The Fundamental Groupoid and Group (Pt. III)
Point of Post: This is a continuation of this post.
What we claim is that, in fact, is a functor (where, of course, is the category whose objects are groupoids and whose arrows are functors). Of course, while we know how acts on objects of we have yet to mention how it acts on arrows. Given two topological spaces and and a continuous map we define to be the functor which takes an object to the object , and which takes an arrow to the arrow . This is well-defined because homotopy respects compositions. Clearly preserves identities, for
for all an arrow in , and , so that is the identity functor . Moreover, it’s clear that preserves composition since if and are continuous maps, we see that for any object in we have that
Thus, we may put, in theorem form, this all as:
Theorem: is a functor .
Pointed Topological Spaces and The Fundamental Group
The issue with the fundamental groupoid is simple–it’s just too much information. The method of finding one-dimensional holes described in the motivation singled out a single point and attempted to find loops anchored at that point which are incontractible (i.e. not null-homotopic). What we have described though is an object which captures (up to homotopy) every single possible path in the space–this is just too much information. So, to get back to our original intent we focus in on one point.
To this end, if is a space and let denote –the arrows in . By definition is the homotopy classes of loops anchored at . Now, while the above may seem like general nonsense, it makes the fact that has extra structure incredibly nice. Indeed:
Theorem: Let be a groupoid and an object of . Then, is a group under the composition of .
Proof: It is simple category theory that a monoid is just a category with a single object, and in particular that is a monoid. The fact that every arrow in is an isomorphism then implies that is, in fact, a group.
Corollary: Let be a topological space and , then is a group.
Proof: By definition is for the groupoid .
We call the group either the first homotopy group of with base point or the fundamental group of with base point .
Now, the issue with passing from the fundamental groupoid to the fundamental groupoid is that now we must feed in more specific information–not only a space, but a space and a specific point. To this end, we define a pointed space to be an ordered pair with a topological space and a distinguished point called the base point. A pointed map between pointed spaces is a continuous map such that –so a continuous map which respects base points. It is easy to see that the collection of pointed spaces and pointed map forms a category, denoted , the category of pointed topological spaces.
Remark: For those who are interested in this kind of thing, is the comma category where is the discrete category with one object.
What we claim is that, like the fundamental groupoid, the fundamental group is a functor, this time to . But, as should be expected, it is a functor not from but from . Indeed, define on objects as . On arrows, define –we shall often denote as and call it the induced map on fundamental groups. Note that the pointedness really only played an assignment role in this functor (domain and codomain assignment) and that for all intents and purposes this is just the functor . Thus, the exact same argument that shows that is a functor shows that
Theorem: is a functor .
This shows us that is, in some sense, a topological invariant. Really (as defined here) is an invariant of pointed spaces, that said if is a homeomorphism, then better be isomorphic to for all . Indeed:
Theorem: Let and be topological spaces. If is a homeomorphism, then for all .
Proof: Since is a homeomorphism it’s obvious that is an isomorphism of pointed spaces. The result then follows from the functorality of .
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