Colin McLarty (born July 12, 1951) 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.
Colin McLarty (born July 12, 1951) 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.
He has written papers about Saunders Mac Lane, one of the founders of category theory.
McLarty's Elementary Categories and Elementary Toposes describes category theory and topos theory at an elementary level. [1]
McLarty worked on establishing that Fermat's Last Theorem can be proven in a setting with much weaker assumptions than the ones used in Wiles's proof of Fermat's Last Theorem, which makes use of involved category theoretical constructions. [2]
He is a member of the Grothendieck Circle, which provides on-line and open access to many writings about the mathematician Alexandre Grothendieck, who revolutionized Banach-space theory and algebraic geometry and whose life has fascinated many biographers and mathematical scientists. [3]
McLarty has also written about the German algebraist Emmy Noether and her involvement with German political history. [4]
At Case Western Reserve University (CWRU), Colin McLarty is the Truman P. Handy Professor of Philosophy and the current Chair of the Philosophy Department. [5]
At CWRU, he is also a professor of mathematics.
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.
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are the total functions from A to B, and the composition of morphisms is the composition of functions.
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
Amalie Emmy Noether was a German mathematician who made many important contributions to abstract algebra. She proved Noether's first and second theorems, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers.
In algebra, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings ; related structures like rngs; as well as an array of properties that prove to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
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.
In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of X. Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.
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. 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.
Francis William Lawvere was an American mathematician known for his work in category theory, topos theory and the philosophy of mathematics.
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, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.
Barry Charles Mazur is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.
In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a local operator or coverage or topology or geometric modality. They were introduced by William Lawvere and Myles Tierney.
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