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Topic: Subobject classifier

	Job Classified
		A linear classifier is a classifier that uses a linear function of its inputs to base its decision on.
		For a two-class classification problem, one can visualize the operation of a linear classifier as splitting a high-dimensional input space with a hyperplane: all points on one side of the hyperplane are classified as "yes", while the others are classified as "no".
		Then π is a local homeomorphism, and the sheaf corresponding is the required subobject classifier (in other words the construction of Ω is by means of its espace étalé).
	aardogs.com /pages10/47/job-classified.html (797 words)

	Subobject classifier (Site not responding. Last check: 2007-11-03)
		In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω.
		As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions: to every subset U of X we can assign the function from X to Ω that maps precisely the elements of U to 1 (see characteristic function).
		The morphism g is then called the classifying morphism for the subobject j.
	bopedia.com /en/wikipedia/s/su/subobject_classifier.html (195 words)

	The Subobject Classifier of the Category of Functional Bisimulations - Watanabe (ResearchIndex) (Site not responding. Last check: 2007-11-03)
		Keywords: nondeterministic dynamical system, functional bisimulation, coalgebra, subobject classifier, dense, AMS Classification: 18B20, 68Q10, 18B25 1 Introduction In [8], we studied the category NDyn of nondeterministic dynamical systems whose morphisms are functional bisimulations.
		A Criterion for the Existence of Subobject Classifiers - Watanabe (1998)
		3 A criterion for the existence of subobject classifiers - Watanabe - 1998
	citeseer.ist.psu.edu /watanabe98subobject.html (514 words)

	Extending Reynolds
		Now if B contains the subobject classifier as a subobject, we can carry out a version of Cantor's diagonal argument in the internal logic of the topos E to conclude that it is degenerate.
		In SET, the subobject classifier coincides with the coproduct, 1+1, of the terminal object with itself; but in a general topos these two objects are very different.
		Note that the "modest sets" model of polymorphism [Hy] shows that even though P(0) cannot have a subobject classifier, it can have many of the properties which are consequences of being a topos (such as being locally cartesian closed, having a natural number object and being finitely cocomplete).
	www.seas.upenn.edu /~sweirich/types/archive/1988/msg00065.html (827 words)

	[No title]
		the point there is that the subobject classifier idea is sort of built on the idea of classifying bundles in topology [17:31]
		and the subobject classifier is a monic from 1 into it [17:42]
		If A and B are equivalent categories prove that subobject classifier for the one gives a subobject classifer for the other [18:13]
	br.endernet.org /~loner/sheaves/topos2.txt (1164 words)

	Subobject - Wikipedia, the free encyclopedia
		(The collection of subobjects of an object may in fact be a proper class; this means that the discussion given is somewhat loose.
		The dual concept to a subobject is a quotient object; that is, to define quotient object replace monomorphism by epimorphism above and reverse arrows.
		In the category Sets, a subobject of A corresponds to a subset B of A, or rather the collection of all maps from sets equipotent to B with image exactly B. The subobject partial order of a set in Sets is just its subset lattice.
	en.wikipedia.org /wiki/Subobject (324 words)

	McLarty: Elementary Categories, Elementary Toposes (Site not responding. Last check: 2007-11-03)
		Given a CCC, we have a subobject classifier iff the functor Sub is isomorphic to a representable functor in Set^{E^op}.
		For two subobjects q:Q>->A and r:R>->A of A and the generalized element w:S->A of A, w is in q=>r iff w(w*q)) is in r.
		This follows since [x,x'fx=fx'] is the kernel pair (subobject of AxA) of f, and the sequent holds iff the kernel pair factors through the diagonal iff f is monic (consider the pullback characterization of monic).
	www.andrew.cmu.edu /user/cebrown/notes/mclarty.html (7588 words)

	Topos Theory at Chicago \| The n-Category Café
		Classifying objects can be found in many categories.
		into a subobject classifier are in bijective correspondence with subobjects of
		a generalized space, is the presence of a subobject classifier.
	golem.ph.utexas.edu /category/2006/10/topos_theory_at_chicago.html (703 words)

	[No title]
		Bill wrote: While the K/S definition is right for the construction of the object classifier over an arbitrary base topos (as Gavin showed) and hence for classifiers for various kinds of finitary algebras over an arbitrary base topos, It isn't, and he didn't.
		Gavin used finite cardinals to construct the object classifier over an arbitrary base topos with NNO (and I subsequently extended the construction to finitary algebraic theories), but it doesn't work over a topos without NNO (and in particular it can't be made to work using K-finiteness).
		Perhaps there is an internal topos object V which is largest with respect to being fully embedded in the given topos E while at the same time having A as its subtopos of internal nonnon sheaves.
	www.mta.ca /~cat-dist/catlist/1999/finite-topos (2322 words)

	Eppendahl: Not v. Non (Site not responding. Last check: 2007-11-03)
		In any topos, logical not is represented by an endomap `not' on the subobject classifier and `not B', where B is some subobject of A, is the greatest subobject of A disjoint from B. In Set, `not' swaps true and false.
		The `not' endomaps in the (presheaf) topos of directed graphs and in the (presheaf) topos of functions - which we draw as tags (elements of the the function's domain) attached to nodes (elements of the function's codomain) - are pictured below.
		However, in the first case the obvious analogue of the above construction gives the wrong endomap (although not an uninteresting one) and in the second case it is not obvious what the analogous construction would be.
	homepage.mac.com /a.eppendahl/work/notes/not-non.html (264 words)

