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Topic: Isomorphism of categories

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	Isomorphism - Wikipedia, the free encyclopedia
		In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich.
		The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures.
		Isomorphic structures are "the same" at some level of abstraction; ignoring the specific identities of the elements in the underlying sets, and focusing just on the structures themselves, the two structures are identical.
	en.wikipedia.org /wiki/Isomorphism (471 words)

	Isomorphism of categories - Wikipedia, the free encyclopedia
		Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms.
		The functor category of all additive functors from this category to the category of abelian groups is isomorphic to the category of left modules over the ring.
		Another isomorphism of categories arises in the theory of Boolean algebras: the category of Boolean algebras is isomorphic to the category of Boolean rings.
	en.wikipedia.org /wiki/Isomorphism_of_categories (636 words)

	PlanetMath: isomorphism
		In the category of vector spaces and linear transformations, a linear transformation is an isomorphism if and only if it is an invertible linear transformation.
		In the category of topological spaces and continuous maps, a continuous map is an isomorphism if and only if it is a homeomorphism.
		This is version 2 of isomorphism, born on 2002-02-13, modified 2002-02-13.
	planetmath.org /encyclopedia/Isomorphism2.html (217 words)

	PlanetMath: category isomorphism
		For example, the category of all finite sets is naturally equivalent to its subcategory of all finite ordinals.
		Isomorphism has a ``size'' restriction, whereas natural equivalence does not.
		This is version 4 of category isomorphism, born on 2004-05-11, modified 2004-05-12.
	planetmath.org /encyclopedia/CategoryIsomorphism.html (115 words)

	20th WCP: Kant's Categories Reconsidered
		This negative thesis suggests a positive one, namely, that categories should be argued for as necessary conditions of the possibility of a knowledge that is distinct from experience, viz., a priori knowledge.
		So let this isomorphism between the categories and the logical functions of judgment be our clue toward discovering what must be done to make the premises given by the metaphysical deduction suitable for the transcendental deduction and its distinctive reliance on the transcendental unity of apperception.
		As noted there, the categories are shown to be necessary for the possibility of experience only in a context in which the objects of possible experience are taken as appearances.
	www.bu.edu /wcp/Papers/Mode/ModeGree.htm (3207 words)

	Isomorphism of categories (Site not responding. Last check: 2007-10-21)
		In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e.
		A functor F : C → D yields an isomorphism of categories if and only if it is bijective on objects and on morphism sets.
		The category of k-linear group representations of G is isomorphic to the category of left moduless over kG.
	www.sciencedaily.com /encyclopedia/isomorphism_of_categories (680 words)

	n-Categories - Sketch of a Definition
		In a similarly appropriate sense the category svf(fam(c)^op) of contra-variant set-valued functors on fam(c) is the free symmetric monoidal small-ly co-complete category on c.
		Definition: the monoidal category of "c-signatures", written "sig(c)", is the category svf(prof(c)); it is a monoidal category because the universal property of svf(fam(c)^op) gives an equivalence of categories between sig(c) and the monoidal category of endomorphisms of the symmetric monoidal small-ly co-complete category svf(fam(c)^op).
		Definition: the category of "c-operads" is the category of monoids in the monoidal category sig(c).
	math.ucr.edu /home/baez/ncat.def.html (2249 words)

	categories: Re: Time for functors to grow up; three queries (Site not responding. Last check: 2007-10-21)
		In general we don't care whether two objects of a category are equal, only whether they're isomorphic; for instance, if a group theorist says that two groups are the same, she doesn't usually know or care whether, by some miracle of set theory, they're actually equal - isomorphism is all that matters.
		So the notion of isomorphism of categories is unreasonably strict, requiring as it does that certain equalities of objects occur.
		So isomorphism of categories seems not to be as unreasonable a notion as the standard rationale would have you believe.
	north.ecc.edu /alsani/ct02(1-2)/msg00061.html (1040 words)

	wikien.info: Main_Page (Site not responding. Last check: 2007-10-21)
		Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945.
		Any directed graph generates a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph.
	www.hostingciamca.com /index.php?title=Category_theory (2374 words)

	Articles - Category theory (Site not responding. Last check: 2007-10-21)
		Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
		Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1945, in connection with algebraic topology.
		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.
	lastring.com /articles/Category_theory?mySession=212cf973f817176716c... (2284 words)

	[No title]
		The category D is triangulated, with distinguished triangles isomorphic to ca* *non- ical cofiber sequences of G-spectra.
		Nevertheless, the smaller collection suffices to generate the associated triang* ulated homotopy category, since that is triangulated equivalent to the homotopy catego ry obtained from the other two model categories*.
		The G-fixed point functor from D to the stable homotopy category of spectra is the composite of i* ** and the G-fixed point spectrum functor from C to spectra.
	hopf.math.purdue.edu /May/WirthRev.txt (3972 words)

