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Topic: Monic morphism

Related Topics

Monomorphism

Epimorphism

Adjoint functors

Preregular space

Subobject

Isomorphism

Topological space

Kolmogorov space

Saunders Mac Lane

Hausdorff space

Curve

	Morphism - ExampleProblems.com
		In the context of universal algebra morphisms are generically known as homomorphisms.
		Despite the abstract nature of morphisms, most people's intuition about them (and indeed much of the terminology) comes from the case of concrete categories where the objects are simply sets with some additional structure and morphisms are functions preserving this structure.
		Morphisms are often depicted as arrows from their domain to their codomain, e.g.
	www.exampleproblems.com /wiki/index.php/Morphism (886 words)

	Morphism Info - Bored Net - Boredom (Site not responding. Last check: 2007-10-02)
		A category is given by two pieces of data: a class of objects and, for any two objects X and Y, a set of morphisms from X to Y.
		Morphisms are often depicted as arrows between those objects.
		However, any morphism that is both an epimorphism and a section, or both a monomorphism and a retraction, must be an isomorphism.
	www.borednet.com /e/n/encyclopedia/m/mo/morphism_1.html (366 words)

	Morphism - Wikipedia, the free encyclopedia
		In mathematics, a morphism is an abstraction of a structure-preserving mapping between two mathematical structures.
		Morphisms with left-inverses are always monomorphisms, but the converse is not always true in every category; a monomorphism may fail to have a left-inverse.
		Morphisms with right-inverse are always epimorphisms, but the converse is not always true in every category; an epimorphism may fail to have a right-inverse.
	en.wikipedia.org /wiki/Morphism (1149 words)

	Projective and Injective Objects (Site not responding. Last check: 2007-10-02)
		An object p is projective if, for any pair of objects a and b, and any epic morphism f from a to b, and any morphism g from p to b, there is at least one morphism h from p to a such that hf = g.
		The composition hf gives a morphism from p to b, and this morphism agrees with g, at least on the generators of p.
		An object j is injective if, for any pair of objects a and b, and any monic morphism f from a to b, and any morphism g from a to j, there is at least one morphism h from b to j such that fh = g.
	www.mathreference.com /cat,proj.html (268 words)

	Monomorphism
		While this is not exactly true for monic maps, it is very close, so this has caused little trouble, unlike the case of epimorphisms.
		Saunders Mac Lane attempted to make a distinction between what he called monomorphisms, which were maps in a concrete category whose underlying maps of sets were injective, and monic maps, which are monomorphisms in the categorical sense of the word.
		Every morphism in a concrete category whose underlying function is injective is a monomorphism.
	www.bssintranet.com /encyclopedia/Genetics/Monomorphism.php (564 words)

	Monic and Epic Morphisms
		When a morphism is monic it probably implements a monomorphism, and when a morphism is epic it probably implements an epimorphism.
		A morphism is monic if it admits a form of cancellation.
		Whenever two morphisms g and h lead into the source of a third morphism f, and gf = hf, then g = h.
	www.mathreference.com /cat,monic.html (778 words)

	The Mizar abstract of CAT_4
		[a,b]; func pr1(a,b) -> Morphism of C equals :: CAT_4:def 6 (the Proj1 of C).
		[a,b]; func pr2(a,b) -> Morphism of C equals :: CAT_4:def 7 (the Proj2 of C).
		[a,b]; func in1(a,b) -> Morphism of C equals :: CAT_4:def 24 (the Incl1 of C).
	www.cs.ualberta.ca /~piotr/Mizar/mirror/http/JFM/Vol4/cat_4.abs.html (92 words)

	Springer Online Reference Works
		Closed imbeddings of schemes or arbitrary morphisms of affine schemes are affine morphisms; other examples of affine morphisms are entire morphisms and finite morphisms.
		Thus the morphism of normalization of a scheme is an affine morphism.
		Under composition and base change the property of a morphism to be an affine morphism is preserved.
	eom.springer.de /a/a011050.htm (162 words)

