{"id":11,"date":"2006-06-08T11:25:00","date_gmt":"2006-06-08T11:25:00","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/06\/08\/diagrams-in-category-theory\/"},"modified":"2006-06-08T11:25:00","modified_gmt":"2006-06-08T11:25:00","slug":"diagrams-in-category-theory","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/06\/08\/diagrams-in-category-theory\/","title":{"rendered":"Diagrams in Category Theory"},"content":{"rendered":"

One of the things that I find niftiest about category theory is category diagrams. A lot of things that normally turn into complex equations or long-winded logical statements can be expressed in diagrams by capturing the things that you’re talking about in a category, and then using category diagrams to express the idea that you want to get accross.<\/p>\n

A category diagram is a directed graph, where the nodes are objects from a category, and the edges are morphisms. Category theorists say that a graph commutes<\/em> if, for any two paths through arrows in the diagram from node A to node B, the composition of all edges from the first path is equal to the composition of all edges from the second path.<\/p>\n

As usual, an example will make that clearer.
\n\"cat-assoc.jpg\"<\/p>\n

This diagram is a way of expression the associativy property of morphisms: f º (g º h) = (f º g) º h. The way that the diagram illustrates this is: (g º h) is the morphism from A to C. When we compose that with f, we wind up at D. Alternatively, (f º g) is the arrow from B to D; if we compose that with H, we wind up at D. The two paths: f º (A → C), and (B → D) &ordm h are both paths from A to D, therefore if the diagram commutes, they must be equal.<\/p>\n

Let’s look at one more diagram, which we’ll use to define an interesting concept, the principal morphism<\/em> between two objects. The principle morphism is a single arrow from A to B, and any composition of morphisms that goes from A to B will end up being equivalent to it.<\/p>\n

In diagram form, a morphism m is principle if (∀ x : A → A) (∀ y : A → B), the following diagram commutes.
\n\"cat-principal.jpg\"<\/p>\n

In words, this says that f is a principal morphism if for every endomorphic arrow x, and for every arrow y from A to B, f is is the result of composing x and y. There’s also something interesting about this diagram that you should notice: A appears twice in the diagram! It’s the same object; we just draw it in two places to make the commutation pattern easier to see. A single object can appear in a diagram as many times as you want to to make the pattern of commutation easy to see. When you’re looking at a diagram, you need to be a bit careful to read the labels to make sure you know what it means. (This paragraph was corrected after a commenter pointed out a really silly error; I originally said “any identity arrow”, not “any endomorphic arrow”.)<\/em><\/p>\n

One more definition by diagram: x and y are a retraction pair<\/em>, and A is a retract<\/em> of B (written A < B) if the following diagram commutes:
\n\"cat-retract.jpg\"<\/p>\n

That is, x : A → B, and y : B → A are a retraction pair if y º x = 1A<\/sub>.<\/p>\n","protected":false},"excerpt":{"rendered":"

One of the things that I find niftiest about category theory is category diagrams. A lot of things that normally turn into complex equations or long-winded logical statements can be expressed in diagrams by capturing the things that you’re talking about in a category, and then using category diagrams to express the idea that you […]<\/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-11","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-b","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/11","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=11"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/11\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=11"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=11"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=11"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}