{"id":88,"date":"2006-07-26T13:57:15","date_gmt":"2006-07-26T13:57:15","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/07\/26\/zeros-in-category-theory\/"},"modified":"2006-07-26T13:57:15","modified_gmt":"2006-07-26T13:57:15","slug":"zeros-in-category-theory","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/07\/26\/zeros-in-category-theory\/","title":{"rendered":"Zeros in Category Theory"},"content":{"rendered":"

Things are a bit busy at work on my real job lately, and I don’t have time to put together as detailed a post for today as I’d like. Frankly, looking at it, my cat theory post yesterday was half-baked at best; I should have held off until I could polish it a bit and make it more comprehensible.
\nSo I’m going to avoid that error today. Since we’ve had an interesting discussion focusing on the number zero, I thought it would be fun to show what zero means in category theory.
\nThere are two zeros it category theory: the zero object, and the zero arrow.
\nZero Objects
\n————–
\nThe zero object is easier. Suppose we’ve got a category, C. Suppose that C has a *terminal object* – that is, an object t for which other object x in C has *exactly* *one* arrow f : x → t. And suppose that C also has an *initial object*: that is, an object i for which every object x in C has *exactly one* arrow, f : i → x. If C has both an initial object i, and a terminal object t, *and* i = t, then it’s a *zero object* for the category. A category can actually have *many* zero objects. For example, in the category of groups, any *trivial* group (that is, a group which contains only one element) is a zero object in the category of groups. For an intuition of why this is called “zero” think of the number zero. It’s a strange little number that sits dead in the middle of everything. It’s the central point on the complex plane, it’s the center of the number line. A zero *object* sits in the middle of a category: everything has exactly one arrow *to* it, and one arrow *from* it.
\nZero Arrows
\n————-
\nThe idea of a zero *arrow* comes roughly from the basic concept of a zero *function*. A zero function is a function f where for all x, f(x) = 0. One of the properties of a zero function is that composing a zero function with any other function results in a zero function. (That is, if f is a zero function, f(g(x))) is also a zero function.)
\nA zero morphism in a category C is part of a set of arrows, called the *zero family* of morphisms of C, where composing *any* morphism with a member of the zero family results in a morphism in the zero family.
\nTo be precise: suppose we have a category C, and for any pair b of objects a and b ∈ Obj(C), there is a morphism 0a,b<\/sub> : a → .
\nThe morphism 0d,e<\/sub> must satisfy one important property:
\nFor any two morphisms m : d → e, and n : f → g, the following diagram must commute:
\n\"zero.jpg\"
\nTo see why this means that any composition with a 0 arrow will give you a zero arrow, look at what happens when we start with an ID arrow. Let’s make n : f → g an id arrow, by making f=g. We get the following diagram:
\n \"zero-id.jpg\"
\nSo – any zero arrow composed with an id arrow is a zero arrow. Now use induction to step up from ID arrows by looking at one arrow composed with an ID arrow and a zero arrow. You’ve still got a zero. Keep going – you’ll keep getting zeros.
\nZero arrows are actually very important buggers: the algebraic and topographical notions of a kernel can be defined in category theory using the zero morphisms.
\nOne last little note: in general, it looks like the zero objects and the zero morphisms are unrelated. They’re not. In fact, if you have a category with a zero object *and* a family of zero morphisms, then you can find the zero morphisms by using the zero object. For any objects a and b, 0a,b<\/sub> = (a → 0) º (0 → b).<\/p>\n","protected":false},"excerpt":{"rendered":"

Things are a bit busy at work on my real job lately, and I don’t have time to put together as detailed a post for today as I’d like. Frankly, looking at it, my cat theory post yesterday was half-baked at best; I should have held off until I could polish it a bit and […]<\/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-88","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-1q","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/88","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=88"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/88\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=88"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=88"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=88"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}