{"id":32,"date":"2006-06-19T10:49:19","date_gmt":"2006-06-19T10:49:19","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/06\/19\/category-theory-natural-transformations-and-structure\/"},"modified":"2006-06-19T10:49:19","modified_gmt":"2006-06-19T10:49:19","slug":"category-theory-natural-transformations-and-structure","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/06\/19\/category-theory-natural-transformations-and-structure\/","title":{"rendered":"Category Theory: Natural Transformations and Structure"},"content":{"rendered":"

The thing that I think is most interesting about category theory is that what it’s really fundamentally about is structure<\/em>. The abstractions of category theory let you talk about structures in an elegant way; and category diagrams let you illustrate structures in a simple visual way. Morphisms express the structure<\/em> of a category; functors are higher level morphisms that express the structure of relationships<\/em> between categories.<\/p>\n

In my last category theory post<\/a>, one of the things I mentioned was how category theory lets you explain the idea of symmetry and group actions – which are a kind of structural immunity to transformation, and a definition of transformation – in a much simpler way than it could be talked about without categories.<\/p>\n

It turns out that symmetry transformations are just the tip of the iceberg of the kinds of structural things we can talk about using categories. In fact, as I alluded to in my last post, if we create a category of categories, we end up with functors as arrows between categories.<\/p>\n

What happens if we take the same kind of thing that we did to get group actions, and we pull out a level, so that instead of looking at the category of categories, focusing on arrows from the specific category of a group to the category of sets, we do it with arrows between members of the category of functors?<\/p>\n

We get the general concept of a natural transformation<\/em>. A natural transformation is a morphism from functor to functor, which preserves the full structure of morphism composition within the categories mapped by the functors.<\/p>\n

Suppose we have two categories, C and D. And suppose we also have two functors, F, G : C → D. A natural transformation from F to G, which we’ll call η maps every object x in C to an arrow ηx<\/sub> : F(x) → G(x). ηx<\/sub> has the property that for every arrow a : x → y in C, ηy<\/sub> º F(a) = G(a) º ηx<\/sub>. If this is true, we call ηx<\/sub> the component<\/em> of η for (or at) x.<\/p>\n

That paragraph is a bit of a whopper to interpret. Fortunately, we can draw a diagram to help illustrate what that means. The following diagram commutes if η has the property described in that paragraph.<\/p>\n

\"natural-transform.jpg\"<\/p>\n

I think this is one of the places where the diagrams really help. We’re talking about a relatively straightforward property here, but it’s very confusing to write about in equational form. But given the commutative diagram, you can see that it’s not so hard: the path ηy<\/sub> º F(a) and the path G(a) º η<sub<\/sub> compose to the same thing: that is, the transformation η hasn’t changed the structure expressed by the morphisms<\/em>.<\/p>\n

And that’s precisely the point of the natural transformation: it’s a way of showing the relationships between different descriptions of structures – just the next step up the ladder. The basic morphisms of a category express the structure of the category; functors express the structure of relationships between categories; and natural transformations express the structure of relationships between relationships<\/em>.<\/p>\n

Coming soon: a few examples of natural transformation, and then… Yoneda’s lemma. Yoneda’s lemma takes the idea we mentioned before of a group being representable by a category with one object, and generalizes all the way up from the level of a single category to the level of natural transformations.<\/p>\n","protected":false},"excerpt":{"rendered":"

The thing that I think is most interesting about category theory is that what it’s really fundamentally about is structure. The abstractions of category theory let you talk about structures in an elegant way; and category diagrams let you illustrate structures in a simple visual way. Morphisms express the structure of a category; functors are […]<\/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":[76,24],"tags":[],"class_list":["post-32","post","type-post","status-publish","format-standard","hentry","category-category-theory","category-goodmath"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-w","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/32","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=32"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/32\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=32"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=32"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=32"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}