	[No title]
		The study of automorphisms of the category $\Theta^0$ is tied to the study of automorphisms of the semigroups $\End W$, $W\in \Ob \Theta^0$.
		This category is regarded as an invariant which is responsible for the geometry in $L$.
		The morphisms of the category of modules induced by inner morphisms of the category of modules with semimorphisms are called {\it semiinner morphisms} of the category of modules.
	www.univie.ac.at /EMIS/journals/ERA-AMS/2002-01-001/2002-01-001.tex.html (4164 words)

	functors and bundles (was Fibered Categories)
		Here's some of what I know about fibered categories: We might say that a category F is a fibered category over A if it acts like a sheaf of categories over A, where I'm thinking of A as a category of schemes of finite type over SpecC.
		A fibered category F over the category of schemes A is a functor p: F -> A. I know that weak stacks are implemented via fibered categories (in the sense of Grothendieck).
		I'm interested in these kinds of thoughts for this reason: Roughly, a natural bundle is a functor from the category of m-dimensional manifolds to the category of fibered manifolds, and a gauge natural (GAN) bundle is a functor from the category of principal G-bundles to the category of fibered manifolds and fiber-respecting mappings.
	www.lns.cornell.edu /spr/2002-09/msg0044057.html (507 words)

	Brain.Save() - Isomorphism
		Isomorphism is a $10 mathematical word that has been on my mind lately.
		The undercurrent of isomorphism is that we have many different ways of talking about the same thing, all of which are basically equivalent on the level that we're interested in.
		The recognition that the canonical form is on equal footing ontologically with all of its isomorphisms is a rejection of the modern neo-Platonic ideal; hence, the postmodern substitution of 'truth' for the much fuzzier idea of 'constructed convention'.
	hyperthink.net /blog/PermaLink,guid,46c804f0-1384-4752-9519-182d407f9253.aspx (2095 words)

	q_duality
		One advantage of the category-theoretic point of view is that it becomes possible to define concretely what it means for isomorphisms to be ``natural''.
		The idea is to look at isomorphisms of whole categories at a time, instead of individual objects; the naturalness consists of compatibility with morphisms between objects.
		There are two concepts: natural isomorphism of two categories, defined with covariant functors, and natural duality, defined with contravariant functors.
	www.math.ucla.edu /~baker/222a/handouts/q_duality/node5.html (107 words)

	categories: Re: Time for functors to grow up; three queries (Site not responding. Last check: 2007-10-21)
		So the notion of isomorphism of categories is unreasonably > strict, requiring as it does that certain equalities of objects occur.
		This is really a consequence of the fact that you're thinking of the category of sets as a unique entity (if not God-given, then at least handed down to us by Zermelo and Fraenkel).
		Of course, if you have two equivalent single-sorted theories, and you consider their models in the same category, you will get isomorphic categories of models.
	north.ecc.edu /alsani/ct02(1-2)/msg00062.html (404 words)

	week76
		In a category, two objects x and y can be equal or not equal, but more interestingly, they can be isomorphic or not, and if they are, they can be isomorphic in many different ways.
		C and D are clearly not isomorphic, because for a functor F: C -> D to be invertible it would need to be one-to-one and onto on objects, and also on morphisms.
		Category theorists generally regard equivalent categories as "the same for all practical purposes".
	math.ucr.edu /home/baez/week76.html (1599 words)

	[No title]
		The equivalence of categories transfers this struct* ure to the globular case - the resulting internal hom in the globular case gives various higher dimensio *nal forms of `lax natural transformation'.
		The rules for the connections are fairly clear extens* ions of the axioms given in [6, 9], given the general notion of thin structure on a double category* disc* *ussed by Spencer in [23].
		Theorem 8.8 The categories of !-categories and of cubical !-categories are equ* *ivalent under the functors and fl.
	hopf.math.purdue.edu /AlAgl-Brown-Steiner/multiplecat.txt (8356 words)

	Sample Chapter for Babb, S.: Managing Mexico: Economists from Nationalism to Neoliberalism.
		Mimetic isomorphism essentially corresponds to the processes identified by world-cultural theorists: organizations in the same "line of business" (i.e., nation-states) share common values and organizational structures, and therefore often imitate one another as a way of minimizing uncertainty.
		Unlike world-cultural isomorphism, which is founded on legitimation and shared values, coercive and normative isomorphism are fundamentally about power: the power of external organizations with resources, in the former, and the power of certified experts, in the latter.
		Unlike mimetic isomorphism, the categories of coercive and normative/expert isomorphism have the potential to incorporate power and resource inequalities between core and periphery.
	www.pupress.princeton.edu /chapters/s7202.html (8137 words)