	Constructing a category Text - Physics Forums Library
		Actually, the first category I want to do this with is already cartesian, and has a subobject classifier.
		To be a topos, it has to be cartesian closed, have equalizers, and a subobject classifier andOmega;.
		(It's subobject classifier is 2 = {true, false}) What I want to do is, given a theory, to build a topos that naturally serves to model that theory, rather than start with my favorite topos and try to build a model of the theory within that topos.
	www.physicsforums.com /archive/index.php/t-80719.html (444 words)

	Geometric and Higher Order Logic in terms of Abstract Stone Duality (Site not responding. Last check: 2007-11-03)
		Conversely, when the adjunction $\Sigma^{(-)}\dashv\Sigma^{(-)}$ is monadic, this equation implies that $\Sigma$ classifies some class of monos, and the Frobenius law $\exists x.(\phi(x)\meet\psi)=(\exists x.\phi(x))\meet\psi)$ for the existential quantifier.
		In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory.
		The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré's theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.
	www.tac.mta.ca /tac/volumes/7/n15/7-15abs.html (243 words)

	Citations: The subobject classifier of the category of functional bisimulations - Watanabe (ResearchIndex)
		Watanabe H., The subobject classifier of the category of functional bisimulations (submitted), 1997.
		Moreover we give the subobject classifier concretely by using the presheaves if there exists a subobject classifier.
		Watanabe, The subobject classifier of the category of functional bisimulations (submitted).
	citeseer.ist.psu.edu /context/865021/217964 (324 words)

	[No title]
		We say that an object Y is a "subobject" of X if Y1 is a subset of X1 and Y2 is a subset of X2.
		Now, to describe a subobject Y of X, we can what's in Y today, and also what's in Y tomorrow.
		This automatically implies a lot of things, such as the existence of the subobject classifier Omega that I was talking about.
	math.ucr.edu /home/baez/twf_ascii/week68 (2852 words)

	Newspaper Classifieds -- Recommendations and Resources (Site not responding. Last check: 2007-11-03)
		Naive Bayes classifier --- assumes independent binomial conditional density models.
		Other topics related to Newspaper Classifieds: Newspaper Clippings
		Categories similar to Newspaper Classifieds: Newspaper Flagstaff
	www.becomingapediatrician.com /health/103/newspaper-classifieds.html (791 words)

	[No title]
		namely a subobject is the pullback of a monic true along a characteristic function [16:10]
		the subobjects of X are isomorphic as a category the arrows from X to the truth object [16:37]
		the subobjects of a representable presheaf are in 1-1 correspondence with the set Omega(C) [16:39]
	br.endernet.org /~loner/sheaves/topos3.txt (1194 words)

	Topos Theory Encyclopedia Article @ Despotism.net (Site not responding. Last check: 2007-11-03)
		All colimits taken over finite index categories exist.
		In many applications, the role of the subobject classifier is pivotal, whereas power objects are not.
		Thus some definitions reverse the roles of what's defined and what's derived.
	www.despotism.net /encyclopedia/Topos_theory (1725 words)

	[No title]
		A belated, somewhat tangential comment, on the distributivity condition A v /\Bi = /\(A v Bi) that Peter mentioned in his posts.
		The subobject classifier, Omega, satisfies this condition (internally) if and only if the topos is boolean.
		See the proof of Theorem 10 in Constructive Complete Distributivity II, Math Proc Cam Phil Soc, (1991) 110, 245-249, by Rosebrugh and Wood, which shows that if Omega^op is Heyting then Omega is Boolean.
	www.mta.ca /~cat-dist/catlist/1999/omega-omega (936 words)

	Klein 2-Geometry VI \| The n-Category Café
		I used `subobject’, but to connect with the Klein geometry, `subfigure’ sounds better (well, more logical to me), especially once it all gets categorified.
		In order to apply the concept of spans in the subobject lattice to 2-geometry, we need to finally figure out what notion of sub-2-object we really need.
		And that this, of course, carries over to constructions like the projective 2-space associated with a vector 2-space, in that there are non-equivalent categories of injective maps of, say, (1,0) into the two different forms of (2,2).
	golem.ph.utexas.edu /category/2006/10/klein_2geometry_vi.html (7445 words)

	Springer Online Reference Works
		For the model theory of higher-order logic one now has recourse to the elementary toposes of F.W. Lawvere and M.
		An (elementary) topos is a Cartesian closed category with a subobject classifier
		This correspondence may be expressed in categorical language, but in the familiar category of sets, where
	eom.springer.de /C/c120060.htm (2234 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Finally, the term subobject classifier is used to simulate set-theoretic concepts like intersection, union, and complement into category theory.
		Now for Topos: A topos is a category C with a terminal object, a subobject classifier, pullbacks for every pair of arrows (morphisms) and exponents for all pairs of objects.
		For category Set, sets are the objects and functions the morphisms.
	online.ohlone.cc.ca.us /computer_st/dtopham/SCU/Monads.doc (3382 words)

	Type Theory (Stanford Encyclopedia of Philosophy)
		We take as morphisms between A, E and B, F the relations R: A→B→o that are functional that is such that for any a: A satisfying E a a there exists one, and only one (modulo F) element b of B such that F b b and R a b.
		For the subobject classifier we take the pair o, E with E: o→o→o defined as
		E M N = and (imply M N) (imply N M)
	plato.stanford.edu /entries/type-theory (6501 words)

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