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	Comma category - Wikipedia, the free encyclopedia
		A comma category (also sometimes called a slice category) is a construction in category theory, a branch of mathematics.
		In either of these two cases, the identity functor may be replaced with some other functor; this yields a family of categories particularly useful in the study of adjoint functors.
		Essentially, we create a category whose objects are cones, and where the limiting cone is a terminal object; then, each universal morphism for the limit is just the morphism to the terminal object.
	en.wikipedia.org /wiki/Comma_category (1162 words)

	Category of topological spaces - Wikipedia, the free encyclopedia
		In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps.
		This is a category because the composition of two continuous maps is again continuous.
		The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.
	en.wikipedia.org /wiki/Top_(category_theory) (526 words)

	All about Comma category - RecipeLand.com Reference library (Site not responding. Last check: 2007-11-04)
		The morphisms of the comma category are mappings on the " $\alpha " and " \beta " parts, producing as a consequence a mapping on the " f " part.$
		The category of pointed sets is a comma category, $(x \downarrow \mathbf{Set}) with x being (a functor selecting) a singleton set, and \mathbf{Set} (the identity functor of) the$ category of sets.
		The category of graphs is $(\mathbf{Set} \downarrow D), with D : \mathbf{Set} \rightarrow \mathbf{Set} the functor taking a set s to s \times s .$
	www.recipeland.com /encyclopaedia/index.php/Comma_category (1520 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		The embeddings category is the right setting for analysing this: each embedding is a tuple of functions mapping the nodes and ports of one action graph into another.
		C, is a category containing (i) the agents that perform reactions and labelled transitions and (ii) the agent contexts that serve as the labels and specify the closure condition for congruence.
		This separation is useful because the category for which we prove a congruence result is typically not one in which RPOs exist, as I show in the next chapter when considering categories of graph contexts.
	join.inria.fr /~leifer/articles/leifer-thesis-tech.txt (22817 words)

	Category of topological spaces -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-04)
		This is a category because the (A mixture of ingredients) composition of two continuous maps is again continuous.
		The study of Top and of properties of ((mathematics) any set of points that satisfy a set of postulates of some kind) topological spaces using the techniques of (Click link for more info and facts about category theory) category theory is known as categorical topology.
		The category of (Click link for more info and facts about pointed topological space) pointed topological spaces is a (Click link for more info and facts about coslice category) coslice category over Top
	www.absoluteastronomy.com /encyclopedia/c/ca/category_of_topological_spaces.htm (455 words)

	Dictionary of Meaning www.mauspfeil.net
		It provides another way of looking at morphisms: instead of simply relating objects of a Category (mathematics) category to one another, they become objects in their own right.
		There are also certain guarantees about the existence of Limit (category theory) limits and colimits in the context of comma categories.
		Locally cartesian closed categories are the classifying category classifying categories of dependent type theory dependent type theories.
	www.mauspfeil.net /Comma_category.html (1195 words)

	Category of topological spaces (Site not responding. Last check: 2007-11-04)
		This is a category because the function compositioncomposition of two continuous maps is again continuous.
		The coproduct (category theory)coproduct is given by the disjoint union (topology)disjoint union of topological spaces.
		It should be noted that the added structure of this subcategory allows for more epics: in fact, the epics in this subcategory are precisely those morphisms with dense setdense image (mathematics)images in their codomains, so that epics need not be surjective/.
	www.infothis.com /find/Category_of_topological_spaces (386 words)

	Practical Foundations of Mathematics (Site not responding. Last check: 2007-11-04)
		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).
		be a diagram in a category with equalisers.
		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)

	Comma category (Site not responding. Last check: 2007-11-04)
		A comma category is a construction in category theory, a branch of mathematics.
		The morphisms of the comma category are mappings on the "
		(T \downarrow A) is the category of graphs whose edges are labelled by elements of
	www.worldhistory.com /wiki/C/Comma-category.htm (1045 words)

	Practical Foundations of Mathematics (Site not responding. Last check: 2007-11-04)
		HSL of Boolean algebras in the category of Heyting semilattices has both a reflection and a co-reflection, and these functors are the same.
		Finally, elementary sketches present equational many-sorted unary theories, and the classifying category is free on the sketch.
		If we take the semantic option, then the universal property of the classifying category is more complicated than Definition 7.1.1: the interpretation functor [[-]] is only unique up to unique isomorphism - if it is defined at all, as some Choice is to be made.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s71.html (1926 words)

	Encyclopedia: Basepoint (Site not responding. Last check: 2007-11-04)
		The pointed set concept is less important; it is anyway the case of a pointed discrete space.
		The coproduct in the category of pointed spaces is the wedge sum, which can be thought of as the one-point union of spaces.
		The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum.
	www.nationmaster.com /encyclopedia/Basepoint (442 words)

	cars - Category of topological spaces (Site not responding. Last check: 2007-11-04)
		The coproduct is given by the disjoint union of topological spaces.
		The forgetful functor has a left adjoint (which equips a given set with the discrete topology) and a right adjoint (which equips a given set with the trivial topology).
		It should be noted that the added structure of this subcategory allows for more epics: in fact, the epics in this subcategory are precisely those morphisms with dense images in their codomains, so that epics need not be surjective.
	www.carluvers.com /cars/Top_%28category_theory%29 (365 words)

	Encyclopedia: Coslice category (Site not responding. Last check: 2007-11-04)
		that are related in some special way (given by the two functors) to the categories
		The morphisms of the comma category are mappings on the "α" and "β" parts, producing as a consequence a mapping on the "f" part.
		Click for other authoritative sources for this topic (summarised at Factbites.com).
	www.nationmaster.com /encyclopedia/Coslice-category (1197 words)

	Practical Foundations of Mathematics (Site not responding. Last check: 2007-11-04)
		B in a category with pullbacks is the pullback square
		Congruences The kernel pair of any map, where it exists in a category, is an equivalence relation (Definition 1.2.3), which we express in the style of Remark 5.2.8.
		This is not finitist dogma: the condition may be formulated abstractly, and is called projectivity, Remark 5.8.4(e).
	www.cs.man.ac.uk /%7Ept/Practical_Foundations/html/s56.html (1603 words)

	Comma category - Enpsychlopedia (Site not responding. Last check: 2007-11-04)
		Suppose that $\mathcal{A}, \mathcal{B}, and \mathcal{C} are$ categories, and $T and S are functors$
		The category of pointed sets is a comma category, $(\bull \downarrow \mathbf{Set}) with \bull being (a functor selecting) any singleton set, and \mathbf{Set} (the identity functor of) the$ category of sets.
		In a similar fashion one can form the category of pointed spaces $(\bull \downarrow \mathbf{Top}) .$
	www.grohol.com /psypsych/Comma_category (1520 words)

	Comma category biography .ms (Site not responding. Last check: 2007-11-04)
		\mathcal{B} The morphisms of the comma category are mappings on the "
		G : \mathcal{D} \rightarrow \mathcal{C} are adjoint if and only if the comma categories
		\mathcal{C} \times \mathcal{D} This allows adjunctions to be described without involving sets, and was in fact the original motivation for introducing comma categories.
	www.biography.ms /Coslice_category.html (1002 words)

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