For me, the frustrating thing about learning category theory was that
\nit seemed to be full of definitions, but that I couldn’t see why I should care.
\nWhat were these category things, and what could I really talk about using this
\nstrange new mathematical language of categories?<\/p>\n
To avoid that in my presentation, I’m going to show you a couple of examples up front of things we can talk about using the language of category theory: sets, partially ordered sets, and groups.<\/p>\n
We can talk about sets using category theory. The objects in the category of sets are, obviously, sets. Arrows in the category of sets are total functions between sets.<\/p>\n
Let’s see how these satisfy our definition of categories:<\/p>\n
Partially ordered sets (that is, sets that have a “<=" operator) can be described as a category, usually called PoSet<\/b>. The objects are the partially ordered sets; the arrows are monotonic functions (a function f is monotonic if (∀ x,y &isin domain(x)) x <= y ⇒ f(x) <= f(y).). Like regular sets, º is function composition.<\/p>\n
It’s pretty easy to show the associativity and identity properties; it’s basically the same as for sets, except that we need to show that º preserves the monotonicity property. And that’s not very hard:<\/p>\n
There is a category Grp<\/b> where the objects are groups; group homomorphisms are arrows. Homomorphisms are structure-preserving functions between sets; so function composition of those structure-preserving functions is the composition operator º. The proof that function composition preserves structure is pretty much the same as the proof we just ran through for partially ordered sets.<\/p>\n
Once you have groups as a category, then you can do something very cool. If groups are a category, then functors over groups are symmetric transformations. Walk it through, and you’ll see that it fits. What took me a week of writing to be able to explain when I was talking about group theory can be stated in one sentence using the language of category theory. That’s a perfect example of why cat theory is useful: it lets you say some very important, very complicated things in very simple ways.<\/p>\n
There’ve been a couple of questions about from category theory skeptics in the comments. Please don’t think I’m ignoring you. This stuff is confusing enough for most people (me included) that I want to take it slowly, just a little bit at a time, to give readers an opportunity to digest each bit before going on to the next. I promise that I’ll answer your questions eventually!<\/p>\n","protected":false},"excerpt":{"rendered":"
For me, the frustrating thing about learning category theory was that it seemed to be full of definitions, but that I couldn’t see why I should care. What were these category things, and what could I really talk about using this strange new mathematical language of categories? To avoid that in my presentation, I’m going […]<\/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-14","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-e","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/14","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=14"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/14\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=14"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=14"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=14"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}