One of the things I found off-putting about category theory when I was first exposed to it was its reliance on the notion of “collections” that are not sets. That seemed to place the entire theory on dubious foundations with paradoxes looming around every corner.<\/p>\n
It turns out you can mostly ignore such issues in application. You can, for example, talk about the forgetful functor that maps a group to the set of its elements, ignoring the group structure, without having to think deeply about the collection<\/em> of all sets, which Russell’s paradox tells us cannot itself be a set.<\/p>\n And yet issues of cardinality are not entirely avoidable. There is a theorem [1] that says in effect that category theory would be uninteresting without collections too large to be sets.<\/p>\n Every category C<\/em> with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions.<\/p><\/blockquote>\n If the collection of arrows (morphisms) between objects in C<\/em> is so small as to be a set, then C<\/em> is a sub-category of the category of sets. As Awodey explains, “the only special properties such categories can possess are ones that are categorically irrelevant, such as features of the objects that do not a\ufb00ect the arrows in any way.”<\/p>\n Most categories of interest have too many objects to be a set, and even more morphisms than objects.<\/p>\n [1] Category Theory by Steve Awodey. Theorem 1.6.<\/p>\n","protected":false},"excerpt":{"rendered":" One of the things I found off-putting about category theory when I was first exposed to it was its reliance on the notion of “collections” that are not sets. That seemed to place the entire theory on dubious foundations with paradoxes looming around every corner. It turns out you can mostly ignore such issues in […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[144],"class_list":["post-246771","post","type-post","status-publish","format-standard","hentry","category-math","tag-category-theory"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Interesting categories are big","description":"You can most ignore matters of cardinality when using category theory, and yet there is a theorem that says category theory would be trivial without such issues.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2025\/12\/28\/interesting-categories-are-big\/","robots":"max-image-preview:large","keywords":"category theory","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Interesting categories are big","og:description":"You can most ignore matters of cardinality when using category theory, and yet there is a theorem that says category theory would be trivial without such issues.","og:url":"https:\/\/www.johndcook.com\/blog\/2025\/12\/28\/interesting-categories-are-big\/","article:published_time":"2025-12-29T01:23:19+00:00","article:modified_time":"2026-01-01T15:25:36+00:00","twitter:card":"summary","twitter:title":"Interesting categories are big","twitter:description":"You can most ignore matters of cardinality when using category theory, and yet there is a theorem that says category theory would be trivial without such issues.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"246771","title":null,"description":"You can most ignore matters of cardinality when using category theory, and yet there is a theorem that says category theory would be trivial without such issues.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2025-12-29 12:47:03","updated":"2026-01-01 15:30:06","ai":null,"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"Related posts<\/h2>\n
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