{U}}$ and
$q^{\mathcal{V}}$ have the same connectivity, and that the other horizontal
maps on the top row are weak equivalences if and only the corresponding maps
on the bottom row are. We have already proven Theorem 2.13 for open covers, so
to complete the proof it suffices to observe that $\mathcal{U}$ satisfies the
hypotheses of the theorem: for each $\mathcal{F}\in\mathcal{N}(\mathcal{U})$,
the components of $\bigcap\mathcal{F}$ are all
$(n-|\mathcal{F}|+1)$–connected, because

$\bigcap_{i\in\mathcal{F}}U(\mathcal{V}(i))=U\left(\bigcap_{i\in\mathcal{F}}\mathcal{V}(i)\right)\simeq\bigcap_{i\in\mathcal{F}}\mathcal{V}(i),$

and the components of $\bigcap_{i\in\mathcal{F}}\mathcal{V}(i)$ are all
$(n-|\mathcal{F}|+1)$–connected by assumption. This completes the proof of
Theorem 2.13.

We record the following statement, proven above, for later reference.

######  Proposition 6.1.

Let $(X,\mathcal{V})$ be a CW cover. Then the natural map
$e\colon\thinspace|\check{\mathcal{C}}(\mathcal{V})|\to X$ is a weak
equivalence.

##  7\. Nerves of simplicial covers

The Nerve Theorem is often applied to covers of _simplicial_ complexes by
subcomplexes. Since every simplicial complex has a natural CW structure with
cells corresponding to simplices, Theorem 2.13 applies to such covers. In the
simplicial case, it was recently observed in [12] that there is an alternative
construction of neighborhoods that can be used in place of those in Section 5
to prove the classical Nerve Theorem, and these neighborhoods can also be used
to prove the simplicial case of Theorem 2.13. See [24, Lemma 70.1] for a
discussion of these neighborhoods.

For the purposes of this section, it will suffice to work with unindexed
simplicial covers – that is, subsets $\mathcal{C}\subseteq 2^{Z}$ such that
each $C\in\mathcal{C}$ is a subcomplex of $Z$ and $\bigcup\mathcal{C}=Z$. If
$\mathcal{C}$ is locally finite (meaning that for each $\sigma\in Z$, the set
$\\{K\in\mathcal{C}\colon\thinspace\sigma\in K\\}$ is finite), then there is a
natural order-reversing map

$\eta\colon\thinspace K\to\hat{\mathcal{N}}(\mathcal{C}),$

defined by $\eta(\sigma)=(\\{K\in\mathcal{C}\colon\thinspace\sigma\in
K\\},[\sigma])$, where
$[\sigma]\in\pi_{0}\left(\bigcap\\{K\in\mathcal{C}\colon\thinspace\sigma\in
K\\}\right)$ is the path component of $\sigma$. When each finite intersection
of sets in $\mathcal{C}$ is contractible, we have
$\hat{\mathcal{N}}(\mathcal{C})=\mathcal{N}(\mathcal{C})$, and in this case,
Björner [3, Proof of Theorem 10.6] proves that $\eta$ is a homotopy
equivalence by observing that the Quillen fiber of $\eta$ over
$\mathcal{F}\in\mathcal{N}(\mathcal{C})$ is exactly $\bigcap\mathcal{F}$.

In order to adapting this observation to the present setting, we need to
recall Björner’s $n$–connected version of Quillen’s Fiber Theorem for posets
[6].

######  Theorem 7.1 (Björner).

Let $f\colon\thinspace P\to Q$ be an order-preserving map of posets and assume
that for each $q\in Q$, the Quillen fiber $f^{-1}(Q_{\geqslant q})$ is
$k$–connected. Then $f$ is $(k+1)$–connected.

######  Remark 7.2.

Some comments are in order regarding the proof of Theorem 7.1, since the
statement above differers from Björner’s original statement in two ways.
First, Björner does not explicitly state the conclusion that $f$ induces a
surjection on homotopy in degree $k+1$, but this does follow from his proof.
His argument proceeds by constructing, via the method of contractible
carriers, a map $g\colon\thinspace(\Delta Q)^{(k+1)}\to\Delta P$ (where
$(\Delta Q)^{(k+1)}$ denotes the $(k+1)$–skeleton) and showing that $f\circ g$
is homotopic to the inclusion $(\Delta Q)^{(k+1)}\hookrightarrow\Delta Q$. By
CW approximation, the inclusion induces a surjection on $\pi_{k+1}$, and hence
$f$ does as well. Second, Björner assumes that $P$ and $Q$ are path connected.
The statement for general posets follows from the path-connected case,
however, since for $k\geqslant-1$ the hypothesis implies that the Quillen
fibers are non-empty, which immediately implies that $f$ is surjective on
$\pi_{0}$, while for $k\geqslant 0$ the hypothesis tells us that each Quillen
fiber lies in a single path component of $P$, so restricting attention to a
given path component of $P$ and its image in $\pi_{0}Q$ does not change the
fibers.

######  Proposition 7.3.

Let $\mathcal{C}$ be a locally finite cover of the simplicial complex $Z$ by
subcomplexes, and fix $n\geqslant 0$. Assume that for each
$\mathcal{F}\in\mathcal{N}(\mathcal{C})$, all path components of
$\bigcap\mathcal{F}$ are $n$–connected. Then $\eta$ is an $(n+1)$–connected
map. If $Z$ is _finite_ and all path components of $\bigcap\mathcal{F}$,
$F\in\mathcal{N}(\mathcal{C})$ are contractible $($“the case $n=\infty$”$)$
then $\eta$ is in fact a _simple_ homotopy equivalence.

###### Proof.

For notational simplicity we will view the path components of
$\bigcap\mathcal{F}$ as subsets of $Z$ (rather than of $|Z|$).

We will compute the Quillen fiber of $\eta$ above
$(\mathcal{F},C)\in\hat{\mathcal{N}}(\mathcal{C})$. Letting $[\sigma]$ denote
the path component of $\bigcap\mathcal{F}$ containing $\sigma$, we have

$\displaystyle\eta^{-1}\left(\hat{\mathcal{N}}(\mathcal{C})_{\geqslant(\mathcal{F},C)}\right)$
$\displaystyle=\\{\sigma\in
Z\colon\thinspace\mathcal{F}\subseteq\\{K\in\mathcal{C}\colon\thinspace\sigma\in
K\\}\textrm{ and }[\sigma]=C\\}$ $\displaystyle=\\{\sigma\in
Z\colon\thinspace\sigma\in\bigcap\mathcal{F}\textrm{ and }[\sigma]=C\\}$
$\displaystyle=C,$

so the connectivity statement in the proposition follows from Theorem 7.1. In
the case $n=\infty$, [1, Theorem 1.2] shows that $\eta$ is in fact a simple
homotopy equivalence so long as $Z$ is finite. ∎

In the case $n=\infty$, this gives another proof of [11, Corollary 3.10].

######  Question 7.4.

Let $\mathcal{C}$ be a locally finite cover of the simplicial complex $Z$. If
$Z$ and $\mathcal{C}$ satisfy the hypotheses of Theorem 2.13, does it follow
that $\eta\colon\thinspace Z\to\hat{\mathcal{N}}(\mathcal{C})$ is an
$(n+1)$–connected map? More specifically, do $\eta$ and the zig-zag in the
proof of Theorem 2.13 induce the same map on homotopy groups?

