{"id":257,"date":"2019-09-13T02:31:55","date_gmt":"2019-09-13T02:31:55","guid":{"rendered":"https:\/\/borovik.net\/selecta\/?p=257"},"modified":"2026-09-13T02:34:43","modified_gmt":"2026-09-13T02:34:43","slug":"how-does-category-theory-relate-to-other-branches-of-mathematics","status":"publish","type":"post","link":"https:\/\/borovik.net\/selecta\/2019\/09\/13\/how-does-category-theory-relate-to-other-branches-of-mathematics\/","title":{"rendered":"How does category theory relate to other branches of mathematics?"},"content":{"rendered":"

My answer to a question on Quora<\/a>:<\/p>\n

An excellent question. Category theory is important on its own, and has important applications in a number of other mathematical theories; however, the crucial and the most fundamental impact of category theory is invisible and under-reported, it is of cultural nature. It can be compared with the influence of set theory: 99% of mathematicians use only a modicum of naive set theory, ignoring deeply penetrating and frequently very hard results of set theory as the live research discipline which continues to develop and flourish.<\/p>\n

There is a telling example: the theory of games of chance was created in the 17th century and gave birth to probability theory; the latter was already quite developed by the time when, in the 20th century, the concept of a deterministic game had finally crystallized – in a paper, of all people, by Zermelo, who proved that chess was a deterministic game: for one of the players, there is a strategy, that is, a function from the set of permissible position to the set of moves, which achieves at least a draw. His paper was published in 1913 (see Zermelo’s theorem (game theory) – Wikipedia<\/a>). Why did this happen so late? The word \u201cset\u201d in the definition of a strategy came to use only in the second half of the 19th century.<\/p>\n

The same is happening with category theory: the vast majority of mathematicians use its ideas and terminology in a very rudimentary and naive form, frequently even without realisation that they are doing so. In the work that I am doing, it had happened to be very important to remember that an algebraic group was a functor from the category of unital commutative rings to the category of groups. Some my colleagues who work in the same theory continue to insist that a group is a fixed set with some operations on it. When my co-author and I recently solved a certain problem which was open since 1999, we were able to do that only because for us a group in question was a functor – not much deeper than that. Why it was not solved by someone else earlier? Because for them a group was just a set.<\/p>\n","protected":false},"excerpt":{"rendered":"

My answer to a question on Quora: An excellent question. Category theory is important on its own, and has important applications in a number of other mathematical theories; however, the crucial and the most fundamental impact of category theory is invisible and under-reported, it is of cultural nature. It can be compared with the influence […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-257","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/257","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/comments?post=257"}],"version-history":[{"count":1,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/257\/revisions"}],"predecessor-version":[{"id":258,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/257\/revisions\/258"}],"wp:attachment":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/media?parent=257"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/categories?post=257"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/tags?post=257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}