{"id":54,"date":"2006-07-04T11:30:21","date_gmt":"2006-07-04T11:30:21","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/07\/04\/categories-and-subthings\/"},"modified":"2006-07-04T11:30:21","modified_gmt":"2006-07-04T11:30:21","slug":"categories-and-subthings","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/07\/04\/categories-and-subthings\/","title":{"rendered":"Categories and SubThings"},"content":{"rendered":"

What’s a subset? That’s easy: if we have two sets A and B, A is a subset of B if every member of A is also a member of B.
\nWhat’s a subgroup? If we have two groups A and B, and the values in group A are a subset of the values in group B, then A is a subgroup of B.
\nFor any kind of thing **X**, what does it mean to be a sub-X? Category theory gives us a way of answering that in a generic way. It’s a bit hard to grasp at first, so let’s start by looking at the basic construction in terms of sets and subsets.
\nThe most generic way of defining subsets is using functions. Suppose we have a set, A. How can we define all of the subsets of A, *in terms of functions*?
\nWe can take the set of all *injective* functions to A (an injective function from X to Y is a function that maps each member of X to a unique member of Y). Let’s call that set of injective functions **Inj**(A). Now, we can define equivalence classes over **Inj**(A), where two functions f : X → A and g : Y → A are equivalent if there is an isomorphism between X and Y.
\nThe domain of each function in one of the equivalence classes in **Inj**(A) is a function isomorphic to a subset of A. So each equivalence class of injective functions defines a subset of A.
\nWe can generalize that function-based definition to categories, so that it can define a sub-object of any kind of object that can be represented in a category.
\nBefore we jump in, let me review one important definition from before; the monomorphism, or monic arrow.
\n>A *monic arrow* is an arrow f : a → b such that (∀ g1<\/sub>,
\n>g2<\/sub>: x → a) f º g1<\/sub> = f º g2<\/sub>
\n>⇒ g1<\/sub> = g2<\/sub>. (That is, if any two arrows composed with
\n>f in f º g end up at the same object only if they are the same.)
\nThe monic arrow is the category theoretic version of an injective function.
\nSuppose we have a category C, and an object a ∈ Obj(C).
\nIf there are are two monic arrows f : x → a and g : y → a, and
\nthere is an arrow h such that g º h = f, then we say f ≤ g (read “f factors through g”). Now, we can take that “≤” relation, and use it to define an equivalence class of morphisms using f ≡ g ⇔ f ≤ g &land; g ≤ f.
\nWhat we wind up with using that equivalence relation is a set of equivalence classes of monomorphisms pointing at A. Each of those equivalence classes of morphisms defines a subobject of A. (Within the equivalence classes are objects which have isomorphisms, so the sources of those arrows are equivalent with respect to this relation.) A subobject of A is the sources of an arrow in one of those equivalence classes.<\/p>\n","protected":false},"excerpt":{"rendered":"

What’s a subset? That’s easy: if we have two sets A and B, A is a subset of B if every member of A is also a member of B. What’s a subgroup? If we have two groups A and B, and the values in group A are a subset of the values in group […]<\/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-54","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-S","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/54","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=54"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/54\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=54"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=54"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=54"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}