##  8\. Partial nerves

The nerve theorems considered so far produce combinatorial approximations to a
space $X$, modeling its homotopy type through degree $n$, from a cover
$\mathcal{V}$ on which one has control over _all_ finite intersections
$\bigcap\mathcal{F}$ with $|\mathcal{F}|\leqslant n$. In this section, we show
that even in cases where the nerve itself has the wrong homotopy type, one can
sometimes still build good combinatorial models for the homotopy type of $X$
using an appropriate collection of intersections from $\mathcal{V}$. As in the
previous section, we will work with unindexed covers $\mathcal{V}\subseteq
2^{X}$ for the sake of simplicity.

######  Definition 8.1.

For a partial cover $\mathcal{V}$ of a space $X$, we denote the set of all
$V\in\mathcal{V}$ containing $x\in X$ by $\mathcal{V}_{x}$. More generally, we
denote the set of all $V\in\mathcal{V}$ containing a subspace $Y\subseteq X$
by $\mathcal{V}_{Y}$ $($so $\mathcal{V}_{x}=\mathcal{V}_{\\{x\\}}$$)$.

######  Definition 8.2.

Let $(X,\mathcal{V})$ be a partial cover of a space $X$. For each $x\in X$,
let $C_{x}=C^{\mathcal{V}}_{x}$ denote the path component of $x$ in
$\bigcap\mathcal{V}_{x}$. We define the _completion_ of the cover
$\mathcal{V}$ to be the cover

$\hat{\mathcal{V}}:=\left\\{C_{x}\colon\thinspace x\in X\right\\}.$

Say $(X,\mathcal{V})$ is a CW cover and $e\subseteq X$ is an open cell. We
denote the path component of $e$ in $\bigcap\mathcal{V}_{e}$ by $C_{e}$.

One should expect in practice that for each $V\in\mathcal{V}$, we have
$\pi_{0}V\subseteq\hat{\mathcal{V}}$, since if $C$ is a path component of
$V\in\mathcal{V}$ and $C\notin\hat{\mathcal{V}}$, then for each $x\in C$ there
must be another set $W\in\mathcal{V}$, $W\neq V$, with $x\in W$. This means
that replacing $V$ by $V\setminus C$ yields a new, simpler, cover of $X$.

Note that for a CW cover $(X,\mathcal{V})$, each point $x\in X$ lies in a
unique open cell $e$ of $X$, and
$\mathcal{V}_{x}=\mathcal{V}_{e}=\mathcal{V}_{\overline{e}}$, where
$\overline{e}$ denotes the closure of $e$ (the closed cell associated to $e$).
Hence $C_{x}=C_{e}=C_{\overline{e}}$ as well.

Before explaining how the completion of a cover can be used to model the
homotopy type of $X$, we note an analogue of Proposition 2.7 in the present
setting.

######  Proposition 8.3.

Let $(X,\mathcal{V})$ be a partial cover of a space $X$, and let
$\widetilde{\mathcal{V}}=\\{\mathcal{V}_{x}\colon\thinspace x\in X\\}$,
considered as a subposet of $2^{\mathcal{V}}$. Assume that for each $x\in X$,
the intersection $\bigcap\mathcal{V}_{x}$ is path connected $($so that
$C_{x}=\bigcap\mathcal{V}_{x}$$)$. Then there is a homotopy equivalence

$c\colon\thinspace\widetilde{\mathcal{V}}^{\hskip
1.4457pt\mathrm{op}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\hat{\mathcal{V}}$

defined by $c(\mathcal{V}_{x})=\bigcap\mathcal{V}_{x}$.

###### Proof.

We claim that for each $x\in X$, the Quillen fiber
$c^{-1}(\hat{\mathcal{V}}_{\geqslant\bigcap\mathcal{V}_{x}})$ has
$\mathcal{V}_{x}$ as its minimum element and hence is contractible. Indeed, if
$\mathcal{V}_{y}$ is in this fiber, then
$c(\mathcal{V}_{y})=\bigcap\mathcal{V}_{y}\supseteq\bigcap\mathcal{V}_{x}$,
and since $x\in\bigcap\mathcal{V}_{x}$ this implies
$x\in\bigcap\mathcal{V}_{y}$ as well. This in turn implies that
$\mathcal{V}_{y}\subseteq\mathcal{V}_{x}$, and hence
$\mathcal{V}_{x}\leqslant\mathcal{V}_{y}$ in $\widetilde{\mathcal{V}}^{\hskip
1.4457pt\mathrm{op}}$. ∎

In the case where each $\mathcal{V}_{x}$ is finite (for instance, if
$\mathcal{V}$ is locally finite), Proposition 8.3 identifies the completion
$\hat{\mathcal{V}}$, up to homotopy, with a subposet of the nerve
$\mathcal{N}(\mathcal{V})$, so we think of the completion $\hat{\mathcal{V}}$
as a _partial_ nerve construction. Note, however, that unlike the case for
ordinary nerves, the poset $\widetilde{\mathcal{V}}$ need not be closed under
passage to subsets (inside of $2^{\mathcal{V}}$), and hence need not be a
subcomplex of the full simplicial complex on the set $\mathcal{V}$, even if
each $\mathcal{V}_{x}$ is finite. Behavior of this sort appears in Example
8.14 below, where a certain (path connected) 2-fold intersection is _not_ part
of $\widetilde{V}$, although larger simplices from the nerve _are_ in
$\widetilde{V}$.

The goal of this section is to prove the following result, along with an
analogous result for certain open covers (Proposition 8.13).

######  Theorem 8.4.

Let $X$ be a CW complex, and let $\mathcal{V}$ be a CW cover of $X$ such that
each $V\in\mathcal{V}$ is $n$–connected. If for each open cell $e\subseteq X$,
the space $C^{\mathcal{V}}_{e}$ is $n$–connected, then there is a zig-zag

(7)
$X\stackrel{{\scriptstyle\simeq}}{{\longleftarrow}}\operatorname*{hocolim}_{W\in\hat{\mathcal{V}}}W\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}|\hat{\mathcal{V}}|$

connecting $X$ to the completion of $\mathcal{V}$, with $\pi$ an
$(n+1)$–connected map.

The homotopy colimit in (7) refers to the tautological diagram
$\hat{\mathcal{V}}\to\underline{\mathrm{Top}}$ sending $W\in\hat{\mathcal{V}}$
to itself, and mapping all morphisms in the poset $\hat{\mathcal{V}}$ to the
corresponding inclusions of spaces.

Before giving the proof of Theorem 8.4, we need some preliminary results on
homotopy colimits and complete covers. It will be convenient to use the
following model for homotopy colimits.

######  Definition 8.5.

For a functor $F\colon\thinspace\mathcal{C}\to\underline{\mathrm{Top}}$, the
homotopy colimit $\operatorname*{hocolim}_{\mathcal{C}}F$ is the _thick_
geometric realization of the simplicial space

$[k]\mapsto\coprod_{c_{0}\to c_{1}\cdots\to c_{k}\in
N_{k}\mathcal{C}}F(c_{0}),$

$($the _simplicial replacement_ of the diagram $F$$)$ with simplicial
structure maps given by those of the nerve $N_{*}\mathcal{C}$ together with
the map $F(c_{0}\to c_{1})\colon\thinspace F(c_{0})\to F(c_{1})$.

