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Topic: Category of pointed spaces

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	PlanetMath: category of pointed topological spaces
		This yields a functor from the category of pointed topological spaces to the category of groups.
		"category of pointed topological spaces" is owned by mathcam.
		This is version 6 of category of pointed topological spaces, born on 2003-10-15, modified 2006-10-07.
	planetmath.org /encyclopedia/BasedTopologicalSpace.html (193 words)

	Pointed space - Wikipedia, the free encyclopedia
		Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint.
		The pointed set concept is less important; it is anyway the case of a pointed discrete space.
		The coproduct in the category of pointed spaces is the wedge sum, which can be thought of as the one-point union of spaces.
	en.wikipedia.org /wiki/Pointed_space (416 words)

	PlanetMath: examples of initial objects and terminal objects and zero objects
		The same is true for the category of abelian groups as well as for the category of modules over a fixed ring.
		In the category of graphs, the null graph is an initial object.
		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.
	planetmath.org /encyclopedia/TerminalObjectsAndZeroObjectsExamplesOfInitialObjects.html (616 words)

	Functor
		We thus obtain a functor from the category of pointed topological spaces to the category of groups.
		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 category with a single object is equivalent to a monoid whose elements are morphisms and whose operation is composition.
	www.brainyencyclopedia.com /encyclopedia/f/fu/functor.html (1725 words)

	[No title]
		TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 21 Lemma 2.16.
		K(F3) TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 29 is null homotopic.
		K(Z; 9) TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 49 is fi1Sq4i1.
	hopf.math.purdue.edu /Rognes/whdiff.txt (15434 words)

	Online Encyclopedia and Dictionary - Category of topological spaces (Site not responding. Last check: 2007-10-23)
		This is a category because the composition of two continuous maps is again continuous.
		The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.
		The coproduct is given by the disjoint union of topological spaces.
	www.fact-archive.com /encyclopedia/Category_of_topological_spaces (383 words)

	Bar Constructions
		Let us define a contractive space to be a space (in a suitable topological category, such as the category of compactly generated Hausdorff spaces) which is an algebra over the cone monad.
		Here, the cone monad means the mapping cone of the map X --> 1 into the one-point space, and this is the monad whose algebras are pointed spaces equipped with a continuous action by the unit interval I, the monoid whose multiplication is "inf", such that multiplication by 0 sends every point to the basepoint.
		An algebra structure may be viewed as a well-behaved homotopy which contracts the space to a point.
	math.ucr.edu /home/baez/universal/bar.html (2334 words)

	Initial object Information
		In the category of pointed sets (whose objects are non-empty sets together with a distinguished element; a morphism from (A,a) to (B,b) being a function f : A → B with f(a) = b), every singleton is a zero object.
		In the category of interpretations of an algebraic model, the initial object is the initial algebra, the interpretation that provides as many distinct objects as the model allows and no more.
		In the category of schemes, the prime spectrum of Z is a terminal object.
	www.bookrags.com /wiki/Initial_object (1037 words)

	Based map - Basedmap
		In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X.
		The pointed set concept is less important; it is anyway the case of a pointed discrete space.
		This functor has a left adjoint which assigns to each topological space X the disjoint union of X and a one point space {-} whose single element is taken to be the basepoint.
	www.kopete.org /Based-map.html (279 words)

	physics - Pointed space
		Maps of pointed spaces are continuous maps preserving basepoints, i.e.
		This functor has a left adjoint which assigns to each topological space X the disjoint union of X and a one point space {•} whose single element is taken to be the basepoint.
		A subspace of a pointed space X is a topological subspace A ⊆ X which shares its basepoint with X so that the inclusion map is basepoint preserving.
	www.physicsdaily.com /physics/Pointed_space (404 words)

	Pointed space (Site not responding. Last check: 2007-10-23)
		Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e.
		Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint.
		The coproduct in the category of pointed spaces is the wedge sum, which can be thought of as the one-point union of spaces.
	www.punweb.com /article/Pointed_topological_space (409 words)

	Functor - Iridis Encyclopedia (Site not responding. Last check: 2007-10-23)
		We thus obtain a functor from the category of pointed topological spaces to the category of groups.
		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 category with a single object is equivalent to a monoid whose elements are morphisms and whose operation is composition.
	www.iridis.com /Functor (1458 words)

