The cosmos is the Universe. Using the word cosmos rather than the word universe implies viewing the universe as a complex and orderly system or entity; the opposite of chaos. The cosmos, and our understanding of the reasons for its existence and significance, are studied in cosmology – a very broad discipline covering any scientific, religious, or philosophical contemplation of the cosmos and its nature, or reasons for existing. Religious and philosophical approaches may include in their concepts of the cosmos various spiritual entities or other matters deemed to exist outside our physical universe.
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.
In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-categoryC is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas:
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms associated with every pair of objects is replaced by an object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint.
In mathematics, a monoidal category is a category C equipped with a bifunctor
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.
Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat.
"Utopia" is an electronic song performed by British group Goldfrapp. The song was written and produced by Alison Goldfrapp and Will Gregory for the duo's debut album Felt Mountain (2000). It was released as the album's second single in November 2000. Although the song did not appear on the UK Singles Chart initially, it reached number 29 on the UK Indie Chart and found minor success in the Netherlands, debuting and peaking at number 94 in January 2001.
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 their fundamental weak ∞-groupoid.
In category theory, a weak n-category is a generalization of the notion of strict n-category where composition and identities are not strictly associative and unital, but only associative and unital up to coherent equivalence. This generalisation only becomes noticeable at dimensions two and above where weak 2-, 3- and 4-categories are typically referred to as bicategories, tricategories, and tetracategories. The subject of weak n-categories is an area of ongoing research.
Gregory Maxwell "Max" Kelly, mathematician, founded the thriving Australian school of category theory.
Isbell conjugacy is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986.
In mathematics, more specifically category theory, a quasi-category is a generalization of the notion of a category. The study of such generalizations is known as higher category theory.
In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies to simplicial objects in Cat. Simplicially enriched categories can, however, be identified with simplicial objects in Cat whose object part is constant, or more precisely, whose all face and degeneracy maps are bijective on objects. Simplicially enriched categories can model -categories, but the dictionary has to be carefully built. Namely many notions, limits for example, are different from the limits in the sense of enriched category theory.
Cosmos Redshift 7 is a high-redshift Lyman-alpha emitter galaxy. At a redshift z = 6.6, the galaxy is observed as it was about 800 million years after the Big Bang, during the epoch of reionisation. With a light travel time of 12.9 billion years, it is one of the oldest, most distant galaxies known.
In mathematics, a 2-functor is a morphism between 2-categories. They may be defined formally using enrichment by saying that a 2-category is exactly a Cat-enriched category and a 2-functor is a Cat-functor.
Philosophical cosmology, philosophy of cosmology or philosophy of cosmos is a discipline directed to the philosophical contemplation of the universe as a totality, and to its conceptual foundations. It draws on several branches of philosophy—metaphysics, epistemology, philosophy of physics, philosophy of science, philosophy of mathematics, and on the fundamental theories of physics. The term cosmology was used at least as early as 1730, by German philosopher Christian Wolff, in Cosmologia Generalis.
