 | An editor has performed a search and found that sufficient sources exist to establish the subject's notability.(July 2024) |
In mathematics, categorical probability is a category-theoretic approach to probability theory. The idea goes back to at least Lawvere's 1962 paper. [1]
Some constructs in the theory are
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 category theory, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1, an NNO N is given by:
This is a glossary of properties and concepts in category theory in mathematics.
In category theory, a branch of mathematics, for every object in every category where the product exists, there exists the diagonal morphism
In category theory, a Lawvere theory is a category that can be considered a categorical counterpart of the notion of an equational theory.
Isbell conjugacy is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986. That is a duality between covariant and contravariant representable presheaves associated with an objects of categories under the Yoneda embedding. Also, Lawvere says that; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics".
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.
The theory of ologs is an attempt to provide a rigorous mathematical framework for knowledge representation, construction of scientific models and data storage using category theory, linguistic and graphical tools. Ologs were introduced in 2012 by David Spivak and Robert Kent.
In category theory, the notion of final functor is a generalization of the notion of final object in a category.
In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a category. For example, given a ring R, let C be a category whose objects are the elements of the set R and whose morphisms are only the identity morphisms. Then C is a categorical ring. But the point is that one can also consider the situation in which an element of R comes with a "nontrivial automorphism".
Higher Topos Theory is a treatise on the theory of ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-topoi, the book is widely considered foundational to higher category theory. Since 2018, Lurie has been transferring the contents of Higher Topos Theory to Kerodon, an "online resource for homotopy-coherent mathematics" inspired by the Stacks Project.
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.
In mathematics, nonabelian algebraic topology studies an aspect of algebraic topology that involves higher-dimensional algebras.
In mathematics, a coherent topos is a topos generated by a collection of quasi-compact quasi-separated objects closed under finite products.
In mathematics, a corestriction of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a subset, the corestriction changes the codomain to a subset. However, the notions are not categorically dual.
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 category of Markov kernels, often denoted Stoch, is the category whose objects are measurable spaces and whose morphisms are Markov kernels. It is analogous to the category of sets and functions, but where the arrows can be interpreted as being stochastic.
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.
In mathematics, especially in topology, a pyknotic set is a sheaf of sets on the site of compact Hausdorff spaces. The notion was introduced by Barwick and Haine to provide a convenient setting for homological algebra. The term pyknotic comes from the Greek πυκνός, meaning dense, compact or thick. The notion can be compared to other approaches of introducing generalized spaces for the purpose of homological algebra such as Clausen and Scholze‘s condensed sets or Johnstone‘s topological topos.
In category theory, a branch of mathematics, there are several ways (completions) to enlarge a given category in a way somehow analogous to a completion in topology. These are:,
 | This category theory-related article is a stub. You can help Wikipedia by expanding it. |