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Topic: Pushout category theory

	Pushout (category theory) - Wikipedia, the free encyclopedia
		In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain.
		The pushout is the categorical dual of the pullback.
		The pushout of f and g is the union of X and Y together with the inclusion morphisms from X and Y.
	en.wikipedia.org /wiki/Pushout_(category_theory) (894 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.
		Partitions of unity in the theory of fibrations.
	hopf.math.purdue.edu /Intermont-JohnsonM/ijxspace.txt (4457 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.
		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.
	math.ucr.edu /home/baez/hda/dimensional_ladder.html (2262 words)

	[No title]
		A major challenge in developing such a unified model theory is in the requirement that it must be able to handle major structural differences between the targeted models as well as the significant differences in the logic bases of their associated constraint languages.
		When we consider a category SPEC of component specifications taken as theories in some logic, properties are expressed as sentences of the underlying logic, and emergence of properties can be characterised by the fact that the morphisms that connect component specifications to the system specification are not conservative.
		The behaviour of structured theory presentations under representation in a logical framework is studied, focusing on the problem of ``lifting'' presentations from the object logic to the metalogic of the framework.
	www.informatik.uni-bremen.de /flirts/flirts.bib (4258 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.
		In the category of sets the pullback of f and g is the set
		The categorical dual of a pullback is a called a pushout.
	www.wikipedia.org /wiki/Fiber_product (334 words)

	PlanetMath: categorical pullback
		Pullbacks and pushouts are unique up to unique isomorphism when they exist.
		pullback square, cone, pushout, pushout square, vertex, base
		(Category theory; homological algebra :: General theory of categories and functors :: Limits and colimits)
	planetmath.org /encyclopedia/Base5.html (107 words)

	MATHS: Category Theory (Site not responding. Last check: 2007-10-20)
		Category Theory is a way for talking about the relationships between the classes of objects modeled by mathematics and logic.
		The original use of the term category was in the idea of a 'categorical' axiom system - an axiom system which defined its objects so exactly that all objects that satisfied the axioms were isomorphic - they mapped into each other, one-to-one, preserving all the axioms and structure.
		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]
		Toposes and abelian categories are striking for the number of elementary properties they have in common (monic+epi = iso, mono/epi is a unique factorization system, etc. etc.) and the paucity of their common models, namely just the final category.
		To the extent that both toposes and abelian categories share much pleasant structure, the models of the intersection of their theories, for a suitable choice of language, would seem to be a nice class in its own right.
		Vaughan wrote: To the extent that both toposes and abelian categories share much pleasant structure, the models of the intersection of their theories, for a suitable choice of language, would seem to be a nice class in its own right.
	www.mta.ca /~cat-dist/catlist/1999/atcat (5103 words)

	[No title]
		In particular, there is a homotopy category of monoids of topological symmetric spectra, and this homotopy category is equivalent to the homotopy category of monoids of simplicial symmetric spectra.
		Y is an isomorphism in the homotopy category.
		Simila* rly, one does not know that a pushout* of a trivial cofibration f in A-algthrough a map g is a trivial cofibration unless both the source of f and the target of g are co* *fibrant in A-mod.
	hopf.math.purdue.edu /Hovey/mon-mod.txt (8088 words)

	Colimit: Definition and Links by Encyclopedian.com - All about Colimit
		In category theory the colimit of a functor, also known as a direct limit, is dual to the notion of a limit (or inductive limit).
		The general definition of a colimit is given on the limit page.
		Special cases of colimits, each dual to a special case of limits, include coproduct[?], coequaliser[?], pushout[?].
	www.encyclopedian.com /co/Colimit.html (82 words)

	Sports Fresh : Article 'Additive category' (Site not responding. Last check: 2007-10-20)
		In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A 1,..., A n of C have a biproduct A 1 ⊕ ··· ⊕ A n in C.
		An additive category is a preadditive category with all finite biproducts.
		A pre-Abelian category is an additive category with all kernels and cokernels.
	www.sports-fresh.net /DisplayArticle861421.html (334 words)

	Further Work
		We have developed a rigorous, systematic theory of blends, based on the rather esoteric branch of mathematics called "category theory," and more specifically, on the category
		of sign systems with semiotic morphisms, which is actually an "order enriched category," because it is enriched with a priority ordering on the morphisms.
		Category theory suggests that the right definition of blend is characterized by the "optimality" or "universal" property of "pushouts" (which must be in some sense "lax" because of the ordering); see [B] for details.
	www-cse.ucsd.edu /users/goguen/papers/sm/node7.html (747 words)

