Morphism (category theory)

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	Image (category theory) - Wikipedia, the free encyclopedia
		In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism f can be expressed as follows:
		In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets.
		The image of f is often denoted by im f.
	en.wikipedia.org /wiki/Image_(category_theory)

	Objects and Morphisms
		If there is a morphism between two equivalence classes, prepend a morphism in one class, and apend a morphism in the other, to get a morphism between the two representatives.
		In groups rings and fields, an equivalence is an isomorphism, and the objects on either side of the equivalence are called isomorphic.
		An equivalence betweeen sets is a bijection, and an equivalence between topological spaces is a homeomorphism.
	www.mathreference.com /cat,def.html

	Kernel (category theory) - Wikipedia, the free encyclopedia
		In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra.
		That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa.
		Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fixed ring (including vector spaces over a fixed field).
	en.wikipedia.org /wiki/Kernel_(category_theory)

	Category theory - Wikipedia, the free encyclopedia
		Such a process is called a functor, and it associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second.
		Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus.
	en.wikipedia.org /wiki/Category_theory (2348 words)

	Cokernel - Wikipedia, the free encyclopedia
		In the category of groups, the cokernel of a group homomorphism f : G → H is the quotient of H by the normal closure of the image of f.
		In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f.
		In a pre-abelian category (a special kind of preadditive category) the existence of kernels and cokernels is guaranteed.
	en.wikipedia.org /wiki/Cokernel (425 words)

	Limit and Colimit
		A morphism in the new category is derived from a morphism from v
		A terminal object t in the new category is the limit of d.
		The initial object in the new category is the colimit of d.
	www.mathreference.com /cat,limit.html (531 words)

	kernel (mathematics)
		Finally, for this last notion of kernel is generalised in a certain sense in category theory; the kernel of a morphism f is the difference kernel of f and the corresponding zero morphism (if this exists).
		But in the case of Mal'cev algebras, it can be replaced by a simpler definition; the kernel of a homomorphism f is the preimage under f of the zero element of the codomain.
		Unrelated to this, if f is any function in any context, then the kernel of f is a certain equivalence relation on the domain of f which is defined in terms of f.
	www.yourencyclopedia.net /Kernel_(mathematics) (531 words)

	Monomorphism - Wikipedia, the free encyclopedia
		In the more general (and abstract) setting of category theory, a monomorphism (also called a monic morphism) is a morphism f : X → Y such that
		However, whereas the difference is more notable in the case of epimorphisms, in "most" naturally occurring categories of algebras the categorical and algebraic meaning coincide because in any concrete category with a free object on a one element set the categorical monomorphisms are all one-to-one.
		Early category theorists argued that the correct category-theoretic generalization of injective (one-to-one) was the definition of monomorphism given above, and simply gave the word this new, somewhat different, meaning; this made the term ambiguous.
	www.wikipedia.org /wiki/Monic_morphism (442 words)

	Science Fair Projects - Epimorphism
		In the more general (and abstract) setting of category theory, an epimorphism (also called an epic morphism) is a morphism f : X → Y such that
		In the category of monoids, Mon, the inclusion function N → Z is a non-surjective monoid homomorphism, and hence not an algebraic epimorphism.
		In the category of rings, Ring, the inclusion map Z → Q is a categorical epimorphism but not an algebraic one.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Epimorphism (398 words)

	Universal Property [Definition]
		Suppose D is a category with zero morphisms In category theory, a zero morphism is a special kind of "trivial" morphism.
		Let J and C be categories with J small Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		The use of this phrase does not mean that mathematicians consider category theory to be fuzzy or non-rigorous, merely that a small minority consider it too abstract to be useful or interesting....
	www.wikimirror.com /Universal_property (398 words)

	Citations: Category Theory for Computing Science - Barr, Wells (ResearchIndex)
		Definition 7.1.5 Let F : P P be an endofunctor on a category P. co algebra for F is a pair (p, m) of an object and a morphism of P such that m : p F (p) Dually, an algebra for F is a pair (p, m) such that m : F (p) p.
		A monoidal category is a category C with a functor : C C C, called the tensor product, an object I in C, called the tensor unit, and natural isomorphisms a, l and r with components B) B I A satisfying the coherence conditions given....
		Definition 1.5 Let F : P P be an endofunctor on a category P. coalgebra for F is a pair (p; m) of an object and a morphism of P such that m : p F (p) Dually, an algebra for F is a pair (p; m) such that m : F (p) p.
	citeseer.lcs.mit.edu /context/1728/0 (398 words)

	Image (category theory) -- Facts, Info, and Encyclopedia article
		Given a (A general concept that marks divisions or coordinations in a conceptual scheme) category C and a (Click link for more info and facts about morphism) morphism in C, the image of f is a (Click link for more info and facts about monomorphism) monomorphism satisfying the following:
		Image (category theory) -- Facts, Info, and Encyclopedia article
	www.absoluteastronomy.com /encyclopedia/I/Im/Image_(category_theory).htm (140 words)

	Normal
		In category theory: a normal morphism is a morphism that arises as the kernel or cokernel of some other morphisms.
		In algebra (in particular, group theory): a normal subgroup is a subgroup that is invariant under conjugation.
		In topology: a normal space is a topological space in which disjoint closed sets can be separated by neighborhoods.
	www.theezine.net /n/normal.html (140 words)

