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Cartesian closed category - Wikipedia, the free encyclopedia The category of finite sets, with functions as morphisms, is cartesian closed for the same reason. The category of all directed graphs is cartesian closed; this is a functor category as explained under functor category. In algebraic topology, cartesian closed categories are particularly easy to work with, and it is regrettable that neither the category of topological spaces with continuous maps nor the category of smooth manifolds with smooth maps is cartesian closed. en.wikipedia.org /wiki/Cartesian_closed_category   (922 words)

Cartesian morphism - Wikipedia, the free encyclopedia In mathematics, in particular category theory, given a functor from a category E to a category C, a morphism in E is cartesian (with respect to p) when for each object Z of E and each morphism en.wikipedia.org /wiki/Cartesian_morphism   (81 words)

Cartesian Cartesian closed category In category theory, a category is cartesian closed if, roughly speaking, any morphism defined... Cartesian diver A Cartesian diver is a classic soft drink bottle, and adjusted so it barely floats at the top of the wat... Cartesian morphism In category theory, given a functor p:E→C from a category E to a category C, a morphism f : X &r... www.brainyencyclopedia.com /topics/cartesian.html   (123 words)

Category theory Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space. If all morphisms have a kernel and a cokernel, and all epimorphism are cokernels and all monomorphisms are kernels, then we speak of an abelian category. A category is called cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors. www.teachersparadise.com /ency/en/wikipedia/c/ca/category_theory.html   (2466 words)

Cartesian closed category. Who is Cartesian closed category? What is Cartesian closed category? Where is Cartesian ... The term "cartesian closed" is used because one thinks of Y×X as akin to the cartesian product of two sets. In cartesian closed categories, a "function of two variables" can always be represented as a "function of one variable". Certain cartesian closed categories, the topoi, have been proposed as a general setting for mathematics. www.knowledgerush.com /kr/encyclopedia/Cartesian_closed_category   (797 words)

Comma category - Enpsychlopedia   (Site not responding. Last check: 2007-10-14) It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, they become objects in their own right. Morphisms are composed by taking $(g,\; h)\; \backslash circ\; (g\text{'},\; h\text{'})$ to\; be$ (g\; \backslash circ\; g\text{'},\; h\; \backslash circ\; h\text{'})$ ,\; whenever\; the\; latter\; expression\; is\; defined.$$$$ The notion of a universal morphism to a particular colimit, or from a limit, can be expressed in terms of a comma category. www.grohol.com /psypsych/Comma_category   (1520 words)

Cartesian closed category - ArtPolitic Encyclopedia of Politics : Information Portal   (Site not responding. Last check: 2007-10-14) The adjointness means that the set of morphisms in C from Y×X to Z is naturally identified with the set of morphism from Y to HOM(X,Z), for any three objects X, Y and Z in C. If C is a small category, then the category of all functors from C to Set (with natural transformations as morphisms) is a cartesian closed category. In algebraic topology, cartesian closed categories are particularly easy to work with, and it is regrettable that neither the category of topological spaces with continous maps nor the category of smooth manifolds with smooth maps is Cartesian closed. www.artpolitic.org /infopedia/ca/Cartesian_closed_category.html   (457 words)

[No title] B are morphisms, a natural transformation from f to g is a vertex of the simplicial set j 2 Hom__(A; B)1=(f; g): In general for a Segal category C, a vertex of C1=(x; y) is the same thing as a morphism I ! In [33] the notion of cartesian family is defined by saying that it is a precartesian family which satisfies a certain quasi-fibrant condition. Note that the morphism in the strictified setup is homotopic to the morphism we have constructed in the original weak situation. hopf.math.purdue.edu /Simpson/giraudH.txt   (11010 words)

Category theory - LearnThis.Info Enclyclopedia   (Site not responding. Last check: 2007-10-14) Relations among morphisms (such as fg = h) can most conveniently be represented with commutative diagrams, where the objects are represented as points and the morphisms as arrows. A morphism from (X,x) to (Y,y) is given by a continuous map f : X → Y with f(x) = y. The solution is to characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. encyclopedia.learnthis.info /c/ca/category_theory.html   (3218 words)

