\na categorical notion of identity, defined using a special object called
\nUnit<\/em> (written “I” or “1”) and two morphisms λ (left identity) and ρ (right identity); and a categorical notion of associativity defined using a morphism α. <\/p>\n
To construct a categorical monoid object, all that we do is take the
\nbasic definitions of an algebraic monoid, and use them to draw a category
\ndiagram.<\/p>\n
So, if we have a monoidal category C, then a monoid in the category is
\nan object M, with two morphisms μ and η. The arrow μ is the
\nrepresentation of multiplication: it’s an arrow μ:M⊗M→M.
\nη is a morphism called unit<\/em> which represents identity; it is a
\nmorphism from the monoidal category’s identity object to M; η:I→M.
\nThe triple (M,μ,η) is a monoid if and only if the following two
\ndiagrams commute:<\/p>\n
![\"monoid-diagram.png\"](\"https:\/\/i0.wp.com\/scientopia.org\/img-archive\/goodmath\/img_299.png?resize=346%2C252\")
<\/p>\n
With this construction, a monoid object in the category of sets is Back when I started to down this categorical path, I said that it helped to understand some of the more complex ideas in abstract algebra. Here’s one example. Take the category of abelian groups. A monoid within the By now, we’ve seen the simple algebraic monoid, which is essentially an abstract construction of a category. We’ve also seen the more complicated, but interesting monoidal category – which is, sort of, a meta-category – a category built using categories. The monoidal category is a fairly complicated object – but it’s useful. What does a […]<\/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":[69,76],"tags":[],"class_list":["post-604","post","type-post","status-publish","format-standard","hentry","category-abstract-algebra","category-category-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-9K","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/604","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=604"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/604\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=604"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=604"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=604"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nexactly<\/em> an algebraic monoid. So we’ve come full circle – we’ve gone
\nfrom the algebraic monoid<\/a>, to the monoidal category<\/a>, to the categorical monoid, back to the algebraic monoid.<\/p>\n
\ncategory of abelian groups is a ring. The rings second operation, and its properties are precisely the set of properties that are implied by the commuting diagram above. And as you’ll see in later posts, we can use that to understand more about rings in terms of categorical definitions and diagrams.<\/p>\n","protected":false},"excerpt":{"rendered":"