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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 categoriesEdit

A category   is filtered when

  • it is not empty,
  • for every two objects   and   in   there exists an object   and two arrows   and   in  ,
  • for every two parallel arrows   in  , there exists an object   and an arrow   such that  .

A filtered colimit is a colimit of a functor   where   is a filtered category.

Cofiltered categoriesEdit

A category   is cofiltered if the opposite category   is filtered. In detail, a category is cofiltered when

  • it is not empty
  • for every two objects   and   in   there exists an object   and two arrows   and   in  ,
  • for every two parallel arrows   in  , there exists an object   and an arrow   such that  .

A cofiltered limit is a limit of a functor   where   is a cofiltered category.

Ind-objects and pro-objectsEdit

Given a small category  , a presheaf of sets   that is a small filtered colimit of representable presheaves, is called an ind-object of the category  . Ind-objects of a category   form a full subcategory   in the category of functors (presheaves)  . The category   of pro-objects in   is the opposite of the category of ind-objects in the opposite category  .

κ-filtered categoriesEdit

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   of the form  ,  , or  . 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   is filtered (according to the above definition) if and only if there is a cocone over any finite diagram  .

Extending this, given a regular cardinal κ, a category   is defined to be κ-filtered if there is a cocone over every diagram   in   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 (co)limit is a (co)limit of a functor   where   is a κ-filtered category.

ReferencesEdit

  • 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.
  • Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2, section IX.1.