	Amazon.com: Triangulated Categories in the Representation of Finite Dimensional Algebras (London Mathematical Society ... (Site not responding. Last check: 2007-10-20)
		Happel presents an introduction to the use of triangulated categories in the study of representations of finit-dimensional algeras.
		In recent years representation theory has been an area of intense research and the author shows that derived categories of finite=dimensional algebras are a useful tool in studying tilting processes.
		Results on the structure of derived categories of hereditary algebras are used to investigate Dynkin algebras and iterated tilted algebras.
	www.amazon.com /exec/obidos/tg/detail/-/0521339227?v=glance (525 words)

	[No title]
		Acknowledgments: The Morita theory in stable model categories which I descri* be in Section 4 is based on joint work with Brooke Shipley spread over many years and * several papers; I would like to take this opportunity to thank her for the pleasant and* * fruitful collaboration.
		Since the category of right Rop-modules is isomorphic to the cate* gory of left R-modules, we can view M as an Sop-Rop-bimodule and N as an Rop-Sop-bimodule, a nd then they provide the equivalence of categories* between Mod-Ropand Mod-Sop.
		The bimodules which induce the equivalences of module categories ca* n both be taken to be Rn, but viewed as `row vectors' (or 1 x n matrices) and `column vec *tors' (or n x 1 matrices) respectively.
	hopf.math.purdue.edu /Schwede/Morita.txt (4235 words)

	AGG Documentation
		The formal semantics of rule application is given in terms of category theory, by a single categorical construction known as a pushout in an appropriate category of attributed graphs with partial morphisms - hence the name Single-Pushout (SPO) approach.
		Category theory and especially the colimit construction, is suitable for both aspects, providing a unifying formal framework for object oriented design.
		Other application areas are Structuring and Refinement of formal specifications based on colimits in the category of signatures, Operational Semantics of functional logic programming languages based on the category of jungles, and Algebraic Development Techniques and their extensions (e.g.
	tfs.cs.tu-berlin.de /agg/docu.html (1590 words)

	A tutorial on graph transformation \| Lambda the Ultimate
		A nice application of category theory to computer science that is rather simpler than its application to semantics tends to get is the single and double pushout approach to graph transformation.
		Categorical pushouts allow patterns and rewrites on many kinds of structure, in particular graphs, to be specified in a simple manner.
		As an educated guess, though, I would imagine that the theory works for general graphs, as they are more natural to treat from a categorical perspective than DAGs (since a category is a graph closed under a path-forming operation).
	lambda-the-ultimate.org /node/view/266 (470 words)

	Practical Foundations of Mathematics (Site not responding. Last check: 2007-10-20)
		Since most of the interesting phenomena may be observed more clearly in the concrete cases of pullback, equaliser, pushout and coequaliser, we postpone the abstract definition to Section 7.3.
		The diversity of the behaviour of finite limits and colimits is striking: the basic features of groups, rings, vector spaces and topology may often be discovered just by looking for the coproducts in these categories.
		However, size issues arise in the case of categories, where they did not for lattices, so we postpone the general result until Theorem 7.3.12.
	www.geocities.com /yury_bendersky/b/f/s50.html (340 words)

	Category Theory Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-20)
		Looking For category theory - Find category theory and more at Lycos Search.
		Find category theory - Your relevant result is a click away!
		Look for category theory - Find category theory at one of the best sites the Internet has to offer!
	www.karr.net /encyclopedia/Category:Category_theory (199 words)

	Category:Category theory - Wikipedia, the free encyclopedia
		Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		There are 126 articles in this section of this category.
		This page was last modified 14:44, 23 October 2005.
	en.wikipedia.org /wiki/Category:Category_theory (55 words)

	Adhesive and quasiadhesive categories (Site not responding. Last check: 2007-10-20)
		We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved, as well as quasiadhesive categories which restrict attention to regular monomorphisms.
		Many examples of graphical structures used in computer science are shown to be examples of adhesive and quasiadhesive categories.
		Double-pushout graph rewriting generalizes well to rewriting on arbitrary adhesive and quasiadhesive categories.
	www.edpsciences.org /articles/ita/abs/2005/03/ita0506/ita0506.html (91 words)

