{"id":592,"date":"2008-02-05T12:17:11","date_gmt":"2008-02-05T12:17:11","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2008\/02\/05\/more-groupoids-and-groups\/"},"modified":"2008-02-05T12:17:11","modified_gmt":"2008-02-05T12:17:11","slug":"more-groupoids-and-groups","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2008\/02\/05\/more-groupoids-and-groups\/","title":{"rendered":"More Groupoids and Groups"},"content":{"rendered":"

In my introduction to groupoids, I mentioned that if you have a groupoid, you can find
\ngroups within it. Given a groupoid in categorical form, if you take any object in the
\ngroupoid, and collect up the paths through morphisms from that object back to itself, then
\nthat collection will form a group. Today, I’m going to explore a bit more of the relationship
\nbetween groupoids and groups.<\/p>\n

Before I get into it, I’d like to do two things. First, a mea culpa: this stuff is out on the edge of what I really understand. My category-theory-foo isn’t great, and I’m definitely
\non thin ice here. I think that I’ve worked things out enough to get this right, but I’m
\nnot sure. So category-savvy commenters, please let me know if you see any major problems, and I’ll do my best to fix them quickly; other folks, be warned that I might have blown some of the details.<\/p>\n

Second, I’d like to point you at Wikipedia’s page on groupoids<\/a> as a
\nreference. That article is quite good. I often look at the articles in Wikipedia and
\nMathWorld when I’m writing posts, and while wikipedia’s articles are rarely bad, they’re also
\noften not particularly good<\/em>. That is, they cover the material, but often in a
\nsomewhat disorganized, hard-to-follow fashion. In the case of groupoids, I think Wikipedia’s
\narticle is the best general explanation of groupoids that I’ve seen – better than most
\ntextbooks, and better than any other web-source that I’ve found. So if you’re interested in
\nfinding out more than I’m going to write about here, that’s a good starting point.<\/p>\n

<\/p>\n

As I’ve explained before, the basic idea of a group is a mathematical structure with an
\noperation that defines a kind of immunity to transformation. Every transformation through the
\ngroup operator preserves some kind of structure. In a groupoid, you don’t have such a strong
\nsymmetry. In a groupoid, you have an operation that includes<\/em> a symmetric
\ntransformation, but not all applications of it are symmetric: the groupoid operator preserves
\nstructure under some conditions<\/em>: that is, there are sequences<\/em> of groupoid
\noperations that result in symmetric, structure-preserving transformation, and it’s easy to
\ndefine just what those sequences of transformations are.<\/p>\n

In categorical terms, that idea of the partially<\/em> symmetric operation of a
\ngroupoid is reflected by the fact that there are many arrows that go from an object (a
\nstructure) to a different<\/em> object – transforming it into something different –
\nsomething distinguishable from the original structure. But there are sequences of operations
\nthat end back at the original object, and those sequences of operations describe symmetric
\ntransformations.<\/p>\n

This leads to the intuition behind the simplest relation between groupoids and groups. If
\nyou have a groupoid with a single object in it, that that groupoid is a group. In a
\nsingle-object groupoid, every application of the operation must<\/em> end back at the
\nobject where it began – so every transformation in a single-object groupoid must be symmetric
\n– so the single-object groupoid is a group.<\/p>\n

That much already shows us some of why the categorical formulation is nice: we’ve gone from something where the relationships and structures are symbolic, to something where we can see what’s going on using a simple diagram: it’s easier to see “all paths from G end at G”
\nwhen you draw the structures, than it is to see the corresponding statement in terms of pure algebra.<\/p>\n

But there’s more to it. Some of the more subtle things are captured very nicely. For
\nexample, there’s a notion of equivalence<\/em> in groupoids that is different from isomorphism, which tells us more about the relationship between groupoids and groups. It comes from the categorical idea of a natural transformation<\/em>. I wrote about
\nnatural transformations here<\/a>. For a quick reminder, if you start with
\na category, you can define structure-preserving transformations between objects using
\nmorphisms; you can define structure-preserving transformations between morphisms using functors; and you can define structure-preserving transformations between functors using
\nnatural transformations.<\/p>\n

