In algebra, given a 2-monad T in a 2-category, a pseudoalgebra for T is a 2-category-version of algebra for T, that satisfies the laws up to coherent isomorphisms. [1]
Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003.
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.
In category theory, a branch of mathematics, a monad is a triple consisting of a functor T from a category to itself and two natural transformations that satisfy the conditions like associativity. For example, if are functors adjoint to each other, then together with determined by the adjoint relation is a monad.
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.
In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by Jonathan Mock Beck in about 1964. It is often stated in dual form for comonads. It is sometimes called the Beck tripleability theorem because of the older term triple for a monad.
This is a glossary of properties and concepts in category theory in mathematics.
In mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory.
In category theory, a branch of mathematics, a monoidal monad is a monad on a monoidal category such that the functor is a lax monoidal functor and the natural transformations and are monoidal natural transformations. In other words, is equipped with coherence maps and satisfying certain properties, and the unit and multiplication are monoidal natural transformations. By monoidality of , the morphisms and are necessarily equal.
In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other.
Undergraduate Texts in Mathematics (UTM) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow books of a standard size.
Alexander A. Beilinson is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1999, Beilinson was awarded the Ostrowski Prize with Helmut Hofer. In 2017, he was elected to the National Academy of Sciences. In 2018, he received the Wolf Prize in Mathematics and in 2020 the Shaw Prize in Mathematics.
In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
Ronald Brown FLSW is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and more than 160 journal articles.
In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".
In algebraic geometry, a quasi-coherent sheaf on an algebraic stack is a generalization of a quasi-coherent sheaf on a scheme. The most concrete description is that it is a data that consists of, for each a scheme S in the base category and in , a quasi-coherent sheaf on S together with maps implementing the compatibility conditions among 's.
Julia Elizabeth Bergner is a mathematician specializing in algebraic topology, homotopy theory, and higher category theory. She is a professor of mathematics at the University of Virginia.
Dmitri Olegovich Orlov, is a Russian mathematician, specializing in algebraic geometry. He is known for the Bondal-Orlov reconstruction theorem (2001).
In mathematics, the category of measurable spaces, often denoted Meas, is the category whose objects are measurable spaces and whose morphisms are measurable maps. This is a category because the composition of two measurable maps is again measurable, and the identity function is measurable.
In mathematics, the Giry monad is a construction that assigns to a measurable space a space of probability measures over it, equipped with a canonical sigma-algebra. It is one of the main examples of a probability monad.