The locally path-connected coreflection (Part III)

This post gives another look at the locally path-connected coreflection that I’ve found quite interesting and useful. For the previous posts see Part I and Part II. In this post, we’ll be motivated by the following two basic observations.

Lemma 1: The topological sum $\coprod_{j\in J}X_j$ of a family of locally path-connected spaces $X_j$ is locally path-connected.

Lemma 2: The quotient of a locally path-connected space is locally path-connected.

Proof. Let $q:X\to Y$ be a quotient map where $X$ is locally path-connected. Let $U$ be an open set in $Y$ and $C$ be a non-empty path-component of $U$ . It suffices to check that $C$ is open in $Y$ . Since $q$ is quotient, we need to check that $q^{-1}(C)$ is open in $X$ . If $x\in q^{-1}(C)\subseteq q^{-1}(U)$ , then there is an open, path connected neighborhood $V$ such that $x\in V\subseteq q^{-1}(U)$ . We claim that $V\subseteq q^{-1}(C)$ . Let $v\in V$ and $\alpha:[0,1]\to V$ be a path from $x$ to $v$ . Then $q\circ\alpha:[0,1]\to q(V)\subseteq U$ is a path from $q(x)\in C$ to $q(v)$ . Since $C$ is the path component of $q(x)$ in $U$ , the path $q\circ\alpha$ must have image entirely in $C$ . Thus $\alpha$ has image in $q^{-1}(C)$ . In particular, $\alpha(1)=v\in q^{-1}(C)$ . $\square$

$\mathbf{Top}$ denotes the usual category of topological spaces and continuous functions. Here is an important definition from Categorical Topology.

Definition 3: Let $\mathscr{C}$ be a subcategory of $\mathbf{Top}$ . The coreflective hull of $\mathscr{C}$ is the full subcategory $H(\mathscr{C})$ of $\mathbf{Top}$ consisting of all spaces which are the quotient of a topological sum (i.e. disjoint union) of spaces in $\mathscr{C}$ .

Certainly $\mathscr{C}\subseteq H(\mathscr{C})$ . Moreover, the coreflective hull is the category “generated” by the collection of spaces $\mathscr{C}$ in the following sense: the inclusion $H(\mathscr{C})\to\mathbf{Top}$ has a right adjoint (called a coreflection) $c:\mathbf{Top}\to H(\mathscr{C})$ . In particular, $c(X)$ is the space with the same underlying set as $X$ but a set $U\subset c(X)$ is open if and only if $f^{-1}(U)$ is open in $Z$ for every map $f:Z\to X$ where $Z\in\mathscr{C}$ . In short, $c(X)$ has the final topology with respect to the collection of all maps from spaces in $\mathscr{C}$ to $X$ .

In the case when $\mathscr{C}=\mathbf{lpcTop}$ is the category of locally path-connected spaces, it follows from Lemmas 1 and 2 that $\mathbf{lpcTop}=H(\mathbf{lpcTop})$ is it’s own coreflective hull. Thus the locally path-connected coreflection $lpc:\mathbf{Top}\to\mathbf{lpcTop}$ discussed in depth in previous posts (Part I and Part II) is precisely the right adjoint. The reason for singling out $\mathbf{lpcTop}$ among other coreflective subcategories of spaces is that (1) local path-connectivity is important in (particularly wild) algebraic topology and (2) the functor $lpc$ has a remarkably simple description: the topology of $lpc(X)$ is generated by the path-components of the open sets of $X$ .

It turns out there is an even simpler way to generate $\mathbf{lpcTop}$ as a coreflective subcategory. I first learned about the following general construction from some fantastic unpublished notes [1] of Jerzy Dydak.

Definition 4: Let $(J,\leq )$ be a directed set. The directed wedge of a collection of spaces $\{(X_j,x_j)|j\in J\}$ indexed by $J$ is the wedge sum $X=\displaystyle\widetilde{\bigvee}_{j\in J}X_j$ (given by identifying the basepoints $x_j$ to a single point $b_0$ ) with the following topology: A set $U\subseteq X$ is open if and only if

$U\cap X_j$ is open in $X_j$ for every $j\in J$ .
if $b_0\in U$ , then there is a $k$ such that $X_j\subset U$ for all $j\geq k$ .

In particular if $(X_j,x_j)=([0,1],0)$ is the unit interval for every $j\in J$ , then $ah(J)=\widetilde{\bigvee}_{j\in J}[0,1]_j$ is the $J$ -arc-hedgehog.

Example 5: If $\omega=\{1,2,...\}$ is the natural numbers, then the $\omega$ -arc hedgehog space is the space $ah(\omega)$ of a sequence of shrinking intervals joined at a point.

Omega arc-hedgehog

In the case that $J=\omega$ and $X_j=S^1$ is the unit circle. The directed wedge $\widetilde{\bigvee}_{\omega}S^1$ is the earring space.