	ATCAT 1999-2000
		The set of isomorphism classes of extensions belonging to the same abelian matched pair carries a Baer-type abelian group structure, and is isomorphic to to the second cohomology group of that matched pair.
		Abstract: It is known that the category of unital commutative C-algebras is equivalent to the category* of compact Hausdorff spaces.
		If the twisting is invertible in the convolution algebra Hom(H,End(A)), then there is an isomorphism of categories betweenthe categories of relative Hopf modules for A and the twisting of A. Thus invertible twistings seem to be very strong.
	www.mscs.dal.ca /~pare/Sem99-00.html (1384 words)

	Lecture and Homework Log, Math 401
		Definition of category, the category of sets, the category of real vector spaces, the category of groups.
		Definition of isomorphism and automorphism in a category.
		Isomorphism of categories, product of categories, the opposite category of a category, natural transformations, natural isomorphisms, equivalences of categories, free objects, all with examples.
	www.math.uiuc.edu /~dan/Courses/1999/Fall/401/log.html (3019 words)

	Deciding Isomorphisms of Simple Types in Polynomial Time - Considine (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Abstract: The isomorphisms holding in all models of the simply typed lambda calculus with surjective and terminal objects are well studied - these models are exactly the Cartesian closed categories.
		Isomorphism of two simple types in such a model is decidable by reduction to a normal form and comparison under a finite number of permutations (Bruce, Di Cosmo, and Longo 1992).
		Unfortunately, these normal forms may be exponentially larger than the original types so this construction decides isomorphism in...
	citeseer.ist.psu.edu /315482.html (545 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		For example, categories of structured sets, such as groups and homomorphisms, but also spatial ones, such as 2-Cob, whose objects are sets of circles and whose arrows are (diffeomorphism) classes of surfaces between them.
		If the first category is a small diagram of arrows, a copy of that diagram within the second category would be a functor.
		At the level of categories, isomorphism is too strong a notion of sameness.
	philsci-archive.pitt.edu /archive/00001959/01/Rome.doc (4405 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Title: Deciding Isomorphisms of Simple Types in Polynomial Time Author: Jeffrey Considine Date: April 2, 2000 Abstract: The isomorphisms holding in all models of the simply typed lambda calculus with surjective and terminal objects are well studied - these models are exactly the Cartesian closed categories.
		We show how using space-sharing/hash-consing techniques and memoization can be used to decide isomorphism in practical polynomial time (low degree, small hidden constant).
		Other researchers have investigated simple type isomorphism in relation to, among other potential applications, type-based retrieval of software modules from libraries and automatic generation of bridge code for multi-language systems.
	www.cs.bu.edu /techreports/abstracts/2000-010 (158 words)

	PH. D. THESIS (Site not responding. Last check: 2007-10-21)
		The isomorphism between the categories UFrm, WUFrm and EUFrm.
		An application: the category UFrm is fully embeddable in a final completion of the category MFrm.
		The category Wnear as a unified theory of (symmetric) topology and uniformity.
	www.mat.uc.pt /~picado/publicat/PhD.html (143 words)

	[No title]
		A monoid is a category with one object.
		all categories: its objects are all categories, its arrows are all functors
		isomorphism of categories, for there may be objects of B not in
	stderr.org /pipermail/inquiry/2003-July.txt (15758 words)

	Pacific Northwest Seminar in Algebraic Geometry (Site not responding. Last check: 2007-10-21)
		The Hilbert scheme $H_{n}$ of $n$ points in the plane ${\bf C}^{2}$ is isomorphic to the Hilbert scheme of regular $S_{n}$ orbits in ${\bf C}^{2n}$.
		It follows from this and a recent theorem of Bridgeland, King and Reid that there is an explicit isomorphism of derived categories between coherent sheaves on $H_{n}$ and $S_{n}$-equivariant ${\bf C}[x_{1},y_{1},\ldots,x_{n},y_{n}]$-modules.
		The isomorphism can be used to prove vanishing theorems on $H_{n}$, derive character formulas, and make computations in the Grothendieck ring and cohomology ring of $H_{n}$.
	www.pims.math.ca /science/2001/pnwag (333 words)

	Practical Foundations of Mathematics
		A category which has all set-indexed limits or colimits is called complete or cocomplete respectively, cf Definition 3.6.12 for lattices.
		A category in which every finite diagram has a cocone (which need not be colimiting) is called filtered; this generalises directedness for posets (Definition 3.4.1).
		Comma categories The next construction is a new kind of limit which arises in 2-categories, just as equalisers appeared when we moved from posets to categories.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s73.html (1987 words)

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