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In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are nonderogatory terms used by mathematicians to describe long, theoretical parts of a proof they skip over when readers are expected to be familiar with them. [1] These terms are mainly used for abstract methods related to category theory and homological algebra. More generally, "abstract nonsense" may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself.
Roughly speaking, category theory is the study of the general form, that is, categories of mathematical theories, without regard to their content. As a result, mathematical proofs that rely on category-theoretic ideas often seem out-of-context, somewhat akin to a non sequitur. Authors sometimes dub these proofs "abstract nonsense" as a light-hearted way of alerting readers to their abstract nature. Labeling an argument "abstract nonsense" is usually not intended to be derogatory, [2] [1] and is instead used jokingly, [3] in a self-deprecating way, [4] affectionately, [5] or even as a compliment to the generality of the argument. Alexander Grothendieck was critical of this notion, and stated that:
The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps... [6]
Certain ideas and constructions in mathematics share a uniformity throughout many domains, unified by category theory. Typical methods include the use of classifying spaces and universal properties, use of the Yoneda lemma, natural transformations between functors, and diagram chasing. [7]
When an audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression "Such and such is true by abstract nonsense" rather than provide an elaborate explanation of particulars. [1] For example, one might say that "By abstract nonsense, products are unique up to isomorphism when they exist", instead of arguing about how these isomorphisms can be derived from the universal property that defines the product. This allows one to skip proof details that can be considered trivial or not providing much insight, focusing instead on genuinely innovative parts of a larger proof.
The term predates the foundation of category theory as a subject itself. Referring to a joint paper with Samuel Eilenberg that introduced the notion of a "category" in 1942, Saunders Mac Lane wrote the subject was 'then called "general abstract nonsense"'. [3] The term is often used to describe the application of category theory and its techniques to less abstract domains. [8] [9]
The term is believed to have been coined by the mathematician Norman Steenrod, [10] [4] [5] himself one of the developers of the categorical point of view.
Alexander Grothendieck, later Alexandre Grothendieck in French was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century.
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.
There have been several attempts in history to reach a unified theory of mathematics. Some of the most respected mathematicians in the academia have expressed views that the whole subject should be fitted into one theory.
This article gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory.
Saunders Mac Lane, born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg.
The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows, where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.
Samuel Eilenberg was a Polish-American mathematician who co-founded category theory and homological algebra.
Henri Paul Cartan was a French mathematician who made substantial contributions to algebraic topology.
The language of mathematics has a wide vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this uses common English words, but with a specific non-obvious meaning when used in a mathematical sense.
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.
In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology, where one studies algebraic invariants of spaces, such as the fundamental weak ∞-groupoid.
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if and are morphisms whose composition is the identity morphism on , then is a section of , and is a retraction of .
Colin McLarty is an American logician whose publications have ranged widely in philosophy and the foundations of mathematics, as well as in the history of science and of mathematics.
In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as its reduced homology groups. The most common version of its proof consists of showing that the composition of the homotopy group functors with the infinite symmetric product defines a reduced homology theory. One of the main tools used in doing so are quasifibrations. The theorem has been generalised in various ways, for example by the Almgren isomorphism theorem.
The article "Sur quelques points d'algèbre homologique" by Alexander Grothendieck, now often referred to as the Tôhoku paper, was published in 1957 in the Tôhoku Mathematical Journal. It revolutionized the subject of homological algebra, a purely algebraic aspect of algebraic topology. It removed the need to distinguish the cases of modules over a ring and sheaves of abelian groups over a topological space.
In mathematics, in the framework of one-universe foundation for category theory, the term conglomerate is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of a Grothendieck universe.
Condensed mathematics is a theory developed by Dustin Clausen and Peter Scholze which, according to some, aims to unify various mathematical subfields, including topology, complex geometry, and algebraic geometry.