Each degeneracy map in this simplicial space has the form $A\hookrightarrow
A\coprod B$, and as noted in the proof of Proposition 2.13, this implies that
the natural projection map from the thick realization to the usual geometric
realization is a weak equivalence. Hence this definition agrees with the usual
one (up to natural weak equivalence). As we will use a result about homotopy
colimits from [9], it is important to note that the this model for the
homotopy colimit always agrees with the other models used in that paper, as is
proven in [9, Appendix].

Natural transformations of functors induce simplicial maps between simplicial
replacements, and hence maps between homotopy colimits. Note that the
simplicial space underlying $\operatorname*{hocolim}_{\mathcal{C}}*$ (where
$*$ denotes the constant functor with value the 1-point space) is exactly the
nerve $N_{*}\mathcal{C}$ of the category $\mathcal{C}$, and this gives rise to
natural maps $\operatorname*{hocolim}_{\mathcal{C}}F\to|N_{*}\mathcal{C}|$.
The map $\pi$ in (7) is obtained in precisely this manner, and the following
lemma shows it is $(n+1)$–connected.

######  Lemma 8.6.

Let $F\colon\thinspace\mathcal{C}\to\underline{\mathrm{Top}}$ be a functor,
and assume that for each $C\in\mathcal{C}$, the space $F(C)$ is $n$–connected.
Then the natural map

$\operatorname*{hocolim}_{\mathcal{C}}F\to\operatorname*{hocolim}_{\mathcal{C}}*=||N_{*}\mathcal{C}||\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}|N_{*}\mathcal{C}|$

is $(n+1)$–connected.

###### Proof.

In each level, the simplicial map inducing
$\operatorname*{hocolim}_{\mathcal{C}}F\to\operatorname*{hocolim}_{\mathcal{C}}*$
is just the projection from a coproduct of $n$–connected spaces to a coproduct
of points, and hence is an $(n+1)$–connected map. So the result follows from
Lemma 1.2. ∎

The proof of Theorem 8.4 will again use the CW neighborhoods considered in the
previous section. We need the following notion, which originated in work of
McCord [20] and was used in the present setting by Dugger and Isaksen [9].

######  Definition 8.7.

A cover $(X,\mathcal{V})$ is said to be _complete_ if for every finite subset
$\mathcal{F}\subseteq\mathcal{V}$, the intersection $\bigcap\mathcal{F}$ is a
union of sets from $\mathcal{V}$.

Note that in the definition of completeness, $\bigcap\mathcal{F}$ is allowed
to be an _infinite_ union of sets from $\mathcal{V}$.

######  Lemma 8.8.

For every cover $(X,\mathcal{V})$, the completion $\hat{\mathcal{V}}$ is, in
fact, complete.

###### Proof.

It suffices to show that for each $C\in\hat{\mathcal{V}}$ with $x\in C$, we
have $C_{x}\subseteq C$. So, say $x\in C\in\hat{\mathcal{V}}$. Then $C=C_{y}$
for some $y\in X$. Now $x\in C_{y}\subseteq\bigcap\mathcal{V}_{y}$ implies
$\mathcal{V}_{y}\subseteq\mathcal{V}_{x}$, so
$\bigcap\mathcal{V}_{x}\subseteq\bigcap\mathcal{V}_{y}$, and comparing the
path components of $x$ in these two spaces yields $C_{x}\subseteq C$. ∎

######  Definition 8.9.

If $\mathcal{V}$ is a set of subcomplexes of the CW complex $X$, then we
define $U(\mathcal{V})=\\{U(V)\colon\thinspace V\in\mathcal{V}\\}$.

######  Lemma 8.10.

Let $(X,\mathcal{V})$ be a complete CW cover. Then $U(\mathcal{V})$ is also
complete.

###### Proof.

Every finite subset of $U(\mathcal{V})$ has the form $U(\mathcal{F})$ for some
finite subset $\mathcal{F}\subseteq\mathcal{V}$, and by completeness there
exists $\mathcal{W}\subseteq\mathcal{V}$ such that
$\bigcap\mathcal{F}=\bigcup\mathcal{W}$. Now Property (2) in Proposition 5.3
yields

$\bigcap
U(\mathcal{F})=U\left(\bigcap\mathcal{F}\right)=U\left(\bigcup\mathcal{W}\right)=\bigcup
U(\mathcal{W}),$

completing the proof. ∎

###### Proof of Theorem 8.4.

Let $\hat{\mathcal{U}}=U(\hat{\mathcal{V}})$, and consider the commutative
diagram

${X}$${\operatorname*{hocolim}_{W\in\hat{\mathcal{V}}}W}$${{|\hat{\mathcal{V}}|}}$${\operatorname*{hocolim}_{U\in\hat{\mathcal{U}}}U}$${{|\hat{\mathcal{U}}|}.}$$\scriptstyle{\simeq}$$\scriptstyle{\cong}$

The map between the homotopy colimits is induced by the poset isomorphism
$U\colon\thinspace\hat{\mathcal{V}}\xrightarrow{\cong}\hat{\mathcal{U}}$ from
Corollary 5.4 together with the natural transformation $W\hookrightarrow
U(W)$. This map is a weak equivalence by Lemma 8.6 and Property (3d) in
Proposition 5.3. Our connectivity assumptions (together with Lemma 8.6) imply
that the maps from the homotopy colimits to the realizations of
$\hat{\mathcal{V}}$ and $\hat{\mathcal{U}}$ are $(n+1)$–connected. Lemmas 8.8
and 8.10 show that $\hat{\mathcal{U}}$ is complete, so the map

$\operatorname*{hocolim}_{U\in\hat{\mathcal{U}}}U\longrightarrow X$

is a weak equivalence by [9, Proposition 4.6(c)]. ∎

In order to formulate the analogous result for open covers of arbitrary
spaces, we will need to restrict to covers in which the path components
$C^{\mathcal{V}}_{x}\subseteq\bigcap\mathcal{V}_{x}$ are open.

######  Definition 8.11.

An open cover $\mathcal{V}$ of a space $X$ is called _thick_ if for each $x\in
X$, the path component $C^{\mathcal{V}}_{x}$ is open in $X$.

######  Example 8.12.

If $\mathcal{V}$ is an open cover of $X$ with each $\mathcal{V}_{x}$ finite
$($for instance, if $\mathcal{V}$ is locally finite$)$ and $X$ is locally path
connected, then $\mathcal{V}$ is automatically thick.

######  Proposition 8.13.

Let $\mathcal{V}$ be a thick open cover of the space $X$, and assume that each
space in $\hat{\mathcal{V}}$ is $n$–connected. Then there is a natural zig-zag

$X\stackrel{{\scriptstyle\simeq}}{{\longleftarrow}}\operatorname*{hocolim}_{W\in\hat{\mathcal{V}}}W\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}|\hat{\mathcal{V}}|$

connecting $X$ to the completion of $\mathcal{V}$, with $\pi$ an
$(n+1)$–connected map.

###### Proof.

As before, Lemma 8.6 implies that $\pi$ is $(n+1)$–connected, and Lemma 8.8
tells us that $\hat{\mathcal{V}}$ is complete. Since $\mathcal{V}$ is thick,
$\hat{\mathcal{V}}$ is an _open_ cover, so [9, Proposition 4.6(c)] shows that
$\operatorname*{hocolim}_{W\in\hat{\mathcal{V}}}W\to X$ is a weak equivalence
as well. ∎

######  Example 8.14.

We now give an example of a CW cover $(X,\mathcal{V})$ for which the nerve
$\mathcal{N}(\mathcal{V})$ is _not_ homotopy equivalent $X$, and yet the
completion $\hat{\mathcal{V}}$ is. The key point in this example is that while
all of the sets in $\mathcal{V}$ and the intersections $\mathcal{V}_{x}$,
$x\in X$, are contractible, one of the pairwise intersections not contractible
$($but is connected$)$.