	Non-abelian homological algebra
		This Bibliography lists some papers and books on multiple categories and groupoids, non-linear homological algebra, and related areas directly relevant to the themes developed in the accompanying article on Higher dimensional group theory.
		R. Brown and T. Porter, `Category theory and higher dimensional algebra: potential descriptive tools in neuroscience', Proceedingsof the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1 (2003) 80-92.
		R. Brown and T. Porter, `Category Theory: an abstract setting for analogy and comparison', Advanced Studies in Mathematics and Logic (to appear) UWB Preprint 05.10.
	www.bangor.ac.uk /%7Emas010/nonlnpub.htm (2829 words)

	Universal property - Wikipedia
		In abstract algebra and category theory, constructions are often defined by an abstract property which requires the existence of unique morphisms under certain conditions.
		It is advisable to study several examples first: product and direct sum, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.
		Let C and D be categories and F : C
	nostalgia.wikipedia.org /wiki/Universal_property (345 words)

	TR-96-17: Algebraic Approaches to Graph Transformation, Part I: Basic Concepts and Double Pushout Approach - ... (Site not responding. Last check: 2007-10-20)
		The algebraic approaches to graph transformation are based on the concept of gluing of graphs, modelled by pushouts in suitable categories of graphs and graph morphisms.
		This allows one not only to give an explicit algebraic or set theoretical description of the constructions, but also to use concepts and results from category theory in order to build up a rich theory and to give elegant proofs even in complex situations.
		The next chapter is devoted instead to the single-pushout approach, and it is closed by a comparison between the two approaches.
	www.di.unipi.it /ricerca/trabstract/TR-96-17.html (186 words)

	Homotopy Theory, and Change of Base for Groupoids and Multiple Groupoids (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Introduction A major problem in homotopy theory is to compute how high dimensional homotopy invariants of a space are affected by low dimensional changes to the space.
		2 The geometric theory of inverse semigroups II: E-unitary cov..
		2 The geometric theory of inverse semigroups I: E-unitary inve..
	citeseer.ist.psu.edu /brown96homotopy.html (804 words)

	Pushout - TheBestLinks.com - Chicago, Illinois, Education, Ontario, ... (Site not responding. Last check: 2007-10-20)
		Pushout - TheBestLinks.com - Chicago, Illinois, Education, Ontario,...
		Pushout, Chicago, Illinois, Education, Ontario, School, Dropout, Violence...
		You can add this article to your own "watchlist" and receive e-mail notification about all changes in this page.
	www.thebestlinks.com /Pushout.html (262 words)

	Math 8306-07: Class Outlines
		The pushout of a cofibration is a cofibration.
		CW pairs, cellular maps, cellular homotopy, the homotopy categories of CW complexes and CW pairs.
		Axiomatic homology theory: axioms for (absolute) homology of CW pairs.
	www.math.umn.edu /~voronov/8306/outline.html (1288 words)

	CASL/RefereeResponse
		This approach ensures that the instantiation of a generic specification is a pushout also in cases where the body and the actual parameter refer to the same specifications, these specifications being in the import.
		It is more relevant to state the relationship between the model-theoretic semantics and theory composition for the specification structuring constructs; this is quite standard, but should indeed be mentioned in the CASL documents (and will be included in future work on proof theory for CASL specifications).
		Defining the result of parameter passing as a pushout does not only provide the result of actualization as "some kind" of union of the actual and body specifications, but it also attaches to the resulting specification a universal property which is very useful for the basic theory.
	www.brics.dk /Projects/CoFI/Documents/CASL/RefereeResponse/index.html (6502 words)

	Re: logical operations and/or category theory?
		To sum up: our familiar picture of the elementary propositional calculus of Boole (and indeed of first and higher order predicate calculus) is greatly generalized and enriched when we study topoi.
		I could go on and yak about how the fundamental constructions of topos theory very neatly encapsulates the distinctions between inner and outer descriptions of subobjects, and internal and external logic, but since this is extensively described in various books I have already listed, I shall cease and desist.
		Remember, topos theory is capable of discussing anything which can be discussed in mathematical terms at all, so there's much more to this categorical reformulation of logic and the foundations of mathematics than merely a reformulation of elementary propositional logic (or even the generalization of this to Heyting algebras rather than just Boolean algebras).
	www.lns.cornell.edu /spr/2001-03/msg0031795.html (487 words)

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