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Topic: Identity morphisms

	Comma category - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-22)
		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.
		In either of these two cases, the identity functor may be replaced with some other functor; this yields a family of categories particularly useful in the study of adjoint functors.
		The notion of a universal morphism to a particular colimit, or from a limit, can be expressed in terms of a comma category.
	en.wikipedia.org /wiki/Comma_category (1153 words)

	Morphism (Site not responding. Last check: 2007-10-22)
		In the context of universal algebra morphisms are generically known as homomorphisms.
		The abstract study of morphisms and the spaces (or objects) on which they are defined forms a branch of mathematics called category theory.
		Despite the abstract nature of morphisms, most people's intution about them (and indeed much of the terminology) comes from the case of the so-called concrete categories where the objects are simply sets with some additional structure and morphisms are functions preserving this structure.
	www.sciencedaily.com /encyclopedia/morphism_1 (848 words)

	Identity (disambiguation)
		In mathematics, the term identity, in distinction to an equation, refers to a equational statement that does always hold.
		For security systems, authentication identity refers to the characteristics that a person has that can be verified.
		In philosophy of mind the identity theory of mind holds that the mind is identical to the brain.
	pedia.newsfilter.co.uk /wikipedia/i/id/identity__disambiguation_.html (211 words)

	Comma category - Encyclopedia, History, Geography and Biography
		The morphisms from $(\alpha, \beta, f)$ to $(\alpha', \beta', f')$ are pairs $(g, h)$ where $g : \alpha \rightarrow \alpha'$ and $h : \beta \rightarrow \beta'$ are morphisms in $\mathcal A$ and $\mathcal B$ respectively, such that the following diagram commutes:
		Morphisms are composed by taking $(g, h) \circ (g', h')$ to be $(g \circ g', h \circ h')$ , whenever the latter expression is defined.
		In a similar way, morphisms like $(g, h) : (B, i_B) \rightarrow (B', i_{B'})$ reduce to simply $h : B \rightarrow B'$ , as $g$ is just the identity morphism on $A$ .
	www.arikah.net /encyclopedia/Coslice_category (1527 words)

	Category (mathematics) - Encyclopedia, History, Geography and Biography
		The morphisms of a category are sometimes called arrows due to the influence of commutative diagrams.
		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.
		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.
	www.arikah.net /encyclopedia/Category_%28mathematics%29 (1213 words)

	Identity (disambiguation) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the term (The distinct personality of an individual regarded as a persisting entity) identity, in distinction to an (A mathematical statement that two expressions are equal) equation, refers to a equational statement that always holds.
		Most commonly identities in this sense are used in (Click link for more info and facts about universal algebra) universal algebra to define a certain class of structures.
		A digital identity can be understood as the set of digital information that is attributable to any given entity.
	www.absoluteastronomy.com /encyclopedia/i/id/identity_(disambiguation).htm (476 words)

	Objects and Morphisms
		The word morphism is derived from the concept of a homomorphism, a group homomorphism or a ring homomorphism or a module homomorphism etc. Yet a morphism is much more general.
		The morphism is completely described by the function on the underlying set, hence the identity map on a set has to be the identity morphism on the corresponding object, and a bijection between sets is usually an equivalence, provided the function and its inverse follow the rules for a morphism in this category.
		The source and target of a morphism m are s(m) and t(m) respectively, and again, s(m) and t(m) are always identity morphisms.
	www.mathreference.com /cat,def.html (1248 words)

	Category theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		The morphisms of a category are sometimes called arrows due to the influence of (Click link for more info and facts about commutative diagram) commutative diagrams.
		Relations among morphisms (such as fg = h) can most conveniently be represented with (Click link for more info and facts about commutative diagram) commutative diagrams, where the objects are represented as points and the morphisms as arrows.
		For example, in the category consisting of two objects A and B, the identity morphisms, and a single morphism f from A to B, f is both epic and monic but is not an isomorphism.
	www.absoluteastronomy.com /encyclopedia/c/ca/category_theory.htm (3054 words)

	Category theory (Site not responding. Last check: 2007-10-22)
		Any monoid forms a small category with a single object x, and where every element of the monoid is a morphism from x to x (the monoid operation yields the categorical composition of morphisms).
		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.
		A morphism from (X,x) to (Y,y) is given by a continuous map f : X → Y with f(x) = y.
	www.sciencedaily.com /encyclopedia/category_theory (3261 words)