	[No title]
		As a consequence, the homotopy category associated to the coarse structure is essentially a subcategory of tha* *t as- sociated to the fine structure.
		Informally, a topological model category is a model category which is enriche* d, tensored and cotensored over o in a way which interacts well with the model structure.
		The pointed space M is considered as a poi* nted G-space with trivial action, all mapping spaces* are considered with actions given by conjugation, and all products have diagonal actions.
	hopf.math.purdue.edu /Intermont-JohnsonM/ijxspace.txt (4457 words)

	Springer Online Reference Works
		is the unit interval and the slant line denotes the operation of identifying a subspace with one point.
		The suspension functor and the loop space functor on the category of pointed spaces are adjoint:
		This adjointness is compatible with the homology and thus also defines an adjunction for the category of pointed topological spaces and homotopy classes of mappings.
	eom.springer.de /s/s091490.htm (185 words)

	Category of topological spaces -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-23)
		This is a category because the (A mixture of ingredients) composition of two continuous maps is again continuous.
		The study of Top and of properties of ((mathematics) any set of points that satisfy a set of postulates of some kind) topological spaces using the techniques of (Click link for more info and facts about category theory) category theory is known as categorical topology.
		The category of (Click link for more info and facts about pointed topological space) pointed topological spaces is a (Click link for more info and facts about coslice category) coslice category over Top
	www.absoluteastronomy.com /encyclopedia/c/ca/category_of_topological_spaces.htm (455 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		That space, which I described as a quotient space of the disc, was really the result of taking a circle and glomming a disc onto it in such a way that the edge of the disc wrapped around the circle 5 times.
		As Rusin points out, K(G,n) is cool because it's a space that knows all about n-dimensional homology with coefficients in the group G. To figure out H_n(X,G) for any space X, we just form the set of homotopy equivalence classes of maps from X to K(G,n), usually written [X,K(G,n)].
		The wedge product is the coproduct in the category of pointed spaces (spaces with one point declared to be a "basepoint") This is simply the coproduct in the category of spaces (or sets), namely disjoint union, modulo an equivalence relation that declares the two basepoints equivalent.
	www.mat.niu.edu /~rusin/known-math/94/holes (2375 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		Gärdenfors' conceptual space formulation has been specifically applied in scene recognition by Chella et al [5] who describe scenes with motion in terms of sets of points in a conceptual space, and interpret linguistic terms as subsets of this space.
		Points in conceptual space are represented as multi-spectral images, where each spectral layer represents a base conceptual space.
		The conceptual space is composed of multiple copies of symbol space DL and concept space D. An attention buffer Att is filled with connected regions of concept space.
	www.newcastle.edu.au /school/design-comm-info/staff/gibbon/iconip02.doc (3294 words)

	Category of topological spaces - Definition, explanation
		In mathematics, the category of topological spaces, often denoted Top, is the category whose objectss are topological spaces and whose morphisms are continuous maps.
		Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e.
		to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function.
	www.calsky.com /lexikon/en/txt/c/ca/category_of_topological_spaces.php (524 words)

	[No title]
		Using one of these structures, one obtains that the localized category is equivalent to the category of n-reduced CW-complexes with dimension less than or equal to m+1 and m-homotopy classes of cellular pointed maps.
		Say that two sketches are K-compatible with respect to base category K just in case in each K-model, the limits for each limit specification in each sketch commute with the colimits for each colimit specification in the other sketch and all limits and colimits are pointwise.
		That is, categories of sketch models have all limits commuting with the sketched colimits and and all colimits commuting with the sketched limits.
	www.mta.ca /~cat-dist/catlist/1999/tac-3 (1480 words)

	Science Fair Projects - Suspension (topology)
		One can also view the suspension as two cones on X glued together at their base (or as a quotient of a single cone).
		Suspension gives rise to a functor, which in rough terms increases dimension of a space by one: it takes an n-sphere to an (n + 1)-sphere for n ≥ 0.
		For sufficiently nice spaces (such as CW complexes) the reduced suspension of X is homotopy equivalent to the ordinary suspension.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Suspension_%28topology%29 (429 words)

	physics - Wedge sum
		In topology, the wedge sum is a "one-point union" of a family of topological spaces.
		Alternatively, the wedge sum can be see as the pushout of the diagram X ← {•} → Y in the category of topological spaces (where {•} is any one point space).
		Van Kampen's theorem gives certain conditions (which are usually fulfilled for well-behaved spaces, such as CW complexes) under which the fundamental group of the wedge sum of two spaces X and Y is the free product of the fundamental groups of X and Y.
	www.physicsdaily.com /physics/Wedge_sum (225 words)

