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Well-pointed Category
In category theory, a category with a terminal object 1 is well-pointed if for every pair of arrows f,g:A\to B such that f\neq g, there is an arrow p:1\to A such that f\circ p\neq g\circ p. (The arrows p are called the global element In category theory, a global element of an object ''A'' from a category is a morphism :h\colon 1 \to A, where is a terminal object of the category.. Roughly speaking, global elements are a generalization of the notion of "elements" from the categor ...s or ''points'' of the category; a well-pointed category is thus one that has "enough points" to distinguish non-equal arrows.) See also * Pointed category References * Category theory {{Categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. 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. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terminal Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * In t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Element
In category theory, a global element of an object ''A'' from a category is a morphism :h\colon 1 \to A, where is a terminal object of the category.. Roughly speaking, global elements are a generalization of the notion of "elements" from the category of sets, and they can be used to import set-theoretic concepts into category theory. However, unlike a set, an object of a general category need not be determined by its global elements (not even up to isomorphism). For example, the terminal object of the category Grph of graph homomorphisms has one vertex and one edge, a self-loop, whence the global elements of a graph are its self-loops, conveying no information either about other kinds of edges, or about vertices having no self-loop, or about whether two self-loops share a vertex. In an elementary topos the global elements of the subobject classifier form a Heyting algebra when ordered by inclusion of the corresponding subobjects of the terminal object.. For example, Grph happens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
