New Levels of Abstraction: Homological Algebra and Category Theory
Overview
Two of the more abstract branches of mathematics are homological algebra and category theory. Important progress was made in both during the first half of the twentieth century. Indeed, since the fields both arose in the latter part of the nineteenth century, virtually all work in them took place in the twentieth. While their practical effects may not be as great as their mathematical importance, research is still worth pursuing because the field of mathematics provides such an accurate description of the universe in which we live. This leads to the assumption that, even ifthese fields are seemingly of little import, the future may hold something more.
Background
According to the website Eric Weisstein's "World of Mathematics," category theory is "the branch of mathematics which formalizes a number of algebraic properties of collections of transformations between mathematical objects (such as binary relations, groups, sets, topological spaces, etc.) of the same type, subject to the constraint that the collections contain the identity mapping and are closed with respect to compositions of mappings. The objects studied in category theory are called categories." This definition includes a link to "category," about which it is said, "A category consists of two things: a collection of objects and, for each pair of objects, a collection of morphisms (sometimes called 'arrows') from one to another." While these definitions sound slightly abstruse, they can be understood without much effort.
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