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 of a family of locally path-connected spaces is locally path-connected.
Lemma 2: The quotient of a locally path-connected space is locally path-connected.
Proof. Let be a quotient map where is locally path-connected. Let be an open set in and be a non-empty path-component of . It suffices to check that is open in . Since is quotient, we need to check that is open in . If , then there is an open, path connected neighborhood such that . We claim that . Let and be a path from to . Then is a path from to . Since is the path component of in , the path must have image entirely in . Thus has image in . In particular, .
denotes the usual category of topological spaces and continuous functions. Here is an important definition from Categorical Topology.
Definition 3: Let be a subcategory of . The coreflective hull of is the full subcategory of consisting of all spaces which are the quotient of a topological sum (i.e. disjoint union) of spaces in .
Certainly . Moreover, the coreflective hull is the category “generated” by the collection of spaces in the following sense: the inclusion has a right adjoint (called a coreflection) . In particular, is the space with the same underlying set as but a set is open if and only if is open in for every map where . In short, has the final topology with respect to the collection of all maps from spaces in to .
In the case when is the category of locally path-connected spaces, it follows from Lemmas 1 and 2 that is it’s own coreflective hull. Thus the locally path-connected coreflection discussed in depth in previous posts (Part I and Part II) is precisely the right adjoint. The reason for singling out among other coreflective subcategories of spaces is that (1) local path-connectivity is important in (particularly wild) algebraic topology and (2) the functor has a remarkably simple description: the topology of is generated by the path-components of the open sets of .
It turns out there is an even simpler way to generate as a coreflective subcategory. I first learned about the following general construction from some fantastic unpublished notes  of Jerzy Dydak.
Definition 4: Let be a directed set. The directed wedge of a collection of spaces indexed by is the wedge sum (given by identifying the basepoints to a single point ) with the following topology: A set is open if and only if
- is open in for every .
- if , then there is a such that for all .
In particular if is the unit interval for every , then is the -arc-hedgehog.
Example 5: If is the natural numbers, then the -arc hedgehog space is the space of a sequence of shrinking intervals joined at a point.
In the case that and is the unit circle. The directed wedge is the earring space.
Lemma 6: If is a collection of path-connected and locally path-connected spaces, then is path-connected and locally path-connected.
In particular, arc-hedgehogs are path-connected and locally-path connected.
Theorem 7: Let be the subcategory of all -arc hedgehogs. Then .
Proof. Since every arc-hedgehog is locally path-connected, have and thus . For the other inclusion, suppose is locally path-connected. Suppose . Clearly if is open, then is open in for every map . For the converse, suppose is not open. There exists a point such that for every path-connected neighborhood of , there is a point . Let be the directed set of path-connected neighborhoods of . For each , find a path from to . Define a map so that the restriction to the -th arc is the path . It is easy to see that is continuous based on how we defined the topology of . Since , we have , however, if denotes the end of the -th arc , then . Thus for all . It follows that cannot be open since is a net in converging to the joining point .
Thus the topology of a locally path-connected space is entirely determined by maps from arc-hedgehog spaces. Notice that if is first countable, then we only need to use the -arc hedgehog .
 Dydak, J. Coverings and fundamental groups: a new approach. Preprint. arXiv:1108.3253
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?
Good question Tim. The coreflective hull of the category whose only object is the unit interval is sometimes called the category of delta()-generated spaces (because it is also the coreflective hull of n-simplices ). Certainly, every -generated space is locally path-connected. While it is true that a first countable, locally path-connected space is -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 is the first uncountable ordinal in which there are no cofinal countable sets, then is locally path-connected but not -generated. To see this, first convince yourself that a path in can only intersect finitely many arcs. This means the subset is not open in even though is open for every path .
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.
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?
Sorry I didn’t see this earlier. It was a typo. The “f” should be “q.” It’s fixed now.
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