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Topic: Pullback (category theory)

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	An information theory Formalisation - IS Foundations Workshop - 1999
		Category Theory is an abstract mathematics aimed at universality.
		Category theory is based on graphs with nodes of objects and edges of arrows.
		The hidden power of this mathematics of category theory is that the assertions have extraordinarily wide application by virtue of their simplicity, their weakness, and thus their generality.
	www.comp.mq.edu.au /isf99/DampneyJohnson.htm (4038 words)

	Pullback (category theory) - Wikipedia, the free encyclopedia
		In category theory, a branch of mathematics, a pullback (also called a fibered product or cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain.
		The associated commutative diagram is a morphism of fiber bundles.
		The categorical dual of a pullback is a called a pushout.
	en.wikipedia.org /wiki/Pullback_(category_theory) (400 words)

	PlanetMath: categorical pullback (Site not responding. Last check: 2007-11-07)
		, or a pullback square, is a commutative diagram
		Pullbacks and pushouts are unique up to unique isomorphism when they exist.
		This is version 2 of categorical pullback, born on 2004-02-14, modified 2004-02-14.
	planetmath.org /encyclopedia/PullbackSquare.html (116 words)

	Categories Home Page
		Category Theory 2000 The international summer conference in category theory was held at Villa Olmo, Como, Italy from Sunday 16th July to Saturday 22nd July 2000.
		Category Theory Symposium The symposium was a special session of the Canadian Mathematical Society Summer 1998 Meeting June 13-15, 1998 at University of New Brunswick (Saint John) Saint John, New Brunswick, Canada and was part of a larger programme including plenary talks, one of which was given by S. Schanuel (SUNY Buffalo).
		Category Theory at the Isle of Thorns was held from July 7 to 12, 1996.
	www.mta.ca /~cat-dist (3516 words)

	The Dimensional Ladder
		Categories from Spaces The fundamental groupoid of a topological space The fundamental group of a pointed space categories from chain complexes: a 2-term chain complex is a category in AbGp.
		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.
		Quivers and Dynkin Diagrams (move to "Categories as Theories", section on "Category Representations".) Representations of quivers: proof that quivers containing certain "bad" subquivers have wild representation type, by noting that the dimension of the space of indecomposable representations exceeds the dimension of the space of intertwiners.
	math.ucr.edu /home/baez/hda/dimensional_ladder.html (2262 words)

	Good Math, Bad Math
		This is one of the last posts in my series on category theory; and it's a two parter.
		Category theory provides a good framework for defining linear logic - and for building a Curry-Howard style type system for describing computations with state that evolves over time.
		For me, the frustrating thing about learning category theory was that it seemed to be full of definitions, but that I couldn't see why I should care.
	scienceblogs.com /goodmath/goodmath/category_theory (1369 words)

	Domain theory (Site not responding. Last check: 2007-11-07)
		Scott noticed in 1969 that in the category of complete lattices and maps preserving directed joins, limit of a sequence of projections (maps with preinverse left adjoints) is isomorphic to the colimit of those left adjoints (called embeddings).
		One of the objectives of category theory is to provide a foundation for itself in particular and mathematics in general which is independent of the traditional use of set theory.
		The answer to this depends upon first being able to express the notion of a ``family'' of sets, and indexed category theory was developed for this purpose.
	www.cs.man.ac.uk /~pt/domains (1096 words)

	Goldblatt. Topoi: The Categorial Analysis of Logic (Site not responding. Last check: 2007-11-07)
		These categories are examples of preorders: categories in which there is at most one arrow between any two objects (identity corresponds to reflexivity and composition corresponds to transitivity).
		Pullbacks of two arrows with a common codomain is probably the most important categorial construction.
		A category is complete if it has all limits, co-complete if it has all colimits, bi-complete if it is complete and co-complete, finitely complete if it has all finite limits, finitely co-complete if it has all finite colimits, and finitely bi-complete if it is finitely complete and finitely co-complete.
	www.andrew.cmu.edu /user/cebrown/notes/goldblatt.html (7165 words)

	Re: Cobig, Coproduct, and Comma (Site not responding. Last check: 2007-11-07)
		He used them in --, The category of categories as a foundation for mathematics, Proceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer-Verlag, New York.
		I gave a brief calculus of comma categories in: --, The categorical comprehension scheme, Category theory, Homology theory and their Applications III, Lecture Notes in Mathematics 99, Springer-Verlag, New York 1969, 242-312.
		The general theory of the properties of lax limits in 2-categories was discussed independently by Street and me in various publications.
	www.cis.upenn.edu /~bcpierce/types/archives/1989/msg00038.html (334 words)