Lemma 6: If $\{X_j|j\in J\}$ is a collection of path-connected and locally path-connected spaces, then $\widetilde{\bigvee}_{j\in J}X_j$ is path-connected and locally path-connected.

In particular, arc-hedgehogs are path-connected and locally-path connected.

Theorem 7: Let $\mathscr{A}$ be the subcategory of all $J$ -arc hedgehogs. Then $H(\mathscr{A})=\mathbf{lpcTop}$ .

Proof. Since every arc-hedgehog is locally path-connected, have $\mathscr{A}\subseteq\mathbf{lpcTop}$ and thus $H(\mathscr{A})\subseteq H(\mathbf{lpcTop})=\mathbf{lpcTop}$ . For the other inclusion, suppose $X$ is locally path-connected. Suppose $U\subseteq X$ . Clearly if $U$ is open, then $f^{-1}(U)$ is open in $ah(J)$ for every map $f:ah(J)\to X$ . For the converse, suppose $U$ is not open. There exists a point $x\in U$ such that for every path-connected neighborhood $V$ of $x$ , there is a point $z_V\in V\backslash U$ . Let $J$ be the directed set of path-connected neighborhoods $V$ of $x$ . For each $V\in J$ , find a path $\alpha_V:[0,1]\to V$ from $x$ to $z_V$ . Define a map $f:ah(J)\to X$ so that the restriction to the $V$ -th arc is the path $\alpha_{V}:[0,1]_{V}\to V$ . It is easy to see that $f$ is continuous based on how we defined the topology of $ah(J)$ . Since $f(b_0)=x$ , we have $b_0\in f^{-1}(U)$ , however, if $1_V$ denotes the end of the $V$ -th arc $[0,1]_V$ , then $f(1_V)=z_V\notin U$ . Thus $1_V\notin f^{-1}(U)$ for all $V\in J$ . It follows that $f^{-1}(U)$ cannot be open since $1_V\to b_0$ is a net in $ah(J)$ converging to the joining point $b_0$ . $\square$

Thus the topology of a locally path-connected space $X$ is entirely determined by maps from arc-hedgehog spaces. Notice that if $X$ is first countable, then we only need to use the $\omega$ -arc hedgehog $ah(\omega)$ .

References:

[1] Dydak, J. Coverings and fundamental groups: a new approach. Preprint. arXiv:1108.3253

8 Responses to The locally path-connected coreflection (Part III)

Tim Campion says:

July 31, 2016 at 5:50 pm

This is beautiful, thanks for writing it!

In a 1963 paper of Gleason (Gleason, A.M., 1963. Universal locally connected refinements. Illinois J. Math. 7, 521–531.), the last section goes further and claims without proof that in fact the locally path-connected spaces are the coreflective hull of the singleton category consisting of the unit interval. I find Gleason’s claim suspicious because this would imply that your arc-hedgehogs are quotients of disjoint unions of intervals. Do you know if this is true?

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- Jeremy Brazas says:
  
  July 31, 2016 at 7:53 pm
  
  Good question Tim. The coreflective hull of the category whose only object is the unit interval is sometimes called the category of delta( $\Delta$ )-generated spaces (because it is also the coreflective hull of n-simplices $\Delta_n$ ). Certainly, every $\Delta$ -generated space is locally path-connected. While it is true that a first countable, locally path-connected space is $\Delta$ -generated, the full converse is not true. So if what I write is correct, then you’re right that an arc-hedgehog had better provide an answer. Indeed, if $\omega_1$ is the first uncountable ordinal in which there are no cofinal countable sets, then $ah(\omega_1)$ is locally path-connected but not $\Delta$ -generated. To see this, first convince yourself that a path in $ah(\omega_1)$ can only intersect finitely many arcs. This means the subset $U=\bigcup_{j\in J}[0,1/2)_j$ is not open in $ah(\omega_1)$ even though $\alpha^{-1}(U)$ is open for every path $\alpha:[0,1]\to ah(\omega_1)$ .
  
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  - Tim Campion says:
    
    July 31, 2016 at 9:19 pm
    
    Ah, thanks! I was indeed interested in this question because it would be remarkable if the $\Delta$-generated spaces coincided with the locally path-connected spaces and yet they had different names! Thinking about it some more, I think another argument would be that since the unit interval has countable tightness — is in fact sequential — and countably tight / sequential spaces are closed under colimits, it follows that every $\Delta$-generated space has countable tightness. But locally path-connected spaces — the arc-hedgehogs for long chains, for instance — can have arbitrarily large tightness.
    
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Jacob Stencil says:

October 26, 2018 at 9:48 pm

In your proof of Lemma 2, you say “Then q\circ\alpha:[0,1]\to q(V)\subseteq U is a path from f(x)\in C to v.” I didn’t see f defined previously in the proof. What does it refer to?

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- Jeremy Brazas says:
  
  August 1, 2019 at 11:37 pm
  
  Sorry I didn’t see this earlier. It was a typo. The “f” should be “q.” It’s fixed now.
  
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