Let $X=S^{2}\cup D\simeq S^{2}\vee S^{2}$, where $S^{2}$ is the unit sphere in
$\use@mathgroup\M@U\symAMSb{R}^{3}$ and $D$ is the unit disk $D$ in the
$x$-$y$ plane, with the CW decomposition shown in Figure 1. This CW structure
has five 2-cells: the upper and lower hemispheres $D_{+}$ and $D_{-}$, and the
three sectors $A$, $B$, $C$ of the disk $D$. Our CW cover is simply the set of
$2$–cells:

$\mathcal{V}=\\{D_{+},D_{-},A,B,C\\}.$

Since the intersection $D_{+}\cap D_{-}=S^{1}$ is not contractible, the
classical Nerve Theorem does not apply $($and Theorem 2.13 applies only with
$n=0$$)$, and we claim that

$\mathcal{N}(\mathcal{V})\simeq S^{2}.$

To see this, first note that by Proposition 2.7 we have
$\mathcal{N}{\mathcal{V}}\simeq P$, where $P\subseteq 2^{X}$ is the poset
consisting of all non-empty intersections of sets from $\mathcal{V}$ $($note
that each of these intersections is path connected$)$. Let

$P_{D}=P_{\leqslant D_{+}}\cup P_{\leqslant D_{-}}$

and note that $\Delta(P_{D})=\Delta(P_{\leqslant
D_{+}})\cup\Delta(P_{\leqslant D_{-}})$. Applying the Nerve Theorem to this
$2$–element cover of $\Delta(P_{D})$ shows that $\Delta(P_{D})$ is
contractible, because

$\Delta(P_{\leqslant D_{+}})\cap\Delta(P_{\leqslant
D_{-}})=\Delta(P_{\leqslant E})\simeq*,$

where $E=D_{+}\cap D_{-}$ is the equator. Now, consider the cover of $\Delta
P$ by the four sets $\Delta(P_{A})$, $\Delta(P_{B})$, $\Delta(P_{C})$, and
$\Delta(P_{D})$. All intersections among these complexes are contractible
except for the four-way intersection, which is empty, so applying the Nerve
Theorem again gives

$\Delta(P)\simeq\partial\Delta^{3}\cong S^{2},$

as claimed.

On the other hand, the completion $\hat{\mathcal{V}}$ is exactly
$P\setminus\\{E\\}$, so each element of $\hat{V}$ is contractible and hence
$|\hat{\mathcal{V}}|\simeq X$ by Theorem 8.4.

$D_{-}$$D_{+}$$A$$B$$C$ Figure 1. A CW decomposition of the sphere union an
equatorial disk

##  9\. Homology and cohomology of (partial) completed nerves

In this section we prove (co)homological versions of Theorem 2.13, Theorem
8.4, and Proposition 8.13. In outline, the proofs are very similar to the
homotopy versions, except that instead of analyzing connectivity of maps
between simplicial spaces, we instead analyze their effect in (co)homology by
comparing the spectral sequences associated to the skeletal filtrations.
Throughout this section, we fix an abelian coefficient group $A$, and all
(co)homology groups are (implicitly) taken with coefficients in $A$.

These spectral sequences were introduced by Segal [30], and are also discussed
in [10]. Briefly, given a good222This assumption, which as mentioned in
Section 4 holds for all Čech complexes and for all simplicial sets, is just
needed to guarantee that the map from the thick realization $||X||$ to the
ordinary realization $|X|$ is a weak equivalence. Without this assumption,
there are similar spectral sequences converging to the (co)homology of
$||X_{*}||$. simplicial space $X_{*}$ there is a strongly convergent
homological spectral sequence

$E^{r}_{p,q}(X_{*})\implies H_{p+q}(|X_{*}|)$

with first page

$E^{1}_{p,q}(X_{*})\cong H_{q}(X_{p}),$

and differentials

$d_{r}\colon\thinspace E^{r}_{p,q}(X_{*})\longrightarrow
E^{r}_{p-r,q+r-1}(X_{*}).$

In cohomology, the situation is the same except that the differentials take
the form

$d_{r}\colon\thinspace E_{r}^{p,q}(X_{*})\longrightarrow
E_{r}^{p+r,q-r+1}(X_{*}).$

######  Definition 9.1.

We say that a space $X$ is $n$–acyclic if $H_{*}(X)=0$ for $0<*\leqslant n$,
and we say $X$ is $n$–coacyclic if $H^{*}(X)=0$ for $0<*\leqslant n$.
Similarly, we say that a map $X\to Y$ is $n$–$($co$)$acyclic if it induces
isomorphism in $($co$)$homology in degrees $0,\ldots,n-1$ and a surjection
$($respectively, an injection$)$ in degree $n$. We say that a zig-zag of maps

$X\longleftarrow Z_{1}\longrightarrow\cdots\longleftarrow
Z_{k+1}\longrightarrow Y$

is $n$–(co)acyclic if all maps in the zig-zag induce isomorphisms in
(co)homology in degrees $0,\ldots,n-1$, and on $H_{n}(-)$ $($respectively,
$H^{n}(-)$$)$, all leftward pointing maps in the zig-zag induce isomorphisms
while all right-pointing maps induce surjections $($respectively,
injections$)$.

Note that an $n$–acyclic zig-zag from $X$ to $Y$ induces a surjection

$H_{n}(X)\twoheadrightarrow H_{n}(Y),$

while an $n$–coacyclic zig-zag induces an injection

$H^{n}(Y)\hookrightarrow H^{n}(X).$

In the $n$–(co)acyclic zig-zags below, all but one map in the zig-zag will
induce isomorphisms in both homology and cohomology in degree $n$.

We will need two elementary algebraic lemmas. The proof of the first one
consists of two short diagram chases.

######  Lemma 9.2.

Consider a commutative diagram

${A}$${B}$${C}$${A^{\prime}}$${B^{\prime}}$${C^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\alpha}$$\scriptstyle{g}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$

in which the rows are chain complexes $($meaning
$gf=g^{\prime}f^{\prime}=0$$)$. If $\beta$ is surjective and $\gamma$ is an
isomorphism, then the map

$\ker(g)/\mathrm{im}(f)\stackrel{{\scriptstyle\beta_{*}}}{{\longrightarrow}}\ker(g^{\prime})/\mathrm{im}(f^{\prime})$

induced by $\beta$ is surjective. Similarly, if $\alpha$ is an isomorphism and
$\beta$ is injective, then $\beta_{*}$ is injective.

The next lemma is proven by a simple induction, using the (strong) 5-lemma.

######  Lemma 9.3.

Let $0=A_{0}\leqslant A_{1}\leqslant\cdots\leqslant A_{n}=A$ and
$0=B_{0}\leqslant B_{1}\leqslant\cdots\leqslant B_{n}\leqslant B$ be filtered
abelian groups, and let $f\colon\thinspace A\to B$ be a homomorphism
respecting the filtrations. Then map of associated graded groups

$f_{*}\colon\thinspace\bigoplus_{i=1}^{n}A_{i}/A_{i-1}\longrightarrow\bigoplus_{i=1}^{n}B_{i}/B_{i-1}$

is surjective $($respectively, injective$)$ if and only if $f$ is surjective
$($injective$)$.