	[No title]
		Its underlying (aspherical) spaces are the sets \overline{\mathcal {M}}^n_0({\mathbb R}) of real points of the moduli space of punctured stable curves of genus zero, which are naturally tiled by Stasheff associahedra.
		When C is the category of pointed topological spaces and G is the suspension, we recover the Bousfield-Friedlander model category of spectra.
		A variant construction requires the unordered points to lie on a cycle $z$ in $V$; this defines a parameterized family of multiplications satisfying the analogue of the WDVV equation.
	www.lehigh.edu /~dmd1/h717 (2063 words)

	[No title]
		We prove that the categories of orthogonal spectra and S-modules are Quillen equivalent and that this equivalence induces Quillen equivalences between the respective categories of ring spectra, of modules over a ring spectrum, and of commutative ring spectra.
		The categories of commutative symmetric ring spectra and commutative orthogonal ring spectra are model categories and are Quillen equivalent.
		In any approach to more general base spaces, one must parametrize changes of fiber representation as one moves around B on the equivariant fundamental groupoid pi(B), which depends on all components of all fixed point spaces and all paths connecting them.
	www.lehigh.edu /dmd1/public/www-data/h17 (2038 words)

	Ottawa Octoberfest 2005
		The endofunctors of the "presheaf" category N obtained by composing the idempotents arising in canonical adjoint cylinders are apparently co-adequate for N-valued logic, at least for N=3.
		Infinitary positive language categories are defined and infintary complements to Robinson consistency from the author's preceding paper are gleaned to present new positive omitting types techniques to infinitary positive fragment higher stratified consistency.
		Thus we consider a category, whose objects are phase spaces, and whose morphisms are canonical transformations (canonical relations).
	aix1.uottawa.ca /~scpsg/Octoberfest05/schedule.html (2397 words)

	Transactions of the American Mathematical Society
		We study this functor from the point of view of orthogonal calculus of functors; we show that it is polynomial of degree
		This space is defined as a colimit of the block structure spaces of projective spaces of finite-dimensional real vector spaces and is closely related to some automorphisms spaces of real projective spaces.
		I Madsen, `On the space of manifold structures for lens spaces', Indian J. of Math.
	www.mathaware.org /tran/2007-359-01/S0002-9947-06-04180-8/home.html (614 words)

	Chu Spaces Live (Site not responding. Last check: 2007-10-23)
		Chu spaces are universal in that all conventionally transformable objects of mathematics are representable by Chu spaces within a single typeless framework.
		The top and bottom points are the top (greatest) and bottom (least) elements of the poset, while the two points in the middle are independent of each other, the whole being in the shape of a diamond.
		The entry at point a and state x is x(a), the value of the homomorphism x at the group element a.
	boole.stanford.edu /live/exercises.html (5754 words)

	Brainstorms: Searching Large Spaces
		In essense Kenyon was hoping the structure of the space of possibilites Omega, was really smaller than all the combinatorial possibilities, because of physical law.
		With respect to David's remarks, issues discussed in the OP of this thread on ISCID bring to the picture a fascinating Darwinian, low-information approach to the "problems" that are mentioned.
		Minor, minor point which I offer as a point of clairfication to pre-empt your less friendly critics from pretending they found a counter example to your work.
	www.iscid.org /boards/ubb-get_topic-f-6-t-000558.html (2031 words)

	The n-Category Café
		On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
		Building a 2-Hilbert space from a finite 2-category C equipped with a 2-functor A: C → U(1)Tor, where U(1)Tor is the 2-group of U(1)-torsors.
		Such algebras (“of observables”, or “of operators”) of field theories (from our point of view: in their decategorified form) are considered as living either in Haag-Kastler nets (in the axiomatic Lorentzian formulation of QFT known as AQFT), or, in Euclidean field theory, in sheaves of algebras (see this).
	golem.ph.utexas.edu /category (5277 words)

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