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| In [[category theory]], '''filtered categories''' generalize the notion of [[directed set]] understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of '''cofiltered''' category which will be recalled below.
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| ==Filtered categories==
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| A [[Category (mathematics)|category]] <math>J</math> is '''filtered''' when
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| * it is not empty,
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| * for every two objects <math>j</math> and <math>j'</math> in <math>J</math> there exists an object <math>k</math> and two arrows <math>f:j\to k</math> and <math>f':j'\to k</math> in <math>J</math>,
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| * for every two parallel arrows <math>u,v:i\to j</math> in <math>J</math>, there exists an object <math>k</math> and an arrow <math>w:j\to k</math> such that <math>wu=wv</math>.
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| A diagram is said to be of cardinality <math>\kappa</math> if the morphism set of its domain is of cardinality <math>\kappa</math>. A category <math>J</math> is filtered if and only if there is a cone over any finite diagram <math>d: D\to J</math>; more generally, for a regular cardinal <math>\kappa</math>, a category <math>J</math> is said to be <math>\kappa</math>-filtered if for every diagram <math>d</math> in <math>J</math> of cardinality smaller than <math>\kappa</math> there is a cone over <math>d</math>.
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| A '''filtered colimit''' is a [[colimit]] of a [[functor]] <math>F:J\to C</math> where <math>J</math> is a filtered category. This readily generalizes to <math>\kappa</math>-filtered limits. An '''ind-object''' in a category <math>C</math> is a presheaf of sets <math>C^{op}\to Set</math> which is a small filtered colimit of representable presheaves. Ind-objects in a category <math>C</math> form a full subcategory <math>Ind(C)</math> in the category of functors <math>C^{op}\to Set</math>. The category <math>Pro(C)=Ind(C^{op})^{op}</math> of pro-objects in <math>C</math> is the opposite of the category of ind-objects in the opposite category <math>C^{op}</math>.
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| ==Cofiltered categories==
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| A category <math>J</math> is cofiltered if the [[opposite category]] <math>J^{\mathrm{op}}</math> is filtered. In detail, a category is cofiltered when
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| * it is not empty
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| * for every two objects <math>j</math> and <math>j'</math> in <math>J</math> there exists an object <math>k</math> and two arrows <math>f:k\to j</math> and <math>f':k \to j'</math> in <math>J</math>,
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| * for every two parallel arrows <math>u,v:j\to i</math> in <math>J</math>, there exists an object <math>k</math> and an arrow <math>w:k\to j</math> such that <math>uw=vw</math>.
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| A '''cofiltered limit''' is a [[limit (category theory)|limit]] of a [[functor]] <math>F:J \to C</math> where <math>J</math> is a cofiltered category.
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| ==References==
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| * Artin, M., Grothendieck, A. and Verdier, J. L. ''Séminaire de Géométrie Algébrique du Bois Marie (SGA 4)''. Lecture Notes in Mathematics 269, Springer Verlag, 1972. Exposé I, 2.7.
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| * {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-98403-2 | year=1998}}, section IX.1.
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| <references/>
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| {{DEFAULTSORT:Filtered Category}}
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| [[Category:Category theory]]
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