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Freyd Cover
In the mathematical discipline of category theory, the Freyd cover or scone category is a construction that yields a set-like construction out of a given category. The only requirement is that the original category has a terminal object. The scone category inherits almost any categorical construct the original category has. Scones can be used to generally describe proofs that use logical relations. The Freyd cover is named after Peter Freyd. The other name, "scone", is intended to suggest that it is like a cone, but with the Sierpiński space in place of the unit interval. Definition Formally, the scone of a category ''C'' with a terminal object 1 is the comma category In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a Category (mathematics), category to one another ... 1_\text \downarrow \operatorname_C(1,-). See also * Artin gl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
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 most 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Relations
Logical relations are a proof method employed in programming language semantics to show that two denotational semantics are equivalent. To describe the process, let us denote the two semantics by type A, there is a particular associated binary relation">relation \sim between [\![A]\!]_1 and [\![A]\!]_2. This relation is defined such that for each program phrase M, the two denotations are related: [\![M]\!]_1 \sim [\![M]\!]_2. Another property of this relation is that related denotations for ''ground types'' are equivalent in some sense, usually equal. The conclusion is then that both denotations exhibit equivalent behavior on ground terms, hence are equivalent. References * https://www.cs.uoregon.edu/research/summerschool/summer16/notes/AhmedLR.pdf * https://www.cs.uoregon.edu/research/summerschool/summer13/lectures/ahmed-1.pdf POPLmark Reloaded Proofs involving logical relations used as a benchmark for proof assistants In computer science and mathematical logic, a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Freyd
Peter John Freyd (; born February 5, 1936) is an American mathematician, a professor at the University of Pennsylvania, known for work in category theory and for founding the False Memory Syndrome Foundation. Mathematics Freyd obtained his Ph.D. from Princeton University in 1960; his dissertation, on ''Functor Theory'', was written under the supervision of Norman Steenrod and David Buchsbaum. Freyd is best known for his Adjoint functors, adjoint functor theorem. He was the author of the foundational book ''Abelian Categories: An Introduction to the Theory of Functors'' (1964). This work culminates in a proof of the Freyd–Mitchell embedding theorem. In addition, Freyd's name is associated with the HOMFLY polynomial, HOMFLYPT polynomial of knot theory, and he and Andre Scedrov originated the concept of (mathematical) allegory (category theory), allegories. In 2012, he became a fellow of the American Mathematical Society. False Memory Syndrome Foundation Freyd and his wif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone (topology)
In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching ''X'' into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by CX or by \operatorname(X). Definitions Formally, the cone of ''X'' is defined as: :CX = (X \times ,1\cup_p v\ =\ \varinjlim \bigl( (X \times ,1 \hookleftarrow (X\times \) \xrightarrow v\bigr), where v is a point (called the vertex of the cone) and p is the projection to that point. In other words, it is the result of attaching the cylinder X \times ,1/math> by its face X\times\ to a point v along the projection p: \bigl( X\times\ \bigr)\to v. If X is a non-empty compact subspace of Euclidean space, the cone on X is homeomorphic to the union of segments from X to any fixed point v \not\in X such that these segments intersect only in v itself. That is, the topological cone agrees with the geometric cone for compact spaces when the latter is defined. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sierpiński Space
In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott topology. Definition and fundamental properties Explicitly, the Sierpiński space is a topological space ''S'' whose underlying point set is \ and whose open sets are \. The closed sets are \. So the singleton set \ is closed and the set \ is open (\varnothing = \ is the empty set). The closure operator on ''S'' is determined by \overline = \, \qquad \overline = \. A finite topological space is also uniquely determined by its specialization preorder. For the Sierpiński space this preorder is actually a partial order and given by 0 \leq 0, \qquad 0 \leq 1, \qquad 1 \leq 1. Topol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation ' is most commonly reserved for the closed interval . Properties The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comma Category
In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a Category (mathematics), category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by William Lawvere, F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some Limit (category theory), limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p. 13). Definition The most general comma ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Gluing
Artin may refer to: * Artin (name), a surname and given name, including a list of people with the name ** Artin, a variant of Harutyun Harutyun ( and in Western Armenian Յարութիւն) also spelled Haroutioun, Harutiun and its variants Harout, Harut and Artin is a common male Armenians, Armenian name; it means "Resurrection of Jesus, resurrection" in Armenian. People with ..., an Armenian given name * 15378 Artin, a main-belt asteroid See also {{disambiguation, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