	FUNCTOR FACTS AND INFORMATION (Site not responding. Last check: 2007-10-22)
		A category with a single object is equivalent to a monoid whose elements are morphisms and whose operation is composition.
		Morphisms in this category are natural_transformations between functors.
		Functors are often defined by universal properties; examples are the tensor_product, the direct_sum and direct_product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
	www.witwib.com /functor (1527 words)

	Functor category - Wikipedia, the free encyclopedia
		their morphism sets are abelian groups and the composition of morphisms is bilinear), then we can consider the category of all additive functors from C to D, denoted by Add(C,D).
		A directed graph consists of a set of arrows and a set of vertices, and two functions from the arrow set to the vertex set, specifying each arrow's start and end vertex.
		The category Cat of all small categories with functors as morphisms is therefore a cartesian closed category.
	www.wikipedia.org /wiki/Functor_category (975 words)

	Identity (disambiguation) - Enpsychlopedia (Site not responding. Last check: 2007-10-22)
		In mathematics, an identity, as distinguished from an equation, is an equality that always holds regardless of the values of any variables that appear in it.
		Digital identity is the representation of identity in terms of digital information.
		In object-oriented programming, identity (object-oriented programming) is a property of objects that allows those objects to be distinguished from each other.
	www.grohol.com /psypsych/Identity_%28disambiguation%29 (392 words)

	Category theory - FreeEncyclopedia (Site not responding. Last check: 2007-10-22)
		Furthermore, different such constructions are often "naturally related" which leads to the concept of natural transformation, a way to "map" one functor to another.
		The morphisms in this category are the continuous maps.
		Another important duality occurs in functional analysis: the category of commutative C-algebras with identity* is contravariantly equivalent to the category of compact Hausdorff spaces.
	openproxy.ath.cx /ca/Category_theory.html (2075 words)

	[No title]
		There is a class of rings known as "rings with identity", such as the "rings with polynomial identity" in which there is an equation of several variables which is to hold for all choices of elements in the ring -- for example, commutative rings are rings satisfying the identity xy-yx=0.
		In the category of rings-with-multiplicative-identity, (homo-) morphisms should carry the identity in the domain to the identity in the co-domain; in particular, a subring (in this category) would have to have the same identity element as the big ring.
		Each of these is an associative algebra with identity over a field F, and is created from a vector space of dimension n and resulting in an algebra of dimension 2^n.
	www.math.niu.edu /~rusin/known-math/95/division.alg (8396 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		We call N the domain (object) and O the codomain of the morphism N --> O. Notice that we may compose two morphisms only when it makes sense to do so: the codomain of the first morphism must be the domain of the second morphism.
		A category in which every morphism is an isomorphism is known as a groupoid.
		A preorder in which the identity morphisms are the only isomorphisms is called a partially ordered set.
	www.math.uregina.ca /~funk/cat.html (250 words)

	PlanetMath: category
		0 is the empty category with no objects or morphisms, 1 is the category with one object and one (identity) morphism.
		Top is the category of all small topological spaces with morphisms continuous functions
		Grp is the category of all small groups whose morphisms are group homomorphisms
	planetmath.org /encyclopedia/Category.html (223 words)

	[No title]
		This fun* ctor is not a (strict) morphism* of clubs since it does not strictly preserve the ope* *rational substitution which is part of the club structure.
		In the case wh* ere A is a strict symmetric monoidal category, this lax functor is in fact a pseudofu *nctor, that associated to A in [Th2] Appendix.
		By [Se] x3 there is a functor is the opposite di* rection, with the two composites linked to the identity* by stable homotopy equivalences.* * - From this it follows that it suffices to link F Sg to Sg by a chain of natural stabl* *e homotopy equivalences of functors from special -spaces to Spectra0.
	hopf.math.purdue.edu /Thomason/thomason_SymMon_equals_Spectra.txt (7868 words)

	[No title]
		B, the fiber Eb over an object b 2 B is the subcategory of objects and morphisms of E that map to b and its identity morphism.
		Observe that the morphism space E (x; y) must be topologized as a subspace of B(ss(x); ss(y)) when E is faithful.
		This implies that the identity functor of UG (n) is itself th* *e uni- versal orientable complex representation.
	hopf.math.purdue.edu /Costenoble-May-Waner/CMWFinal.txt (16893 words)