######  Proposition 9.4.

Fix $n\geqslant 0$, and let $f_{*}\colon\thinspace X_{*}\to Y_{*}$ be a map of
good simplicial spaces and assume that for each $k\geqslant 0$, the component
map $f_{k}\colon\thinspace X_{k}\to Y_{k}$ is $(n-k)$– $($co$)$acyclic. Then
$|f_{*}|\colon\thinspace|X_{*}|\to|Y_{*}|$ is $n$–$($co$)$acyclic.

###### Proof.

We give the argument for homology; the case for cohomology is nearly
identical. Since $|f_{*}|$ respects the skeletal filtrations on $|X_{*}|$ and
$|Y_{*}|$, it induces a maps between the associated homology spectral
sequences. On the first page, the map of spectral sequences has the form

$E^{1}_{p,q}(X_{*})=H_{q}(X_{p})\xrightarrow{(f_{p})_{*}}H_{q}(Y)=E^{1}_{p,q}(Y_{*}),$

meaning that it is an isomorphism when $p+q\leqslant n$ and a surjection when
$p+q=n+1$. By Lemma 9.2 and induction on $r$, it follows that for each
$r\geqslant 1$, the induced maps

$E^{r}_{p,q}(X_{*})\longrightarrow E^{r}_{p,q}(Y_{*}),$

are isomorphisms when $p+q<n$ and surjections when $p+q=n$. For fixed $p$ and
$q$, the groups $E^{r}_{p,q}(X_{*})$ and $E^{r}_{p,q}(Y_{*})$ stabilize (as
$r$ increases) once $r>p+q$, because all differentials in and out
$E^{r}_{p,q}$ are zero for dimensional reasons (for $r>p+q$). Hence
$\bigoplus_{p}E^{k+1}_{p,k-p}(X_{*})$ is the associated graded group of the
skeletal filtration on $H_{k}(|X_{*}|)$, and similarly for $Y_{*}$. Lemma 9.3
now completes the proof. ∎

######  Remark 9.5.

The proof of Proposition 9.4 shows that the result also holds for maps of
arbitrary semisimplicial spaces $($using thick realizations$)$, since as
mentioned above, the spectral sequences exist in that more general setting.

######  Theorem 9.6.

Consider an object $(X,\mathcal{V})$ in either $\underline{\mathrm{LOCov}}$ or
$\underline{\mathrm{CWCov}}$. Assume that for some $n\geqslant 0$ and each set
$\mathcal{F}\in\mathcal{N}(\mathcal{V})$ with cardinality
$0<|\mathcal{F}|\leqslant n$, every path component of $\bigcap\mathcal{F}$ is
$(n-|\mathcal{F}|+1)$– $($co$)$acyclic. Then the natural zig-zag $(\ref{zz})$
is $(n+1)$– $($co$)$acyclic.

Before giving the proof, we note that for covers of simplicial complexes by
subcomplexes a slightly weaker version of this result appears as [23,
Corollary 4.4]. In the CW case, one can deduce an algebraic version of Theorem
9.6, comparing the (co)homology groups of $X$ and of
$\mathcal{N}(\mathcal{V})$, from the Mayer–Vietoris spectral sequence
constructed in Brown [8, Section VII.4]. One advantage of the approach taken
here is that it will be apparent from the proof that the zig-zag (4) induces a
highly (co)acyclic map, whereas from the algebraic perspective it seems harder
to obtain this type of conclusion. The spectral sequence used in the proof
below should be thought of as a more geometric version of the Mayer–Vietoris
spectral sequence; in particular, the groups on the $E^{1}$ pages of these
spectral sequences are isomorphic.

###### Proof.

The proof of Theorem 2.13 shows that all maps in the zig-zag (4) are weak
equivalences, except for the map $\pi_{*}$; note that connectivity hypotheses
in Theorem 2.13 were needed only for analyzing the connectivity of $\pi$. It
remains to show that $\pi$ is $n$–(co)acyclic. This follows from Proposition
9.4, since our hypotheses exactly guarantee that for each $p\geqslant 0$, the
map

$\check{\mathcal{C}}_{p}(\mathcal{V})\to\check{\mathcal{C}}^{\delta}_{p}(\mathcal{V})$

is $(n-p+1)$– $($co$)$acyclic. ∎

######  Proposition 9.7.

Let $(X,\mathcal{V})$ be either a thick open cover of the space $X$ or a CW
cover of the CW complex $X$, and assume that each space in $\hat{\mathcal{V}}$
is $n$–$($co$)$acyclic. Then then the natural zig-zag $(\ref{zz2})$ is
$(n+1)$– $($co$)$acyclic.

###### Proof.

The proofs of Propositions 8.4 and 8.13 show that the natural map

$\operatorname*{hocolim}_{W\in\hat{\mathcal{V}}}W\longrightarrow X$

is a weak equivalence, and it remains to show that the projection

(8)
$\operatorname*{hocolim}_{W\in\hat{\mathcal{V}}}\longrightarrow\operatorname*{hocolim}_{W\in\hat{V}}*\longrightarrow|N_{*}\mathcal{V}|$

is $(n+1)$– $($co$)$acyclic. This follows from Proposition 8.6 together with
the Relative Hurewicz Theorem ([32, Theorem 7.5.4] or [14, Theorem 4.3.7]), or
alternatively one may apply Proposition 9.4. ∎

######  Remark 9.8.

As discussed in [10], the spectral sequences used above can also be
constructed for homology with arbitrary local coefficients. Beginning with a
local coefficient system on $X$, thought of as a functor
$\mathcal{L}\colon\thinspace\pi(X)\to\underline{\mathrm{Ab}}$ out of the
fundamental groupoid of $X$, one would like to compare $H_{*}(X;\mathcal{L})$
$($at least in a range of dimensions$)$ with the homology of some local system
on $|\hat{\mathcal{N}}(\mathcal{V})|$. If we assume that each
$V\in\mathcal{V}$ is simply connected, so that the zig-zig $(\ref{zz})$
induces isomorphisms on fundamental groupoids, then $\mathcal{L}$ corresponds
under these isomorphisms to some local system $\mathcal{L}^{\prime}$ on
$|\hat{\mathcal{N}}(\mathcal{V})|$. The arguments above then lead to an exact
analogue of Theorem 9.6, comparing $H_{*}(X;\mathcal{L})$ and
$H_{*}(\hat{\mathcal{N}}(\mathcal{V});\mathcal{L}^{\prime})$. One can extend
Proposition 9.7 to homology with local coefficient similarly, assuming each
$W\in\hat{\mathcal{V}}$ is simply connected.

##  10\. Cutsets in partially ordered sets

As an application of the simplicial case of Theorem 2.13, we can extend
several results on crosscuts in posets. The notion of a cutset generalizes
that of a crosscut.

######  Definition 10.1.

Given a poset $P$ and an element $x\in P$, the _star_ of $x$ is the subposet
of $P$ defined by

$\mathrm{St}(x)=P_{\leqslant x}\cup P_{\geqslant x}.$

A _cutset_ of a poset $P$ is a subset $X\subseteq P$ such that

$\mathcal{V}_{X}:=\\{\Delta(\mathrm{St}(x)):\,x\in X\\}$

is a $($simplicial$)$ cover of $\Delta P$.

The definition of a cutset can also be phrased as saying that for each finite
chain $\sigma\subseteq P$, there is some $x\in X$ such that
$\sigma\cup\\{x\\}$ is still a chain (in which case
$\sigma\in\Delta(\mathrm{St}(x))$).