	Categories and Relations - Datamaster User's Manual (Site not responding. Last check: 2007-10-02)
		Category structures are often illustrated using diagrams: morphisms are drawn as arrows, and composition is shown as two arrow placed with the head of the first joined to the tail of the second.
		The identity morphisms in A are denoted by ∣A∣.
		Proof: If x is monic, the pullback source of x with itself it isomorhic to R. Since, in the presence of equalizers, covers are epic, the columns of R°R to B are equal, and therefore monic.
	uniquesoftwaredesigns.com /datamaster_demo/doc/user_manual_catrel.html (5463 words)

	PlanetMath: supplemental axioms for an Abelian category
		Every morphism has a kernel and a cokernel.
		Coproducts exist and the coproduct of monics is a monic.
		This is version 7 of supplemental axioms for an Abelian category, born on 2001-12-12, modified 2004-04-07.
	planetmath.org /encyclopedia/Complete8.html (223 words)

	The Mizar abstract of CAT_1
		f) and (for f,g being Element of the Morphisms of C st (the Dom of C).
		g) and (for f,g,h being Element of the Morphisms of C st (the Dom of C).
		f); theorem :: CAT_1:100 for T being Function of the Morphisms of C,the Morphisms of D for F being Function of the Objects of C, the Objects of D st (for c being Object of C holds T.
	markun.cs.shinshu-u.ac.jp /Mirror/mizar.org/JFM/Vol1/cat_1.abs.html (1116 words)

	Morphisms (Site not responding. Last check: 2007-10-02)
		The endomorphism ring of E is isomorphic to an order in either a quadratic number field (if E is ordinary) or a quaternion algebra (if E is supersingular).
		For an integer n, this returns the multiplication by n morphism as an element of the endomorphism ring H=End_K(E).
		In particular, for n = 1 the identity morphism is returned as the map [x, 1, 1, 1], for n = 0 the zero map [0, 0, 1, 0], and for n = (-1) the inversion map [x, 1, - 1, 1].
	www.math.wisc.edu /help/magma/text616.html (1619 words)

	Monic - Wikipedia, the free encyclopedia
		monic morphism - a special kind of morphism in category theory.
		monic polynomial - a polynomial whose leading coefficient is one.
		This disambiguation page lists articles associated with the same title.
	en.wikipedia.org /wiki/Monic (97 words)

	[No title]
		This domain is noncanonical in that $f\ f^{-1}$ will be the identity function but won't be equal to 1.
		morphism: (R -> R, R -> R) -> % ++ morphism(f, g) returns the invertible morphism given by f, where ++ g is the inverse of f..
		morphism: ((R, Integer) -> R) -> % ++ morphism(f) returns the morphism given by \spad{f^n(x) = f(x,n)}.
	wiki.axiom-developer.org /Ore/src (1581 words)

	Amazon.com: "split monic": Key Phrase page (Site not responding. Last check: 2007-10-02)
		We say that f is a split monic if there is a morphism g: B, A for which g f = -idA, and that f is a...
		A monic f is called a split monic if there is a map j:...
		Hence F --M is a split monic, hence TF -ETA is monic, hence TF-4F is monic and by minimality it is an isomorphism.
	www.amazon.com /phrase/split-monic (533 words)

	Citations: Heldermann Verlag - Herrlich, Strecker, Second (ResearchIndex)
		....but the pairs of net morphisms for which the pullback exists are characterized, and the pullback construction is given for these cases.
		We say that the functor r : G R is a re ection of G in R. In case every r G is monic in G, R is said to be monore ective, and r a monore ection.
		E is a collection of sinks, and M is a class of X morphisms such that: 0) each of E and M is closed under compositions with isomorphisms.
	citeseer.ist.psu.edu /context/2788/0 (2641 words)