[No title] The claim in Lemma 2 that the diagram in the statement of the Lemma is homotopy cartesian refers to the closed model structure on the cate- gory of bisimplicial sets for which the cofibrations are the inclusions and the weak equivalences are the maps X ! Ob (A)________//BA is a homotopy cartesian diagram of bisimplicial presheaves. Mor(A0) ____t_____//Ob(A) is a homotopy cartesian diagram of simplicial presheaves. hopf.math.purdue.edu /Jardine/diagrams.txt   (6047 words)

Function (mathematics) A morphism is then an ordered triple (X, Y, f), where f is a "function" with domain X and codomain Y. Since X and Y do not necessarily correspond to a set of objects, however, morphisms do not always behave like functions, and, for example, enlarging the codomain (which does nothing to a function) gives a different morphism which you cannot identify with the original one. Ordinary functions are sometimes referred to as morphisms when they are morphisms in a concrete category. www.sciencedaily.com /encyclopedia/function__mathematics_   (2922 words)

Category Theory Composition of morphisms corresponds to multiplication of elements of the monoid. For instance, given two sets A and B, set theory allows us to construct their cartesian product A X B. For an example of the second sort, given a finite abelian group, it can be decomposed into a product of some of its subgroups. Indeed, from a categorical point of view, a set-theoretical cartesian product, a direct product of groups, a direct product of abelian groups, a product of topological spaces and a conjunction of propositions in a deductive system are all instances of a categorical concept: the categorical product. plato.stanford.edu /entries/category-theory   (7029 words)

Universal Property [Definition]   (Site not responding. Last check: 2007-10-14) If, however, a universal morphism (A, φ) does exists then it is unique up toIn mathematics, the term "up to xxxx" is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. The objects of C are morphisms f : X → Y in D, and a morphism from f : X → Y to g : S → T is given by a pair (α,β) of morphisms α : X → S and β : Y → T such that βf = gα. Dually, the colimit of F is a universal morphism from F to Δ. www.wikimirror.com /Universal_property   (3619 words)

Articles - Function (mathematics)   (Site not responding. Last check: 2007-10-14) A generalisation of the notion of function is morphism in the context of category theory. A category is a collection of objects and morphisms, each morphism is an ordered triple (X, Y, f), where f is a rule connecting domain X and codomain Y, and X and Y are objects in the collection. Ordinary functions are sometimes referred to as morphisms in a concrete category. www.anfolk.com /articles/Function_(mathematics)   (2874 words)

fibration   (Site not responding. Last check: 2007-10-14) We are not in this case given a local cartesian product structure (which defines the more restricted fiber bundle case), but something possibly weaker that still allows 'sideways' movement from fiber to fiber. One of the main desirable properties of the Serre spectral sequence is to account for the action of the fundamental group of the base X on the homology of the total space Y. In the category theory, a functor p : E → C from a category E to a category C is a fibration iff for every object X of E and every map γ into pX in C there exists a cartesian morphism into X over γ (see also semidirect product). www.yourencyclopedia.net /fibration.html   (396 words)

Category (mathematics) - Enpsychlopedia   (Site not responding. Last check: 2007-10-14) (a, b)) to denote the hom-class of all morphisms from a to b. The morphisms of a category are sometimes called arrows due to the influence of commutative diagrams. Any directed graph generates a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph. www.grohol.com /wiki/Object_(category_theory)   (1247 words)

Re: SUO: Re: IFF LOT Glossary This means that there is a (unique) theory morphism from the direct image dir(f)(T) = (K, B) to (M, C). PS: The direct and inverse operations between LOTs is equivalent to the concept morphisms between the equivalent truth concept lattices (TCLs). Secondly, for any > >morphism of languages f : L1 --> L2 in the category of languages, there are > >direct and inverse image functions > > dir(f) : LOT(L1) --> LOT(L2) > > inv(f) : LOT(L2) --> LOT(L1) > >that are adjoint monotonic functions. suo.ieee.org /email/msg10125.html   (1061 words)

