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.
==Filtered categories==
A category <math>J</math> is '''filtered''' when * it is not empty, * 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>, * 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>.
A '''filtered colimit''' is a colimit of a functor <math>F:J\to C</math> where <math>J</math> is a filtered category.
==Cofiltered categories== 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 * it is not empty, * 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>, * 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>.
A '''cofiltered limit''' is a limit of a functor <math>F:J \to C</math> where <math>J</math> is a cofiltered category.
==Ind-objects and pro-objects==
Given a small category <math>C</math>, a presheaf of sets <math>C^{op}\to Set</math> that is a small filtered colimit of representable presheaves, is called an '''ind-object''' of the category <math>C</math>. Ind-objects of a category <math>C</math> form a full subcategory <math>Ind(C)</math> in the category of functors (presheaves) <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>.
==κ-filtered categories== There is a variant of "filtered category" known as a "κ-filtered category", defined as follows. This begins with the following observation: the three conditions in the definition of filtered category above say respectively that there exists a cocone over any diagram in <math>J</math> of the form <math>\{\ \ \}\rightarrow J</math>, <math>\{j\ \ \ j'\}\rightarrow J</math>, or <math>\{i\rightrightarrows j\}\rightarrow J</math>. The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist for ''any'' finite diagram; in other words, a category <math>J</math> is filtered (according to the above definition) if and only if there is a cocone over any ''finite'' diagram <math>d: D\to J</math>.
Extending this, given a regular cardinal κ, a category <math>J</math> is defined to be κ-filtered if there is a cocone over every diagram <math>d</math> in <math>J</math> of cardinality smaller than κ. (A small diagram is of cardinality κ if the morphism set of its domain is of cardinality κ.)
A κ-filtered colimit is a colimit of a functor <math>F:J\to C</math> where <math>J</math> is a κ-filtered category.
==References== <references/> {{refbegin}} * 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. * {{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. {{refend}}
{{DEFAULTSORT:Filtered Category}} Category:Category theory