{"id":1585,"date":"2022-11-30T13:05:50","date_gmt":"2022-11-30T19:05:50","guid":{"rendered":"https:\/\/sites.imsa.edu\/hadron\/?p=1585"},"modified":"2023-11-09T21:50:50","modified_gmt":"2023-11-10T03:50:50","slug":"category-theory","status":"publish","type":"post","link":"https:\/\/sites.imsa.edu\/hadron\/2022\/11\/30\/category-theory\/","title":{"rendered":"Category Theory"},"content":{"rendered":"<p style=\"text-align: center\"><span style=\"font-weight: 400\">By: Andrew D. Katson<\/span><\/p>\n<p><span style=\"font-weight: 400\">Definition<\/span><\/p>\n<p><span style=\"font-weight: 400\">Category theory, as invented in 1945, is a field of mathematics that works with objects and their relationships. A category is a set made up of multiple elements, called \u201cobjects\u201d, and their \u201cmappings\u201d (5, 11). In figure 1, we see that X, Y, and Z are elements, and f, g, and their composition, g(f), are mappings. There are properties that every category must exhibit. These include (5, 11):<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">An identity mapping, <\/span><i><span style=\"font-weight: 400\">e<\/span><\/i><span style=\"font-weight: 400\">, which maps an element to itself, see Fig. 2<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Mappings must exhibit the associative property, namely, a(bc)=(ab)c<\/span><\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1586\" src=\"http:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-1-286x300.png\" alt=\"\" width=\"286\" height=\"300\" srcset=\"https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-1-286x300.png 286w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-1-52x55.png 52w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-1.png 314w\" sizes=\"auto, (max-width: 286px) 100vw, 286px\" \/><\/p>\n<p><b>Fig. 1: A basic category diagram (Source: Wikipedia Commons.)<\/b><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1587\" src=\"http:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-2-300x269.png\" alt=\"\" width=\"300\" height=\"269\" srcset=\"https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-2-300x269.png 300w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-2-61x55.png 61w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-2.png 381w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><b>Fig. 2: A basic category diagram with identity mappings 1a, 1b, and 1c labeled (Source: Wikipedia Commons.)<\/b><\/p>\n<p><span style=\"font-weight: 400\">Category theory further defines other abstract ideas with more specific characteristics, such as \u201cabstract\u201d categories and \u201cconcrete\u201d categories, but all in all: they are just sets with specific characteristics.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400\">N-dimensional Categories<\/span><\/p>\n<p><span style=\"font-weight: 400\">We may define a category of n dimension by considering the conceptual ideas of a standard category. In a normal category, we have objects, the rules they obey, and the relationships they exhibit. If we say a set contains all the real multiples of 2, then that may be our relationship. Hence, we define these terms (6): 0-Cells, the objects of the category, 1-Cells, the relationship between the objects, 2-Cells, the relationship between the relationships of the objects, 3-Cells, the relationship between relationships between relationships of the objects, and rules, which the objects must obey. But of course, rules too may be subject to rules. <\/span><span style=\"font-weight: 400\">In a given system, we may say \u201cthe rules bounding the objects must all be of this form\u201d, thus providing restrictions for the rules.<\/span><span style=\"font-weight: 400\"> Hence, a similar concept must apply to the rules. Furthermore, the amount of rules needed for rules depends on how <\/span><i><span style=\"font-weight: 400\">\u201cstrict\u201d <\/span><\/i><span style=\"font-weight: 400\">the rules are (6). And all this from the simple realization of how sets work.\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400\">Examples of categories<\/span><\/p>\n<p><span style=\"font-weight: 400\">We have now defined a category, but we still don\u2019t know why this abstraction is useful. It may help us to see what sets may be defined as categories. It may be proved that all of the following are categories (5, 11):<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1588\" src=\"http:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/2022-11-27-13_22_15-EDITED-Akatson-Mathematics-Google-Docs-300x113.png\" alt=\"\" width=\"552\" height=\"208\" srcset=\"https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/2022-11-27-13_22_15-EDITED-Akatson-Mathematics-Google-Docs-300x113.png 300w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/2022-11-27-13_22_15-EDITED-Akatson-Mathematics-Google-Docs-768x289.png 768w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/2022-11-27-13_22_15-EDITED-Akatson-Mathematics-Google-Docs-146x55.png 146w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/2022-11-27-13_22_15-EDITED-Akatson-Mathematics-Google-Docs-400x150.png 400w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/2022-11-27-13_22_15-EDITED-Akatson-Mathematics-Google-Docs.png 976w\" sizes=\"auto, (max-width: 552px) 100vw, 552px\" \/><\/p>\n<p><b>Fig. 3: Examples of Categories, along with their objects and mappings.