We need the following well-known fact.

######  Lemma 10.2.

Let $P$ be a poset. Then for each $x\in P$, the star $\mathrm{St}(x)$ is
contractible.

For completeness, we sketch two proofs. First, the constant map
$\mathrm{St}(x)\to\mathrm{St}(x)$, $p\mapsto x$, is point-wise comparable to
the identity, and hence homotopic to the identity by Björner–Walker [7,
Proposition 3.1]. Alternatively, $\mathrm{St}(x)$ is the join of the posets
$P_{\leqslant x}$ and $P_{>x}$, so by [27, Proposition 1.9] we have
$|\mathrm{St}(x)|\cong|P_{\leqslant x}|*|P_{>x}|$, which is contractible since
$P_{\leqslant x}$ is contractible [21, Lemma 2.3].

######  Definition 10.3.

If $P$ is a poset, we write $\pi_{0}(P)$ for the set of equivalence classes of
$P$ under the equivalence relation generated by the order relation on $P$.

Note that there is a natural bijection $\pi_{0}(P)\to\pi_{0}|P|$ sending the
equivalence class of $p$ to the path component of the corresponding vertex in
$|P|$.

In the previous section, we defined the notion of an $n$–(co)acyclic zig-zag
of maps between spaces (Definition 9.1). The notion of an $n$–connected zig-
zag is defined analogously.

######  Proposition 10.4.

Let $X$ be a cutset of a poset $P$.

If $P$ is connected, then the fundamental group of $P$ is isomorphic to the
fundamental group of the connected, 2-dimensional regular CW–complex $R(P,X)$
constructed as follows: the vertex set of $R(P,X)$ is $X$; the set of edges
connecting $x,x^{\prime}\in X$ $($$x\neq x^{\prime}$$)$ is
$\pi_{0}(\mathrm{St}(x)\cap\mathrm{St}(x^{\prime}))$, and for each 3–element
set $\\{x,x^{\prime},x^{\prime\prime}\\}\subseteq X$ and each path component
$C\in\pi_{0}(\mathrm{St}(x)\cap\mathrm{St}(x^{\prime})\cap\mathrm{St}_{x^{\prime\prime}})$,
there is a 2–cell in $R(P,X)$ attached $($by a homeomorphism$)$ to the
triangle whose edges correspond to the components of $C$ in the pairwise
intersections $\mathrm{St}(x)\cap\mathrm{St}(x^{\prime})$,
$\mathrm{St}(x^{\prime})\cap\mathrm{St}(x^{\prime\prime})$, and
$\mathrm{St}(x)\cap\mathrm{St}(x^{\prime\prime})$.

In general, there is a natural zig-zag connecting $|P|$ to the geometric
realization of the poset

$\Gamma(P,X):=\\{C\subseteq P\colon\thinspace C\in\pi_{0}(D)\textrm{ for some
}D\in\mathcal{N}(\mathcal{V}_{X})\\}.$

Assume that for every $D\in\mathcal{N}(\mathcal{V}_{X})$, each
$C\in\pi_{0}(D)$ is $(n-|D|+1)$–connected $($respectively, $(n-|D|+1)$–
$($co$)$acyclic for $A$–coefficients$)$. Then this zig-zag is
$(n+1)$–connected $($respectively, $(n+1)$– $($co$)$acyclic for
$A$–coefficients$)$.

When $\mathrm{St}(x)\cap\mathrm{St}(x^{\prime})$ is path connected for each
$x,x^{\prime}\in X$ (and $X$ is the set minimal elements of $P$), this result
reduces to the one in [6, Example 9]. When all $C\in\Gamma(P,X)$ are
contractible, it reduces to [26, Theorem 3.12]. For crosscuts in lattices, the
acyclic case ($n=\infty$) recovers Folkman’s crosscut theorem [13].

###### Proof.

Let $\mathcal{C}_{X}=\\{|\mathrm{St}(x)|\colon\thinspace x\in X\\}$ be the CW
cover of $|P|$ associated to $\mathcal{V}_{X}$. Lemma 10.2 shows that the
hypotheses of Theorem 2.13 are satisfied for $n=1$, and we obtain an
isomorphism $\pi_{1}P\cong\pi_{1}\hat{\mathcal{N}}(\mathcal{C}_{X})$ (note
that the theorem also implies that $\hat{\mathcal{N}}(\mathcal{C}_{X})$ is
path connected, although this can also be seen directly from connectedness of
$P$). The complex $R(P,X)$ is just 2-skeleton of the canonical CW complex
structure on the geometric realization of the simplicial set in Proposition
2.4, so $\pi_{1}\hat{\mathcal{N}}(\mathcal{C}_{X})\cong\pi_{1}R(P,X)$.

In general, we have the natural zig-zag (4) connecting $|P|$ to
$|\hat{\mathcal{N}}(\mathcal{C}_{X})|$, and under the assumptions of the
Proposition this zig-zag is $(n+1)$–connected by Theorem 2.13 (or
$(n+1)$–(co)acyclic by Theorem 9.6), and by Proposition 2.7, there is a weak
equivalence

$\hat{\mathcal{N}}(\mathcal{C}_{X})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\overline{\mathcal{V}}_{X}=\left\\{K\subseteq|P|:\,K\in\pi_{0}\left(\bigcap\mathcal{F}\right)\textrm{
for some }\mathcal{F}\in\mathcal{N}(\mathcal{C}_{X})\right\\}.$

Finally, the map $C\mapsto|C|$ is a natural isomorphism of posets

$\Gamma(P,X)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\overline{\mathcal{V}}_{X}.$

∎

##  11\. Detection of $n$–connected maps and variations on Quillen’s Poset
Fiber Theorem

In [6], Björner proved an $n$–connected version of Quillen’s Poset Fiber
Theorem, showing that a map of posets $f\colon\thinspace P\to Q$ is
$(n+1)$–connected if each Quillen fiber $f^{-1}(Q_{\geqslant q})$ is
$n$–connected. With some additional hypotheses, we can prove a version of the
Poset Fiber Theorem that closely mimics our $n$–connected version of the Nerve
Theorem, relaxing the connectivity requirements on the fibers as one moves
deeper into $Q$. The same strategy leads to a rather different method for
detecting highly connected poset maps, by examining inverse images of chains
rather than cones (Proposition 11.12). In fact, we will deduce both of these
results from a very general detection result for $n$–connected maps,
generalizing [34, Theorem 6.7.9].

As in previous sections, the results below have (co)homological analogues,
which can all be proven in the same manner as the connectivity statements
(using Proposition 9.4 in place of Lemma 1.2). Alternatively, the
(co)homological versions may be proven using the Mayer–Vietoris spectral
sequence [8, Section VII.4] instead of the spectral sequence used in
Proposition 9.4. For these reasons we state and prove only the connectivity
results.

We begin by formulating a general detection result for $n$–connected maps
between CW complexes.

######  Proposition 11.1.

Let $X$ and $Y$ be CW complexes and let $f\colon\thinspace X\to Y$ be a map.
Let $\mathcal{V}\colon\thinspace I\to 2^{Y}$ be a CW cover of $Y$ such that
$f^{-1}(\mathcal{V}(i))$ is a subcomplex of $X$ for each $i\in I$. If

(9) $f\colon\thinspace
f^{-1}(\mathcal{V}(i_{1})\cap\cdots\cap\mathcal{V}(i_{k}))\longrightarrow\mathcal{V}(i_{1})\cap\cdots\cap\mathcal{V}(i_{k})$

is $(n-k+1)$–connected for each $(i_{1},\ldots,i_{k})\in I^{k}$, then $f$ is
$n$–connected.