	Pullbacks (Site not responding. Last check: 2007-11-07)
		1) " Pullbacks" -- In the context of Pullbacks
		Fibration 16: nder composition and pullback (category theory)pullbacks.
		Pullback (category theory) 19: sub> are the natural projections.
	www.lottery-news.net /dust6611-pullbacks.html (120 words)

	Wikinfo \| Functor (Site not responding. Last check: 2007-11-07)
		In category theory, a functor is a special type of mapping between categories.
		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.wikinfo.org /wiki.php?title=Functor (1552 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Functors between categories "& all that" didn't play an obvious role, although I think it was of psychological help to know that (most of) what I was doing could be placed in the birational category, where up to birational equivalence the singularities aren't a problem (algebraic varieties are birationally equivalent to nonsingular ones).
		I suppose some of it could be interpreted as relating the category of varieties over the rationals with the category of varieties over the rationals with a G action on them, and the group action can itself be thought of as a functor.
		It would not be a sensible task to rewrite the paper without category theory (considered as a unifying principle, as a mode for efficient calculation in certain algebraic structures, and as a supplier of new algebraic structures).
	www.math.niu.edu /~rusin/known-math/00_incoming/why_cat (1202 words)

	The Rising Sea » Mathematics Notes
		Derived Functors: (DF) (co)chain complexes in an abelian category, (co)homology, projective and injective resolutions, left and right derived functors of additive functors between abelian categories, long exact (co)homology sequences, long exact sequences of derived functors, dimension shifting and acyclic resolutions, change of base, homology and colimits, cohomology and limits, delta functors.
		Ext: (EXT) Ext in general abelian categories, using injectives and projectives and balancing the two, Ext for linear categories, dimension shifting, Ext and coproducts, Ext for commutative rings, another characterisation of derived functors.
		Linearised Categories: (LC) Generalise the group ring construction to the linearisation of any small category with respect to a sheaf of rings, the graded version of this construction.
	therisingsea.org /?page_id=3 (1462 words)

	Limit (category theory) - Wikipedia, the free encyclopedia (via CobWeb/3.1 planet2.scs.cs.nyu.edu) (Site not responding. Last check: 2007-11-07)
		In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits.
		Consequently, the category J is usually a small category and has fewer elements than the category C.
		Typical examples of categories that are not complete are categories with some "size restriction": the category of finite groups or the category of finite-dimensional vector spaces over a fixed field.
	en.wikipedia.org.cob-web.org:8888 /wiki/Limit_(category_theory) (1960 words)

	Good Math, Bad Math : Arrow Equality and Pullbacks
		A pullback is a way of describing a kind of equivalence of arrows, which gets used a lot in things like interesting natural transformations.
		But, before we get to pullbacks, it helps to understand the equalizer of a pair of morphisms, which is a weaker notion of arrow equivalence.
		One of the things that's hardest to wrap your head around with category theory is that the deep structure isn't there: it's deliberately been abstracted away.
	scienceblogs.com /goodmath/2006/07/arrow_equality_and_pullbacks_1.php (1135 words)

	UCSC General Catalog 2006-08 - Programs and Courses
		Field theory: field extensions, algebraic and transcendental extensions, splitting fields, algebraic closures, separable and normal extensions, the Galois theory, finite fields, Galois theory of polynomials.
		The goal is to provide the student with foundations suitable for further work in advanced number theory, in conformal field theory, and in the theory of Riemann surfaces.
		The aim of this course is to acquaint the participants with basic concepts of category theory and homological algebra, as follows: chain complexes, homology, homotopy, several (co)homology theories (topological spaces, manifolds, groups, algebras, Lie groups), projective and injective resolutions, derived functors (Ext and Tor).
	reg.ucsc.edu /catalog/html/programs_courses/mathCourses.htm (3953 words)

	Abstracts (Site not responding. Last check: 2007-11-07)
		To the best of my knowledge, there has not previously been a full study of any such situation in a model categorical framework, and the theory gives a striking illustration of both the power and the intrinsic limitations of model category theory.
		Pullback base change functors have both left and right adjoints.
		In fact, it is nontrivial to show that both point-set level adjunctions descend to adjunctions relating the derived homotopy categories.
	jdc.math.uwo.ca /homotopy/lectures.php?email=may@math.uchicago.edu (189 words)