[No title]   (Site not responding. Last check: 2007-10-14) A {\em bifunction} $f$ is said to be {\em covariant} if there is a function $g$ from $A$ into $B$ that $f$ is the Cartesian square of $g$ and $f$ is {\em contravariant} if there is a function $g$ such that $f(o_1,o_2) = \langle g(o_2),g(o_1) \rangle$. Eventually, the morphism map of a functor from $C_1$ into $C_2$ is a transformation from the arrows of the category $C_1$ into the composition of the object map of the functor and the arrows of $C_2$.\par Several kinds of functor structures have been defined: one-to-one, faithful, onto, full and id-preserving. We were pressed to split property that the composition be preserved into two: comp-preserving (for covariant functors) and comp-reversing (for contravariant functors). merak.pb.bialystok.pl /mizarbib/functor0.bib   (219 words)

The Monoidal Category of Hilbert Spaces The reason is that in quantum theory, the states of a system are no longer described by a set, but by a Hilbert space. In other words, composing the top morphism with the right-hand one gives the same result as composing the left-hand one with the bottom one. This compatibility condition expresses the fact that no arbitrary choices are required to define the associator: in particular, it is defined in a basis-independent manner. math.ucr.edu /home/baez/quantum/node4.html   (2679 words)

Concurrency Abstracts   (Site not responding. Last check: 2007-10-14) We apply Cartesian logic to reject not only divine intervention, preordained synchronization, and the eventual mass retreat to monism, but also an assumption Descartes himself somehow neglected to reject, that causal interaction within these planes is an easier problem than between. Uncertainty arises when we define a measurement to be a morphism and notice that increasing structure in the observed object reduces clarity of observation. Its algebraic structure is essentially that of linear logic, with its morphisms being consequence-preserving renamings of propositions, and with its operations forming the core of a natural concurrent programming language. boole.stanford.edu /abstracts.html   (9620 words)

Cartesian closed category - InfoSearchPoint.com   (Site not responding. Last check: 2007-10-14) In category theory, a category is cartesian closed if any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. The adjointness condition means that the set of morphisms in C from Y×X to Z is naturally identified with the set of morphisms from Y to HOM(X,Z), for any three objects X, Y and Z in C. What other "equations" are valid in all cartesian closed categories? www.infosearchpoint.com /display/Cartesian_closed_category   (662 words)

Citations: A coherence thorem for canonical morphism in cartesian closed categories - Babaev, Solovjev (ResearchIndex) Babaev, A. and Solovjev, S. A coherence thorem for canonical morphism in cartesian closed categories. It seems to be possible to extend the above uniqueness result of the implicational fragment to the implication conjunction fragment. Babaev, S. Solovjev, A coherence thorem for canonical morphism in cartesian closed categories, Journal of Soviet Mathematics, 20 (1982), pp. citeseer.ist.psu.edu /context/48505/0   (405 words)

[No title]   (Site not responding. Last check: 2007-10-14) Tensor products: If C denotes the category of vectorspaces over a fixed field, with linear maps as morphisms, then the tensor product VW defines a functor C x C G(X) in D such that for every morphism f : X D which assigns every object to itself and every morphism to itself) and \GF is naturally isomorphic to I www.informationgenius.com /encyclopedia/c/ca/category_theory.html   (2864 words)

week135 This is just an abstraction of the properties of the usual Cartesian product of sets, which is why we call a category "Cartesian" if any pair of objects has a product. Anyway, this has made me feel for a while that topos theory isn't sufficiently "quantum" to be useful in understanding the peculiar special features of quantum physics. Now, presheaves over any category form a topos, so this means we should be able to think of a topological quantum field theory as a "Hilbert space object" in the topos of presheaves over nCob. math.ucr.edu /home/baez/week135.html   (2547 words)

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