###### Proof.

Let $f^{-1}\mathcal{V}\colon\thinspace I\to 2^{X}$ be the cover $i\mapsto
f^{-1}(\mathcal{V}(i))$. Then $f^{-1}(\mathcal{V})$ is a CW cover of $X$, and
we have a map of covers
$(f,\mathrm{Id}_{I}):(X,f^{-1}\mathcal{V})\to(Y,\mathcal{V})$. This gives rise
to a commutative diagram

${X}$${Y}$${{|\check{\mathcal{C}}(f^{-1}\mathcal{V})|}}$${{|\check{\mathcal{C}}(\mathcal{V})|},}$$\scriptstyle{f}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

in which the vertical maps are weak equivalences by Proposition 6.1 and the
bottom map is induced by $(f,\mathrm{Id}_{I})$. By Lemma 1.2, to complete the
proof it suffices to check that the map

$\check{\mathcal{C}}_{m}(f^{-1}\mathcal{V})\longrightarrow\check{\mathcal{C}}_{m}(\mathcal{V})$

is $(n-m)$–connected for each $m\geqslant 0$. But this map is simply the
coproduct, over all $(i_{1},\ldots,i_{m+1})\in I^{m+1}$, of all the maps (9),
and hence its connectivity is at least $n-(m+1)+1=n-m$. ∎

######  Remark 11.2.

Note that in the above proof, it was important to view $f^{-1}(\mathcal{V})$
as a cover indexed by the same set as $\mathcal{V}$ itself, since otherwise
when $f$ is non-surjective the spaces
$\check{\mathcal{C}}_{m}(f^{-1}\mathcal{V})$ could have fewer path components
than $\check{\mathcal{C}}_{m}(\mathcal{V})$: for instance, we could have
$f^{-1}(\mathcal{V}(i))=f^{-1}(\mathcal{V}(j))$ for some $i\neq j$.

Before specializing to the case of posets, we record a version of Proposition
11.1 for general topological spaces.

######  Proposition 11.3.

Let $f\colon\thinspace X\to Y$ be a map of topological spaces, and let

$\mathcal{V}\colon\thinspace I\to 2^{Y}$

be a cover of $Y$ such the interiors of the sets in $\mathcal{V}(I)$ form an
open cover of $Y$. If the map

$f\colon\thinspace
f^{-1}(\mathcal{V}(i_{1})\cap\cdots\cap\mathcal{V}(i_{k}))\longrightarrow\mathcal{V}(i_{1})\cap\cdots\cap\mathcal{V}(i_{k})$

is $(n-k+1)$–connected for each $(i_{1},\ldots,i_{k})\in I^{k}$, then $f$ is
$n$–connected.

The proof is essentially the same as above, using Proposition 4.1 and noting
that since the interiors of the sets $\mathcal{V}(i)$ cover $Y$, the interiors
of the $f^{-1}(\mathcal{V}(i))$ cover $X$.

We now consider some special cases of Proposition 11.1 for posets. Note that
when $f\colon\thinspace X\to Y$ is the geometric realization of a map between
posets, or more generally a map of simplicial sets, the inverse image of a
subcomplex of $Y$ is automatically a subcomplex of $X$, so this hypothesis in
Proposition 11.1 holds automatically.

######  Definition 11.4.

Let $Q$ be a poset. We say that a subset $S\subseteq Q$ is _bounded above_ if
there exists an element $q\in Q$ such that $S\subseteq Q_{\leqslant q}$.
Recall that the _join_ of $S$, if it exists, is the $($unique$)$ minimum
element of the set $\\{q\in Q:\,S\subseteq Q_{\leqslant q}\\}$. We denote the
join of $S$ by $\vee S$.

Let $\mathcal{M}(Q)\subseteq Q$ denote the set of all minimal elements in $Q$.
We say that $Q$ is _coherent_ if

  * •

$Q$ is bounded below $($that is, every element $q\in Q$ lies above some
minimal element $m\in\mathcal{M}$$)$, and

  * •

Every finite set $\mu\subseteq\mathcal{M}$ that is bounded above has a join
$\vee\mu\in Q$.

When $Q$ is coherent, we denote the set of all joins of finite subsets of
$\mathcal{M}$ by $\widetilde{\mathcal{M}}$.

There are many examples of coherent posets: for instance, every finite lattice
is coherent, and if $Q\subseteq 2^{X}$ is finite and closed under either
unions or intersections, then $Q$ is coherent. There are many infinite
examples as well, such as the coset poset of an arbitrary group or the poset
of finite, non-trivial subgroups in an arbitrary group.

######  Proposition 11.5.

Let $f\colon\thinspace P\to Q$ be a map of posets, and assume that $Q$ is
coherent. If $f^{-1}(Q_{\geqslant m_{1}\vee\cdots\vee m_{k}})$ is
$(n-k+1)$–connected for every
$m_{1}\ldots,m_{k}\in\widetilde{\mathcal{M}}(Q)$, then $f$ is
$(n+1)$–connected.

###### Proof.

For every list of elements $q_{1},\ldots,q_{k}\in\mathcal{M}(Q)$, the
intersection

$Q_{\geqslant q_{1}}\cap\cdots\cap Q_{\geqslant q_{k}}=Q_{\geqslant
q_{1}\vee\cdots\vee q_{k}}$

is contractible, so the map from the $(n-k+1)$–connected space
$f^{-1}(Q_{\geqslant m_{1}\vee\cdots\vee m_{k}})$ to $Q_{\geqslant
q_{1}\vee\cdots\vee q_{k}}$ is $(n-k+2)$–connected. The result now follows by
applying Proposition 11.1 to the cover of $|Q|$ given by $\mathcal{M}(Q)\to
2^{|Q|}$, $q\mapsto|Q_{\geqslant q}|$. ∎

######  Remark 11.6.

There is a dual version of Theorem 11.5 in which minimal elements and joins
are replaced by maximal elements and meets, and upwards cones are replaced by
downward cones.

From the perspective above, the Poset Fiber Theorem arises from studying
covers of posets by cones. If $P$ is a poset in which every chain is finite,
then the above arguments can also be run for the cover of $P$ by _maximal
chains_. It turns out that intersections of maximal chains have an intrinsic
description. We explain this, we will need some terminology and notation.

######  Definition 11.7.

Let $P$ be a poset. We write the equivalence relation generated by the order
relation on $P$ as $\sim$. Given a subset $S\subseteq P$ and an element $p\in
P$, we write $p\sim S$ to mean that $p\sim s$ for all $s\in S$.

The _neighborhood in_ $P$ of a subset $S\subseteq P$ is defined by

$N_{P}(S)=\\{p\in P:\,p\sim S\\},$

and the _core_ of $S$ is defined by

$C(S)=N_{S}(S).$

We say that a subset $S\subseteq P$ is _essential_ if $S=C(N_{P}(S))$.

From here on all neighborhoods will be taken with respect to the ambient poset
$P$, so we drop $P$ from the notation.

We record some simple consequences of these definitions.

######  Lemma 11.8.

Let $P$ be a poset, and let $S$ and $T$ be subsets of $P$. Then the following
statements hold:

  1. (1)

$S\subseteq C(N(S))$.

  2. (2)

$C(N(S))\subseteq N(N(S))$

  3. (3)

$S\subseteq T\implies N(T)\subseteq N(S)$.