	Product (category theory) - Wikipedia, the free encyclopedia (via CobWeb/3.1 planet2.scs.cs.nyu.edu) (Site not responding. Last check: 2007-11-07)
		In category theory, one defines products to generalize constructions such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces.
		These properties are formally similar to those of a commutative monoid; a category with its finite products and terminal object constitutes a symmetric monoidal category.
		A distributive category is one in which this morphism is actually an isomorphism.
	en.wikipedia.org.cob-web.org:8888 /wiki/Product_(category_theory) (635 words)

	Category Theory for Computer Science
		Cartesian closed categories and the simplytyped lambda calculus.
		Using Category Theory to Design Implicit Conversions and Generic Operators.
		Category theory in programming language semantics and design
	www.daimi.au.dk /~nygaard/CTfCS (620 words)

	[Giuseppe.Longo@THEORY.CS.CMU.EDU: Re: the cmu workshop]
		This question can be formalized through results about the relationships (such as conservative extension) between equational theories.
		These form a cartesian closed category which has fixed points for domain equations.
		We also show that our uniformity condition arises already in the restricted setting of coherent spaces and embeddings: it is equivalent to the pullback condition for elements of variable type, an essential ingredient in Girard's coherent semantics.
	www.cis.upenn.edu /~bcpierce/types/archives/1988/msg00043.html (1437 words)

	[No title]
		One problem in the usual version of the Pro category of towers is that, whil* e fi- nite limits and colimits exist, and may be constructed in a straightforward (le velwise) manner, the same does not hold for infinite colimits; and these were needed for * the application we had in mind in [BT ].
		A category C is called pointed if it has a zero object (i.e., one which is b* *oth initial and terminal).
		Note that in general our category C will not be locally generated b* y the subcategory F, in the sense of [GU, xx7,9], because C need not be cocomplete - and we are interested precisely in such cases, because only then will the cocomplet *ion of C be of interest.
	www.math.purdue.edu /research/atopology/Blanc/Blanc_towers.txt (5443 words)

	Preliminary Examinations And Basic Graduate Sequences
		Students in the Ph.D. program who do not pass the examinations by the end of their third year should expect to be transferred to the M.A. program, or be subject to dismissal.
		Review of the basic theory of one complex variable, the Cauchy-Riemann equations, Cauchy's theorem, power series expansions, the maximum modulus principle, Classification of singularities, Residue theorem, argument principle, harmonic functions, linear fractional transformations, Conformal mappings, Riemann mapping theorem, Picard's theorem, introduction to Riemann surfaces.
		Theory of manifolds: Definitions of manifolds, tangent bundle, inverse and implicit function theorems, transversality, Sard’s theorem and the Whitney embedding theorem, differential forms, exterior derivative, Stokes’ theorem, integration, vector fields, flows, Lie brackets, Frobenius’ theorem
	www.math.ucsc.edu /graduate/preexam_seq.html (1021 words)

	Citations: Practical Foundations of Mathematics - Taylor (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		The theory could be developed using both sytactic and semantic hyperdoctrines with more notation but essentially along the same lines.
		A Study of Categories of Algebras and Coalgebras - Hughes (2001)
		See also [Coc93] for a discussion of extensive categories, a subject we return to in Section 4.1, where we show that E # is extensive, given that E is extensive and # preserves regular monos and pullbacks along regular monos.
	citeseer.ist.psu.edu /context/1320546/0 (759 words)

	2-Equivariance and "Weak Pullback" \| The String Coffee Table
		Maybe my use of ‘pullback’ is a red herring, I wonder.
		This lead me, maybe unwisely, to mention the term pullback, even though there is no pullback cone involved here.
		So the triangle diagram satisfied by them lives in the functor category, which, for the case I discussed, is a 1-category.
	golem.ph.utexas.edu /string/archives/000674.html (1503 words)

	An Adjoint Characterization of the Category of Sets - Rosebrugh, Wood (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		We will see that it suffices to assume that B has an adjoint string V a W a X a Y with V pullback preserving.
		We have expressed the construction in a form we believe begs generalization to, for example,...
		Robert Rosebrugh and Richard Wood, An adjoint characterization of the category of sets, Proceedings AMS 122 (1994) 409-413.
	citeseer.ist.psu.edu /166172.html (515 words)

	Category Theory Definition List
		These definitions can all be found in Herrlich and Strecker's nice introduction to Category Theory.
		If you came here hoping to find some definitions instead of lists of terms, see my page of links to on-line category thory resources.
		Natural Functor from a Category to its Quotient Category
	www.maths.tcd.ie /~icecube/Cats/defs.html (48 words)

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