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Topic: Object category theory

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	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.
		Such a process is called a functor, and it associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second.
		Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus.
	en.wikipedia.org /wiki/Category_theory (2348 words)

	PlanetMath: category theory
		Category theory gives us tools for analyzing such functors: we can talk about natural transformations of functors, and in fact we can use these to assemble the category of functors from one category to another into a category, provided certain set-theoretic constraints are met (universes are a tool used to address these set-theoretic difficulties).
		The fundamental theorem of Galois theory is that the functor from a subgroup of the Galois group of a field to its fixed field is an equivalence of categories.
		One finds a functor from the category the objects are in to some simpler category; if the images are not isomorphic, then the objects were not isomorphic.
	planetmath.org /encyclopedia/CategoryTheory.html (1646 words)

	Category Theory
		Category theory is a general mathematical theory of structures and sytems of structures.
		Category theory reveals that many of these constructions are in fact special cases of objects in a category with what is called a "universal property".
		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.
	www.science.uva.nl /~seop/archives/spr1999/entries/category-theory (3066 words)

	Category Theory
		Category theory studies structural aspects of mathematics that are common to many fields of mathematics: e.g., algebra, topology, functional analysis, logic, and computer science.
		Category theory is a mathematical language which arose in the study of limits for universal coefficient theorems in Cech cohomology by Eilenberg and Mac Lane (1942); so the topic has its origins in some sophisticated topology.
		The category version of the definition of a group: A group is a category with one object in which all the morphisms are isomorphisms.
	education.wichita.edu /alagic/nextpage/categories.htm (1383 words)

	Category Theory: The Language of Mathematics*
		Since category theory, or more precisely since the theory of the category of categories, is first-order, it cannot, either as a language or foundation, capture the "central dogma of the axiomatic method: that isomorphic structures are mathematically indistinguishable in their essential properties".
		It eliminatory role is stifled by an inability to capture the category of categories as an object of mathematics.
		We say that category theory is the language of mathematical theories and their relations because it allows us to talk about their general structure in terms of "objects" and "functors", wherein such terms are likewise taken as "syntactic assemblages waiting for a structure of the appropriate sort to give them formulas meaning".
	www.math.mcgill.ca /rags/seminar/Landry.html (4212 words)

	Category theory - FreeEncyclopedia (Site not responding. Last check: 2007-10-21)
		Category theory is also used in a foundational way in functional programming, for example to discuss the idea of typed lambda calculus in terms of cartesian-closed categories.
		Then it becomes possible to relate different categories by functors, generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second.
		One of the central themes of algebraic geometry is the equivalence of the category C of affine schemes and the category D of commutative rings.
	openproxy.ath.cx /ca/Category_theory.html (2075 words)

	Category Theory (Reading Course)
		Category theory is a kind of network algebra which provides a general framework for describing mathematical objects and their interrelations.
		It is equivalent to suppose the existence of a (non-null) family of objects (associated with the identity arrows) so that every arrow a may be regarded as an arrow from the source object Sa of a to the target object Ta of a.
		The standard example of a category is the family Ens of sets and functions with composition defined as ordinary composition of functions and the identity arrow is the identity function.
	www.georgetown.edu /faculty/kainen/category.html (1030 words)

	Abstract algebra:Category theory - Wikibooks
		Category theory is the study of categories, which are collections of objects and morphisms (or arrows), or from one object to another.
		The category whose objects are smooth (differentiable,topological) manifolds, and morphisms are smooth (differentiable,continuous) maps.
		is a category with the same objects, and all the arrows reversed.
	en.wikibooks.org /wiki/Abstract_algebra:Category_theory (309 words)

	Intermediate Depth Representations
		Rosch's classification theory [26] proposes that the most appropriate level of category abstraction for an object is the most cognitively economic one - which she calls the basic level.
		Gluck and Corter [14] propose two metrics for Category Utility which is a context-sensitive measure of the predictive ability of a level of categorization based upon the structure theory.
		Categories which are very accurate but apply to few individuals are not favoured, nor are those that cover a large proportion of the population but because of generality have poor accuracy.
	www.coiera.com /papers/aimj2/aimj.doc.html (6861 words)

	Functor
		In category theory, a functor is a special type of mapping between categories.
		Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
		A category with a single object is equivalent to a monoid whose elements are morphisms and whose operation is composition.
	www.free-download-soft.com /info/net-tools.html (1432 words)