<\/b><\/p>\n<p><span style=\"font-weight: 400\">And many, many more! The reader need not know what each of these examples are, just that they are all categories: they all have objects with mappings that move between these objects. However, the examples above do demonstrate that most mathematical ideas may be expressed as some form of a category. It is here that the true elegance of the category is found: the ability to express mathematical fields as their own separate structure, and then combine them via an overarching idea. For example, in Fig 3, we link linear algebra (<\/span><b><i>Vect\u2096<\/i><\/b><span style=\"font-weight: 400\">), abstract algebra (<\/span><b><i>Grp<\/i><\/b><span style=\"font-weight: 400\">), topology (<\/span><b><i>Top<\/i><\/b><span style=\"font-weight: 400\">),\u00a0 and foundational mathematics (<\/span><b><i>Set<\/i><\/b><span style=\"font-weight: 400\">)<\/span> <span style=\"font-weight: 400\">under <\/span><i><span style=\"font-weight: 400\">one <\/span><\/i><span style=\"font-weight: 400\">unifying property\u2013 they\u2019re all categories.\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400\">Applications<\/span><\/p>\n<p><span style=\"font-weight: 400\">The applications of category theory are growing constantly. As of now, category theory has found applications in data modeling (4), machine learning (7), linguistics, computer science, physics (2, 3), biology (9, 10), mathematics, economics (8), and even in defining the scientific method itself (1). Understanding \u201cthe math of mathematics\u201d helps us develop any science that may include the presence of mathematics. Category theory also uses simplistic diagrams to express complex ideas. These diagrams happen to look very similar to that of other sciences. For example, in data modeling, we may use a diagram such as in Fig. 4. In category theory, we define a \u201cdirected multigraph\u201d to help us transform something like Fig. 5, into a mathematical object because many types of graphs happen to be categories (4).\u00a0<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1592\" src=\"http:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-4-300x232.png\" alt=\"\" width=\"309\" height=\"239\" srcset=\"https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-4-300x232.png 300w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-4-71x55.png 71w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-4-400x310.png 400w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-4-195x150.png 195w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-4.png 469w\" sizes=\"auto, (max-width: 309px) 100vw, 309px\" \/><\/p>\n<p><b>Fig. 4: An example of a basic Data Modeling diagram. (4)<\/b><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1593\" src=\"http:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-5-300x290.png\" alt=\"\" width=\"300\" height=\"290\" srcset=\"https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-5-300x290.png 300w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-5-57x55.png 57w, https:\/\/sites.imsa.edu\/hadron\/files\/2022\/11\/Fig-5.png 378w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><b>Fig 5: A possible directed multigraph, which is a category, for data modeling. (4)<\/b><\/p>\n<p><span style=\"font-weight: 400\">Because of the similarity between these diagrams, one may rapidly construct a category and hence apply category theory to determine the significance of their diagram. Since categories can take many, many forms, we can explore many fields of science and mathematics, link them together, and push the boundaries of our understanding.\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400\">Summary<\/span><\/p>\n<p><span style=\"font-weight: 400\">Today, category theory is a promising field of mathematics, with modern applications to just about any field of science. Its abstractive properties allow for the joining of vastly distinct mathematics. And despite these achievements, its definitions remain clear, elegant, and straightforward: a category is nothing but a simple set with a few key characteristics. Category theory is, in essence, the expression of complex ideas as simple collections of objects to unite mathematics and science as a whole.