	Descent Equivalence (ResearchIndex)
		Abstract: For a C-indexed category A, an A -descent equivalence is a morphism of bundles in C which induces an equivalence between the A -descent categories of its domain and codomain.
		Key words: internal category, indexed category, (effective) A -descent morphism, A...
		In this note, properties of such morphisms are studied, and those morphisms which are A -descent equivalences for all C-indexed categories A are fully characterized.
	citeseer.ist.psu.edu /417670.html (277 words)

	Limit and Colimit
		Bringing in this implied morphism, the third side of the triangle, does not affect commutativity, and the new edge becomes the third morphism in our new category.
		As you might surmise, we can create yet another category on d by letting the arrows run from g out to v.
		If d is merely g with no edges, the colimit is the coproduct in the original category.
	www.mathreference.com /cat,limit.html (277 words)

	Kernel (mathematics) - Wikipedia, the free encyclopedia
		Kernels in abstract algebra are general constructions which measure the failure of a homomorphism or function to be injective.
		The kernel pair of a morphism f is defined as a pullback of f with itself.
		In set theory, the kernel of a function f : X → Y is an equivalence relation on X which is defined in terms of f.
	en.wikipedia.org /wiki/Kernel_(mathematics) (277 words)

	Normal morphism - Wikipedia, the free encyclopedia
		In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism.
		The category of abelian groups is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.
		A normal category is a category in which morphisms are normal, whenever reasonable.
	en.wikipedia.org /wiki/Binormal (292 words)

	Product (category theory) - Wikipedia, the free encyclopedia
		In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces.
		The product construction given above is actually a special case of a limit in category theory.
		Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
	en.wikipedia.org /wiki/Product_(category_theory) (292 words)

	Pre-Abelian category - Wikipedia, the free encyclopedia
		Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.
		In mathematics, specifically in category theory, a pre-Abelian category is an additive category that has all kernels and cokernels.
		For example, in the category of topological Abelian groups, the image of a morphism actually corresponds to the inclusion of the closure of the range of the function.
	en.wikipedia.org /wiki/Pre-Abelian_category (882 words)

	Kernel (category theory)
		In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra.
		That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa.
		The dual concept to that of kernel is that of cokernel.
	www.worldhistory.com /wiki/K/Kernel-(category-theory).htm (888 words)

	The Dimensional Ladder
		Categories of Mathematical Objects Definition of category There are many categories of mathematical gadgets, but we'll consider three: Set, Vect and Top, the latter two because they don't arise very quickly from category theory itself.
		Functors Definition Examples: a functor from the free category on an object, a morphism, an isomorphism, an endo, an auto Example: a functor from a group G to Set is a set acted on by G, or G-set.
		Quantum Groups algebras, coalgebras, bialgebras (in a general monoidal category) the category of representations of an algebra the monoidal category of representations of a bialgebra the braided monoidal category of representations of a quasitriangular bialgebra the symmetric monoidal category of representations of a triangular bialgebra the monoidal (resp.
	math.ucr.edu /home/baez/hda/dimensional_ladder.html (2262 words)

	Abstract algebra:Category theory - Wikibooks
		Category theory is the study of categories, which are collections of objects and morphisms (or arrows), or from one object to another.
		The category whose objects are smooth (differentiable,topological) manifolds, and morphisms are smooth (differentiable,continuous) maps.
		is a category with the same objects, and all the arrows reversed.
	en.wikibooks.org /wiki/Abstract_algebra:Category_theory (309 words)

	Injective cogenerator
		In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality.
		For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a free abelian group).
		As an example of a cogenerator in the same category, we have Q/Z, the rationals modulo the integers, which is a divisible abelian group.
	www.guajara.com /wiki/en/wikipedia/i/in/injective_cogenerator.html (557 words)

	Kernel (category theory) - InfoSearchPoint.com
		In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra.
		That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa.
		Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fixed ring (including vector spaces over a fixed field).
	www.infosearchpoint.com /display/Kernel_(categories) (822 words)

	Kernel (category theory) : Kernel of a morphism
		In category theory and its applications to other branches of mathematics, a kernel is a type of limit that generalises the notion of kernel from algebra in certain contexts.
		That is, the kernel of a morphism is its cokernel in the opposite category[?], and vice versa.
		Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fixed ring (including vector spaces over a fixed field).
	www.fastload.org /ke/Kernel_of_a_morphism.html (851 words)

	PlanetMath: examples of initial objects and terminal objects and zero objects
		Similarly, the category of all small categories with functors as morphisms has the empty category as initial object and the one-object-one-morphism category as terminal object.
		(Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
		The same is true for the category of abelian groups as well as for the category of modules over a fixed ring.
	planetmath.org /encyclopedia/TerminalObjectsAndZeroObjectsExamplesOfInitialObjects.html (851 words)

	Product (category theory) - Wikipedia, the free encyclopedia
		In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces.
		The product construction given above is actually a special case of a limit in category theory.
		Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
	en.wikipedia.org /wiki/Product_(category_theory) (399 words)

	Product (category theory)
		In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces.
		The product construction given above is actually a special case of a limit in category theory.
		Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
	www.freeglossary.com /Categorical_product (399 words)

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Topic: Morphism (category theory)

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