	Categories and functors for the structural-phenomenol,ogical modeling (Site not responding. Last check: 2007-10-22)
		A collection of phenomenological senses is a category if there exists morphisms among these objects, and if the composition map of morphisms and the identity morphism, like for any category [11], are respected.
		The phenomenological morphism may be conceived, in an abstract way, as a non-structural, non-formal, process of transformation from one phenomenological sense to another.
		Each state is a structure, a set, and the morphisms between the objects are therefore also functions (from a functional point of view, relations and functions among sets were named formal functions [10]).
	www.racai.ro /MD-Web/Categories.html (2762 words)

	Category Theory
		What matters is the way an object is related to the other objects of the category, that is, the morphisms going in and the morphisms going out, or, put differently, how certain structures can be mapped into it and how it can map its structure into other structures of the same kind.
		This simply means that, given two categories C and D, a functor F from C to D, should send objects of C to objects of D and morphisms of C to morphisms of D in such a way that composition of morphisms in C is preserved, i.e.
		For instance, given a category C, there is always the identity functor from C to C which sends every object of C to itself and every morphism of C to itself.
	www.science.uva.nl /~seop/archives/win2003/entries/category-theory (3074 words)

	Commutative Diagrams
		Within the context of a category, a diagram is a digraph with objects as vertices and morphisms as edges.
		c, there is an implied morphism from a to c, produced by composing the two morphisms in the chain.
		The composition is the identity map, the morphisms are inverses, and the points are equivalent.
	www.mathreference.com /cat,cdiag.html (506 words)

	MORPHISMS
		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.websters-online-dictionary.org /definition/MORPHISMS (2839 words)

	Category theory
		Any partially ordered set (P, ≤) forms a small category, where the objects are the members of P, and the morphisms are arrows pointing from x to y precisely when x ≤ y.
		For example, every singleton (set with one element) is a terminal object in the category of sets.
		A morphism from (X,x) to (Y,y) is given by a continuous map f : X
	www.fact-index.com /c/ca/category_theory.html (2886 words)

	Category theory : Identity morphism (Site not responding. Last check: 2007-10-22)
		X called the identity morphism for X, such that for every morphism f : A -> B we have id
		If C and D are categories, one can form the product category C x D: the objects are pairs consisting of one object from C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D.
		It uses material from the wikipedia article Category theory : Identity morphism.
	www.eurofreehost.com /id/Identity_morphism_3.html (486 words)

	Category Theory
		For every pair a, b of objects, there is a collection Mor(a, b), namely, the morphisms from a to b in C (when f is a morphism from a to b, we write f: a → b);
		Identity: if f: a → b, then (idb o f) = f and (f o ida) = f.
		For, given two elements p, q of the preordered set, there is a morphism f: p → q if and only if p is less than or equal to q.
	plato.stanford.edu /entries/category-theory (7029 words)

	[No title]
		Omitting identity morphisms, our little category C looks like this: S --------> V E --------> T Now let's work out what a presheaf on C is. It's a contravariant functor F: C -> Set.
		We could list all the morphisms and the rules for composing them explicitly, but there is a much slicker way to describe them.
		Namely, it's a morphism F: I x C -> C' for which the composite i F C ----> I x C ----> C' equals f, and the composite j F C ----> I x C ----> C' equals g.
	www.math.niu.edu /%7Erusin/known-math/98/homotopy_thy (3995 words)

	Categories
		Given a morphism g in Hom(X,Y) and a morphism f in Hom(Y,Z), there is morphism which we call fg in Hom(X,Z).
		The objects in the category Tang are {0,1,2,...} and the morphisms in Hom(m,n) are (isotopy classes of) tangles with m strands going in and n strands coming out.
		That means that in addition to objects and morphisms, it has "2-morphisms", that is, morphisms between morphisms.
	math.ucr.edu /home/baez/categories.html (2546 words)

	MATHS: Category Theory
		Since the kernel of the identity map for #{a,b} is {()} we say that #{a,b} is free of all constraints.
		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)

	Physics Help and Math Help - Physics Forums - isham new quantization
		02-12-2004 12:55 PM a category is a collection of objects along morphisms between the objects such that the morphisms satisfy associativity (with the appropriate domains and codomains) and for every object there is an identity morphism.
		it is possible to forget the objects and work only with the morphisms since identity morphisms are in one to one correspondence with objects.
		a morphism is called an isomorphism if there is a left and right inverse (with appropriate domains and codomains).
	www.physicsforums.com /printthread.php?t=14371&pp=40 (1217 words)

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