\u00a0\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400\">References<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">Category theory for Scientists (old version) &#8211; MIT mathematics<\/span><\/i><span style=\"font-weight: 400\">. (n.d.). Retrieved October 24, 2022, from https:\/\/math.mit.edu\/~dspivak\/CT4S.pdf<\/span><\/p>\n<p><span style=\"font-weight: 400\">D\u00f6ring, A., &amp; Isham, C. J. (1970, January 1). <\/span><i><span style=\"font-weight: 400\">A Topos Foundation for theories of physics: I. Formal languages for physics<\/span><\/i><span style=\"font-weight: 400\">. AIP Publishing. Retrieved October 24, 2022, from https:\/\/aip.scitation.org\/doi\/abs\/10.1063\/1.2883740<\/span><\/p>\n<p><span style=\"font-weight: 400\">Guts, A. K., &amp; Grinkevich, E. B. (1996, October 31). <\/span><i><span style=\"font-weight: 400\">Toposes in general theory of relativity<\/span><\/i><span style=\"font-weight: 400\">. arXiv.org. Retrieved October 24, 2022, from https:\/\/arxiv.org\/abs\/gr-qc\/9610073<\/span><\/p>\n<p><span style=\"font-weight: 400\">Lippe, E., &amp; Ter Hofstede, A. H. M. (1996, January 1). <\/span><i><span style=\"font-weight: 400\">A category theory approach to Conceptual Data Modeling<\/span><\/i><span style=\"font-weight: 400\">. RAIRO &#8211; Theoretical Informatics and Applications &#8211; Informatique Th\u00e9orique et Applications. Retrieved October 24, 2022, from http:\/\/www.numdam.org\/item\/ITA_1996__30_1_31_0\/<\/span><\/p>\n<p><span style=\"font-weight: 400\">Marquis, J.-P. (2019, August 29). <\/span><i><span style=\"font-weight: 400\">Category theory<\/span><\/i><span style=\"font-weight: 400\">. Stanford Encyclopedia of Philosophy. Retrieved October 24, 2022, from https:\/\/plato.stanford.edu\/entries\/category-theory\/<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">\u00a0Orew &#8211; Eugenia Cheng<\/span><\/i><span style=\"font-weight: 400\">. (n.d.). Retrieved October 25, 2022, from https:\/\/eugeniacheng.com\/wp-content\/uploads\/2017\/02\/cheng-architecture.pdf<\/span><\/p>\n<p><span style=\"font-weight: 400\">Shiebler, D., Gavranovi\u0107, B., &amp; Wilson, P. (2021, June 13). <\/span><i><span style=\"font-weight: 400\">Category theory in machine learning<\/span><\/i><span style=\"font-weight: 400\">. arXiv.org. Retrieved October 24, 2022, from https:\/\/arxiv.org\/abs\/2106.07032<\/span><\/p>\n<p><span style=\"font-weight: 400\">Tran, C. S., Nicolau, D., Nayak, R., &amp; Verhoeven, P. (2021, July 1). <\/span><i><span style=\"font-weight: 400\">Modeling credit risk: A category theory perspective<\/span><\/i><span style=\"font-weight: 400\">. MDPI. Retrieved October 24, 2022, from https:\/\/www.mdpi.com\/1911-8074\/14\/7\/298<\/span><\/p>\n<p><span style=\"font-weight: 400\">Tuy\u00e9ras, R. (2018, August 1). <\/span><i><span style=\"font-weight: 400\">Category theory for genetics II: Genotype, phenotype and haplotype<\/span><\/i><span style=\"font-weight: 400\">. arXiv.org. Retrieved October 24, 2022, from https:\/\/arxiv.org\/abs\/1805.07004v2<\/span><\/p>\n<p><span style=\"font-weight: 400\">Tuy\u00e9ras, R. (2020, April 4). <\/span><i><span style=\"font-weight: 400\">Category theory for Genetics I: Mutations and sequence alignments<\/span><\/i><span style=\"font-weight: 400\">. arXiv.org. Retrieved October 24, 2022, from https:\/\/arxiv.org\/abs\/1805.07002<\/span><\/p>\n<p><i><span style=\"font-weight: 400\">\u00a0What is category theory anyway?<\/span><\/i><span style=\"font-weight: 400\"> RSS. (n.d.). Retrieved October 24, 2022, from https:\/\/www.math3ma.com\/blog\/what-is-category-theory-anyway\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>By: Andrew D. Katson Definition Category theory, as invented in 1945, is a field of mathematics that works with objects and their relationships. A category is a set made up of multiple elements, called \u201cobjects\u201d, and their \u201cmappings\u201d (5, 11). In figure 1, we see<\/p>\n","protected":false},"author":821,"featured_media":1586,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[11],"tags":[],"class_list":["post-1585","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/posts\/1585","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/users\/821"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/comments?post=1585"}],"version-history":[{"count":6,"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/posts\/1585\/revisions"}],"predecessor-version":[{"id":1634,"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/posts\/1585\/revisions\/1634"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/media\/1586"}],"wp:attachment":[{"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/media?parent=1585"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/categories?post=1585"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.imsa.edu\/hadron\/wp-json\/wp\/v2\/tags?post=1585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}