Let’s look at that notion of loose equivalence. Suppose we have a groupoid, O, which is
\nconnected – that is, where in the category for G, given any two objects a and b in O, there
\nis an arrow from a to b. We can form a group G out of the groupoid, by collapsing<\/em>
\nthe connected groupoid to a single object. Loosely speaking, you can pick an arbitrary object from the groupoid as a representative, and collapse the arrows – each set of arrows
\nfrom the representative to a particular other object becomes an arrow in the collapsed groupoid; those arrows become the objects of the group, and arrow composition is the group operator. The end result is a group formed from the collapsed arrows of any connected groupoid. <\/p>\n

What if the groupoid isn’t connected? You take the connected components of the groupoid, and each of them can be collapsed into a group – so you can collapse any groupoid into a collection of disjoint groups. <\/p>\n

Here’s where it gets interesting. The collapse of a groupoid isn’t unique. It’s determined by the selection of the representative objects. The collapsed groupoids are all
\nequivalent. In fact, you can define an isomorphism between them – they’re all ultimately the
\nsame structure; so you can define the isomorphic mappings between objects in the different
\ncollapsed versions.<\/p>\n

But suppose you don’t define the mappings between objects. Suppose all you look at are
\nthe arrows that form the groups. So you’ve discarded the objects from the original groupoid,
\nand all you have left are single objects corresponding to the different connected components, each of which is a group. What you’ve done, then, is to create a mapping from a groupoid to a collection of groups. It is not<\/em> an isomorphism – but it is a mapping from the groupoid to a collection of groups that is, in some sense, equivalent to the original groupoid. (Note that I said “collection”, not<\/em> set. In fact, it’s a multiset; multiple components can collapse down to the same group, but you need to keep an instance of that group for each of the components.) The structure of that collection of groups is determined by the original groupoid. <\/p>\n

But it’s not an isomorphism. That transformation has discarded some information about the original structure. There are multiple groupoids that can all be collapsed into the same
\ncollection of groups. Because you didn’t keep the mappings that allow you to identify the arrows to the original groupoid objects, you can’t go back. It’s a one-way transformation.<\/p>\n

Category theory allows you to define that loss of information in a very convenient way. Any high-level transformation like that – where you’re basically transforming a category into a different category (in this case, from a groupoid category to a category containing a collection of groups) can be described by categorical transformations. The structure-preserving relations between arrows in the original category – the functors<\/em> of the original category – in a categorical sense define the structural information contained in the category. When you map from category to category, there
\nis a mapping between the functors that define the informational relationships between
\nthe two categories. If you can map from one category C to another category D without losing any structural information, that is reflected in category theory by the existence of
\na natural transformation<\/em> from C to D. In the case of the groupoid-to-groups collapse, there is no natural transformation. The non-existence of a natural transformation
\nmeans that there is going to be a significant loss of structural information.<\/p>\n

What does that mean?<\/p>\n

It means that groupoids are richer<\/em> structures that groups. There is interesting
\ninformation about symmetry which we can capture in terms of the algebraic structure of
\na groupoid which we cannot<\/em> capture in terms of the groups that make up the components of a groupoid. Without understanding the groupoids, we are missing something about
\nthe meaning of symmetry!<\/p>\n","protected":false},"excerpt":{"rendered":"

In my introduction to groupoids, I mentioned that if you have a groupoid, you can find groups within it. Given a groupoid in categorical form, if you take any object in the groupoid, and collect up the paths through morphisms from that object back to itself, then that collection will form a group. Today, I’m […]<\/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":[76,26],"tags":[],"class_list":["post-592","post","type-post","status-publish","format-standard","hentry","category-category-theory","category-group-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-9y","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/592","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=592"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/592\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=592"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=592"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=592"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}