In my last topology post, I started talking about the fundamental group of a topological space. What makes the fundamental group interesting is that it tells you interesting things about the structure
\nof the space in terms of paths that circle around and end where they started. For example, if you’re looking at a basic torus, you can go in loops staying in a euclidean-looking region; you can loop around the donut hole, or you can loop around the donut-body.
\nOf course, in the comments, an astute reader (John Armstrong) leapt ahead of me, and mentioned the fundamental group*oid* of a topological space, and its connection with category theory. That’s
\nsupposed to be the topic of this post.<\/p>\n
In my last topology post, I started talking about the fundamental group of a topological space. What makes the fundamental group interesting is that it tells you interesting things about the structure of the space in terms of paths that circle around and end where they started. For example, if you’re looking at a basic […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[65],"tags":[],"class_list":["post-240","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-3S","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/240","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=240"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/240\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=240"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=240"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=240"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nTo start, what’s a groupoid? Basically, take the basic concept of a group, and expand it to cases
\nwhere the group operation *isn’t* total. To be formal, a groupoid consists of:
\n1. A set of values, *G*.
\n2. A *partial* operation × : *G* × *G* → *G*, such that:
\n* for all x, y, z ∈ *G*, if (x×y)×z is defined, and x×(y×z) is defined, then they’re equal.
\n3. A *total* inverse operation ()-1<\/sup> : *G* → *G*, such that:
\n* for all *x*, *y* ∈ *G*, if *x*×*y* is defined, then
\n*x*-1<\/sup>×*x*×*y* = *y*.
\n* for all *x*, *y* ∈ *G*, if *x*×*y* is defined, then *x*×*y*×*y*-1<\/sup> = *x*.
\n* for all x ∈ *G*, x×x-1<\/sup> and x-1<\/sup>×x are defined.
\nThe fundamental groupoid of a topological space is similar in concept to the fundamental group, but instead of being based on the set of *loops* from a point to itself, it’s based on the set of *all paths*. The groupoid operation is the same as the group operation: path concatenation. So given two paths, *x* and *y*, the *x×y* is defined if the end-point of *x* is the start-point of *y*. And for a path *x* from a point *a* to a point *b*, *x-1<\/sup>* is the reverse path from *b* to *a*. So the fundamental groupoid is formed
\nfrom the equivalence classes of paths in the topological space formed by homotopies.
\nWhere is gets kind of neat is when you look at a groupoid in category theory. In category theoretic terms, a groupoid is just a category where every morphism is *iso*. (As a reminder, *that* means that every morphism is reversable.) Composition of the morphism is the group operation. Much simpler definition, eh?
\nNow, let’s play some tricks. Suppose we have a topological space, **T**, and its fundamental groupoid, *G*. We’ll think about *G* in its categorical form, because that’s easier to talk about. So the category form of the groupoid has the same objects as **T**, with morphisms between objects *x* and *y*if there is a *continuous* path from *x* to *y*.
\nWe can say that the groupoid *G* is connected if for all *x,y ∈ G*, there is a morphism *m : x → y*. If the groupoid is connected, then the topological space is *path*-connected. (And that’s a much easier definition of path-connected to get than the non-categorical one, which is based on another of those “isomorphism from [0,1]” beasts!)
\nSo, suppose we have a connected groupoid, *O*. If we keep looking at it in terms of categories, we see something interesting about groupoids pretty easily. We can describe
\na connected groupoid using a group. The way we do that is: take the set of
\nobjects of *O*, and for all pairs of points x and y ∈ *O*, the set of *morphisms* from x to y is an object of the group. The group operation is just composition of the morphisms. So for any connected groupoid, there’s a really simple representation of the groupoid as a category-based group.
\nWhat if the groupoid *O* is *not* connected? Then it can still be represented in terms of groups, but you need more than one. You can describe the groupoid by a set of pairs (Gi<\/sub>,Oi<\/sub>), where Oi<\/sub> is a subset of the objects of *O*, and
\nGi<\/sub> is the category-based group describing Oi<\/sub>. The strange thing
\nabout this is that there *isn’t* a canonical group-based description of a non-connected groupoid for a topological space. There can be many different ways of describing a groupoid in
\nterms of groups over subsets of its elements, and there is *no* way to choose one
\nof those as being canonical – they’re different, but equivalent.
\nIf you think about it, that’s pretty strange. Most of the time in math, we can find ways of
\ncreating canonical definitions: for some mathematical construct *X*, we can generally find
\nsome way of saying “This is what X is, and any other way of defining X can be reduced to another way of restating this.” And yet, for something deeply fundamental about the shapes of spaces, *we can’t do that*. That’s strange, and fascinating.<\/p>\n","protected":false},"excerpt":{"rendered":"