	Category Theory
		Category theory now occupies a central position not only in contemporary mathematics, but also in theoretical computer science and even in mathematical physics.
		After their 1945 paper, it was not clear that the concepts of category theory would be more than a convenient language and so it remained for approximately fifteen years.
		For it is in his thesis that Lawvere proposed the idea of developing the category of categories as a foundation for category theory, set theory and, thus, the whole of mathematics, as well as using categories for the study of theories, that is the logical aspects of mathematics.
	plato.stanford.edu /entries/category-theory (7029 words)

	[Inquiry] Re: Category Theory (Site not responding. Last check: 2007-10-21)
		A 'category' is a graph with two additional functions:
		for all objects a in O and all composable pairs
		for "c is an object of C" and "f is an arrow of C",
	stderr.org /pipermail/inquiry/2003-July/000623.html (298 words)

	Re: category theory ?
		The formal prerequisites for beginning to study category theory are shockingly few: all you really need is a smidgen of knowledge about sets and functions, just as you'd need for any branch of math these days.
		The way I'm describing it here, category theory is a kind of math for sophisticated people who have "seen it all, done it all" and are tired of doing the same thing over and over again.
		To check that a category with one object is "essentially just a monoid", note that if our category C has one object x, the set hom(x,x) of all morphisms from x to x is indeed a set with an associative binary product, namely composition, and a unit element, namely 1_x.
	www.lns.cornell.edu /spr/1999-09/msg0017954.html (4467 words)

	MATHS: Category Theory
		Category Theory is a way for talking about the relationships between the classes of objects modeled by mathematics and logic.
		A Category is a mixture of an algebra and a directed graph.
		Category theory shows that in most known algebras and logistic systems, there is a way to construct an equivalent.
	www.csci.csusb.edu /dick/maths/math_25_Categories.html (3607 words)

	[No title]
		This is an interesting project, since category theory is all about objects and morphisms.
		There are lots of different topoi; you can do a lot of the same mathematics in all of them; but there are also lots of differences between them: for example, the axiom of choice need not hold in a topos, and the law of the excluded middle ("either P or not(P)") need not hold.
		C) says that there is an object called the "subobject classifier" Omega, which acts like {0,1}, in that functions from any set x into {0,1} are secretly the same as subsets of x.
	www.math.niu.edu /~rusin/known-math/00_incoming/topos (912 words)

	Elementary Category Theory \| Lambda the Ultimate
		About the closest you can get in pure CT and using a generic notion of Set Theory (and, possibly intuitionism and Choice are also relevant), is that a subset is iso to a `subobject', which is an equivalence class of monos.
		So, turning that around: the equalizer of two functions is the subset of their domain on which they agree, plus the inclusion function from the subset to the larger set.
		For those interested in a brief introduction to category theory, James Cheney has recently posted some PDF slides titled Category Theory for Dummies on his home page.
	lambda-the-ultimate.org /node/view/39 (1228 words)

	[Inquiry] Re: Category Theory (Site not responding. Last check: 2007-10-21)
		Mac Lane uses a symbol for the one object and one (identity) arrow category that looks like a dot with a sling out of it and an arrow back into it.
		$0$ is the empty category (no objects, no arrows).
		$3$ is the category with three objects whose non-identity arrows
	stderr.org /pipermail/inquiry/2003-July/000624.html (148 words)

	Categories
		Quantum theory can be thought of as the generalization of classical mechanics you get by dropping the assumption that observable quantities like position and momentum commute.
		The example to think of is the category in which the objects are sets and the morphisms are functions.
		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.
	math.ucr.edu /home/baez/categories.html (2546 words)

	Basic Category Theory (Site not responding. Last check: 2007-10-21)
		This course was given to advanced undergraduate and beginning Ph.D. students in the fall of 1994 in Aarhus, as part of Glynn Winskel's semantics course.
		It is, in the author's view, the very minimum of category theory one needs to know if one is going to use it sensibly.
		These are there to give the reader at least a very rough idea of how the theory ``works''.
	www.brics.dk /LS/95/1/BRICS-LS-95-1/BRICS-LS-95-1.html (141 words)

	Prototyping a Formal Object-Oriented Database in P/FDM (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Category theory is used to define the Product model, a formal notation for representing features of an object based database.
		In particular, we will examine how this model deals with three of the most important problems inherent in object databases, those of queries, closure and views, as well as how our model deals with more common database concepts, such as keys, relationships, aggregation, etc. We will implement a...
		Object Orientation In Database Interoperation Case Study Of..
	citeseer.ist.psu.edu /279868.html (466 words)