  4. (4)

$S\subseteq T\implies C(N(S))\cap N(T)\subseteq C(N(T))$.

######  Lemma 11.9.

If $\epsilon_{1}$ and $\epsilon_{2}$ are essential chains in $P$, then so is
$\epsilon_{1}\cap\epsilon_{2}$.

###### Proof.

We will use the properties listed in Lemma 11.8. In light of Property (1), it
suffices to show that for $i=1,2$, we have

$C(N(\epsilon_{1}\cap\epsilon_{2}))\subseteq\epsilon_{i}=C(N(\epsilon_{i})).$

By Property (4), we have

$C(N(\epsilon_{1}\cap\epsilon_{2}))\cap N(\epsilon_{i})\subseteq
C(N(\epsilon_{i})),$

so to complete the proof we just need to show that
$C(N(\epsilon_{1}\cap\epsilon_{2}))\subseteq N(\epsilon_{i})$. Since
$\epsilon_{i}$ is a chain, we have $\epsilon_{i}\subseteq
N(\epsilon_{1}\cap\epsilon_{2})$, and now Properties (2) and (3) yield

$C(N(\epsilon_{1}\cap\epsilon_{2}))\subseteq
N(N(\epsilon_{1}\cap\epsilon_{2}))\subseteq N(\epsilon_{i}),$

completing the proof. ∎

######  Definition 11.10.

We say a poset $P$ is _locally finite-dimensional_ if every chain in $P$ is
finite.

######  Proposition 11.11.

Let $P$ be a locally finite-dimensional poset. Then a chain $\epsilon\subseteq
P$ is essential if and only if it is the intersection of some finite
collection of maximal chains.

###### Proof.

First, note that all maximal chains are essential, since if $\mu\subseteq P$
is a maximal chain, then $N(\mu)=\mu$ and hence $C(N(\mu))=C(\mu)=\mu$. It
follows from Lemma 11.9 and induction that every finite intersection of
maximal chains is essential.

Conversely, say $\epsilon\subseteq P$ is an essential chain. Let
$\mu(\epsilon)$ be the set of all maximal chains containing $\epsilon$. Since
each element of $\mu(\epsilon)$ is a finite set, $\bigcap\mu(\epsilon)$ is
equal to $\bigcap\mu^{\prime}(\epsilon)$ for some finite subset
$\mu^{\prime}(\epsilon)\subseteq\mu(\epsilon)$. Hence it suffices to show that
$\bigcap\mu(\epsilon)=\epsilon$. We have
$\epsilon\subseteq\bigcap\mu(\epsilon)$ by definition. To prove the reverse
inclusion, consider an element $p\in\bigcap\mu(\epsilon)$. We need to prove
that $p\in\epsilon=C(N(\epsilon))$. Certainly $p\in N(\epsilon)$, so it
suffices to show that $p\in N(N(\epsilon))$. Say $n\in N(\epsilon)$. We want
to show that $p\sim n$. Since $n\sim\epsilon$, we know that
$\\{n\\}\cup\epsilon$ is a chain, and applying Zorn’s Lemma to the poset
$\\{D\in\Delta P:\,\\{n\\}\cup\epsilon\subseteq D\\}$ shows that there exists
a maximal chain $M$ containing $\\{n\\}\cup\epsilon$. Now $M\in\mu(\epsilon)$,
so $p\in M$ and hence $p\sim n$. ∎

The following theorem is proven in the same manner as Proposition 11.5.

######  Proposition 11.12.

Let $f\colon\thinspace P\to Q$ be a map of posets, with $Q$ locally finite-
dimensional. If $f^{-1}(m_{1}\cap\cdots\cap m_{k})$ is $(n-k+1)$–connected for
all maximal chains $m_{1},\ldots,m_{k}\subseteq Q$, then $f$ is
$(n+1)$–connected.

In particular, if $f^{-1}(\epsilon)$ is contractible for each essential chain
$\epsilon\subseteq Q$, then $f$ is a homotopy equivalence.

######  Example 11.13.

There is a natural way to build examples of poset maps for which the inverse
image of every essential chain is highly connected $($or highly acyclic$)$.
Let $P$ be a poset and let $Q_{p}$, $p\in P$ be a collection posets indexed by
$P$. Define a new poset

$P^{Q}:=\\{(p,q)\in P\times\bigcup_{p\in P}Q_{p}:\,q\in Q_{p}\\}$

with order relation $(p,q)\leqslant(p^{\prime},q^{\prime})$ if and only if
either

  * •

$p^{\prime}=p$ and $q\leqslant q^{\prime}\in Q_{p}$, or

  * •

$p<p^{\prime}$.

Note that the natural projection $\pi\colon\thinspace P^{Q}\to P$,
$\pi(p,q)\mapsto p$, is order-preserving, and the inclusions
$Q_{p}\hookrightarrow P^{Q}$, $q\mapsto(p,q)$, are order-isomorphisms onto
their images. If $\sigma\subseteq P$ is a chain, then
$\pi^{-1}(\sigma)=\\{(p,q)\in P^{Q}:\,p\in\sigma\\}$ is the join of the posets
$Q_{p}$, $p\in\sigma$. Letting $c(Q_{p})$ denote the connectivity of $Q_{p}$
$($that is, the maximum $n$ such that $Q_{p}$ is $n$–connected, or $\infty$ if
$Q_{p}$ is contractible$)$, it follows from [27, Proposition 1.9] and [21,
Lemma 2.3] that the connectivity of $\pi^{-1}(\sigma)$ is $($at least$)$
$2\dim(\sigma)+\sum_{p\in\sigma}c(Q_{p})$ $($note that since we have defined
the empty space to be $(-2)$–connected, this formula holds even if some of the
posets $Q_{p}$ are empty$)$.

As a simple example, let $P=\\{0,0^{\prime},1,2\\}$ with the usual numerical
ordering, except that $0$ and $0^{\prime}$ are incomparable. Both maximal
chains in $P$ contain $1$, so if $Q_{1}$ is contractible then
$\pi^{-1}(\sigma)\simeq*$ for each essential chain $\sigma\in P$. Note,
however, that if the other $Q_{i}$ are not contractible, then $\pi$ does not
satisfy the hypotheses of the ordinary Poset Fiber Theorems, as
$\pi^{-1}(P_{\geqslant 2})=Q_{2}$ and $\pi^{-1}(P_{\leqslant 0})=Q_{0}$ $($and
similarly for $0^{\prime}$$)$.

######  Example 11.14.

The construction in Example 11.13 gives examples of covers in which the
connectivity of intersections decreases as in the hypotheses of Theorem 2.13.
Starting with a poset $P$, we can set $Q_{p}=\mathcal{S}^{0}$ for each $p\in
P$, where $\mathcal{S}^{0}$ is the 2-element poset whose elements are
incomparable $($so that $|\mathcal{S}^{0}|=S^{0}$$)$. Let $\mathcal{V}$ be the
cover of $P^{Q}$ by the inverse images of maximal chains in $P$ under the
projection $\pi\colon\thinspace P^{Q}\to P$. Then each intersection of
elements from $\mathcal{V}$ has the form $W=\pi^{-1}(p_{1}<\cdots<p_{k})$ for
some chain $p_{1}<\cdots<p_{k}$ in $P$. It follows that $W$ is a join of $k$
copies of $\mathcal{S}^{0}$, so $|W|\cong S^{k-1}$ and has connectivity $k-2$.

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