	MATHS: Category Theory
		Any commutative diagram can be taken as a template or pattern for a set of objects in a new category and then morphisms between diagrams defined as lists of morphisms between corresponding nodes in the diagrams.
		The type of all Subsets (@T) of a given type(T) of element equipped with all mappings between subsets as morphisms is a category.
		A pushout is a kind of loose co-product: a pair of objects have a push out which has a an overlap allowed on a third object.
	www.csci.csusb.edu /dick/maths/math_25_Categories.html (3607 words)

	[No title]
		A morphism 4 Dwyer and Wilkerson from f : V R to g : W R is then a commutative diagram V f R h # k W g R in which "commutativity", of course, means h*g = f ; sometimes we will even refer to f as the "composite" of g and h.
		The object f is monic in the above sense iff (any representative of) f is a group monomorphism, and central iff the image of f lies in the center of G. In particular, the following is true.
		Then ker(f) is a subgroup of V, and f extends uniquely to a monic map g : V = ker(f) R. Proof: Let W be a maximal subgroup of V such that f extends to an object g : V =W R. It is clear that W ker(f).
	hopf.math.purdue.edu /Dwyer-Wilkerson/cohomology-decompositions/jm.txt (6117 words)

	Good Math, Bad Math : A First Glance at Category Theory
		A morphism is an abstraction of the concept of homomorphism, which I talked about a bit when I was writing about group theory.
		To talk about a specific morphism from a to b, we write it as "name : a → b"; as in "f : int → int".
		1b is the identity morphism for the object b: the unique morphism such that for all other arrows f, if f : a → b, then 1b º f = f.
	scienceblogs.com /goodmath/2006/06/a_first_glance_at_category_the.php (3222 words)

	Monomorphism - Article from FactBug.org - the fast Wikipedia mirror site
		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 f o g
		To see this, note that if q o f = q o g for some morphisms f,g: G → Q where G is some divisible abelian group then q o h = 0 where h = f - g (this makes sense as this is an additive category).
		An extremal monomorphism is a monomorphism that has no epimorphism as a first factor, unless that epimorphism is an isomorphism.
	www.factbug.org /cgi-bin/a.cgi?a=59538 (324 words)

	Amazon.com: "linear monic polynomials": Key Phrase page (Site not responding. Last check: 2007-10-02)
		See all pages with references to linear monic polynomials.
		Algebraic interlude matrix: thus, the entries of XI - AI(H) are elements of A or linear monic polynomials in X: X - all -a12.
		l To see how this is done, we consider the elements of Z as roots of linear monic polynomials, namely if a E Z, then a is a root of f (x) = x - a.
	www.amazon.com /phrase/linear-monic-polynomials (396 words)

	ISO monoid counterexamples [Math] (Site not responding. Last check: 2007-10-02)
		Does anyone know of examples of either (1) a monoid monomorphism
		whose kernel is not 1, or (2) of a non-monic morphism of monoids
		> whose kernel is not 1, or (2) of a non-monic morphism of monoids
	www.adras.com /ISO-monoid-counterexamples.t11334-92.html (576 words)

	Abstract (Site not responding. Last check: 2007-10-02)
		Kock and Moerdijk proved that each etendue is generated by a site in which every morphism is monic.
		In this paper we provide an alternative characterisation of etendues in terms of ordered groupoids.
		As a byproduct of our main investigation, we obtain a definition of an inverse semigroup acting on a presheaf of structures which has wide applicability.
	www.math.carleton.ca /~bsteinbg/abstracts/abs33.html (127 words)

	MathAction and Axiom ('', 'mathaction', 'Ore')
		)abbrev domain AUTOMOR Automorphism Automorphism(R:Ring): Join(Group, Eltable(R, R)) with morphism: (R -> R) -> % ++ morphism(f) returns the non-invertible morphism given by f.
		compiling exported morphism : (R,Integer) -> R -> $ AUTOMOR;morphism;M$;9 is replaced by f Time: 0 SEC.
		compiling exported morphism : R -> R -> $ Time: 0 SEC.
	wiki.axiom-developer.org /Ore (2906 words)

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