	Limit (category theory) (Site not responding. Last check: 2007-10-21)
		In category theory, the limit of a functor generalizes the notions of inverse limit and product used in various parts of mathematics.
		If J is a small category and every functor from J to C has a limit, then the limit operation forms a functor from the functor category (see category theory) C
		In the category Ab of abelian groups, this for example shows that the kernel of a product of homomorphisms is naturally identified with the product of the kernels.
	www.explainthis.info /li/limit-(category-theory).html (877 words)

	Diary for mwh
		Some people of course do both category theory and logic, but I'm not one of them.
		The cycle detector crucially depends on the reference counts to determine when a bunch of objects are an unreachable cycle.
		I was briefly interested in logic, but now anytime I end up worrying about my set theory, I take it as a sign I've wandered into the wrong ends of the subject.
	www.advogato.org /person/mwh/diary.html?start=106 (1310 words)

	topos
		Unfortunately, if you don't know some category theory, the above definition will be mysterious and will require a further sequence of definitions to bring it back to the basic concepts of category theory - object, morphism, composition, identity.
		This is a great introduction to category theory via the topos of sets: it describes ordinary set theory in topos-theoretic terms, making it clear which axioms will be dropped when we go to more general topoi, and why.
		Don't be scared by the title: it starts at the beginning and explains categories before going on to topoi and their relation to logic.
	math.ucr.edu /home/baez/topos.html (1773 words)

	Category Theory for Computer Science (Site not responding. Last check: 2007-10-21)
		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)

	mbox: Re: Mechanization of category theory
		Maybe in reply to: Clemens Ballarin: "Mechanization of category theory"
		I'm about to implement basic parts of category theory in a tactical theorem
		This first part is described in the paper "Constructive Category Theory"
	www-unix.mcs.anl.gov /qed/mail-archive/volume-3/0140.html (399 words)

	Talk:Category theory
		Can it assign the same object in D to many objects in C, or many objects in D to a single object in C? I can't see that this is ruled out by the requirements on a functor, but maybe I'm just not being smart enough.
		It isn't actually a function because functions are defined on sets and categories are generally 'bigger' than sets.
		XXXV All day I hear the noise of waters Sad as the sea-bird is when, going.
	www.termsdefined.net /ta/talk:category-theory.html (389 words)

	PLT Online
		This is a collection of programming language theory texts and resources, all of which are freely available over the Internet.
		Many valuable reference texts on programming language theory, previously only available in paper form, have in recent years become publicly accessible from the net.
		Part of the reason PL theory and advanced programming languages seem impenetrable to other communities is that learning materials are hard to obtain, or demand a sizeable investment of resources (time, money,...) even if the potential reader is only exploring the subject.
	www.cs.uu.nl /~franka/ref (985 words)

	Connected category - Wikipedia, the free encyclopedia
		In category theory, a branch of mathematics, a connected category is a category in which, for every two objects, there is at least one morphism connecting them.
		An example of a simple connected category containing two objects and a single (non-identity) morphism is as follows:
		This page was last modified 11:00, 2 October 2005.
	en.wikipedia.org /wiki/Connected_(category_theory) (71 words)

	week73
		Now, there are lots of things one can do with sets, which lead to all sorts of interesting examples of categories, but in a sense the primordial category is Set, the category of sets and functions.
		But this in turn makes the collection of all categories into a "two-dimensional" structure, a 2-category having objects, morphisms between objects, and 2-morphisms between morphisms.
		To study sets carefully we need categories, to study categories well we need 2-categories, to study 2-categories well we need 3-categories, and so on...
	math.ucr.edu /home/baez/week73.html (2116 words)

	M584: Category theory (Site not responding. Last check: 2007-10-21)
		both be terminal objects of a given category C.
		is a natural transformation from FG to the identity functor on the domain category A of G.
		Verify that for each object X of B and A of A,
	orion.math.iastate.edu /jdhsmith/class/M584F02.htm (305 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Language of category theory (object, morphism, universal object, functor, coproducts, products, free objects).
		Ring theory (definitions, UFD, PID, fraction field, polynomials).
		Field theory (algebraic extensions, Galois theory, finite fields, cyclotomic extensions, solvability, algebraic closure, transcendental extensions, separability and inseparability).
	www.math.hawaii.edu /~tom/syllabi/611_syl.html (318 words)

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