Filtered category

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.

A category J {\displaystyle J} is filtered when

• it is not empty,
• for every two objects j {\displaystyle j} and j ′ {\displaystyle j'} in J {\displaystyle J} there exists an object k {\displaystyle k} and two arrows f : j → k {\displaystyle f:j\to k} and f ′ : j ′ → k {\displaystyle f':j'\to k} in J {\displaystyle J} ,
• for every two parallel arrows u , v : i → j {\displaystyle u,v:i\to j} in J {\displaystyle J} , there exists an object k {\displaystyle k} and an arrow w : j → k {\displaystyle w:j\to k} such that w u = w v {\displaystyle wu=wv} .

A filtered colimit is a colimit of a functor F : J → C {\displaystyle F:J\to C} where J {\displaystyle J} is a filtered category.

A category J {\displaystyle J} is cofiltered if the opposite category J o p {\displaystyle J^{\mathrm {op} }} is filtered. In detail, a category is cofiltered when

• it is not empty,
• for every two objects j {\displaystyle j} and j ′ {\displaystyle j'} in J {\displaystyle J} there exists an object k {\displaystyle k} and two arrows f : k → j {\displaystyle f:k\to j} and f ′ : k → j ′ {\displaystyle f':k\to j'} in J {\displaystyle J} ,
• for every two parallel arrows u , v : j → i {\displaystyle u,v:j\to i} in J {\displaystyle J} , there exists an object k {\displaystyle k} and an arrow w : k → j {\displaystyle w:k\to j} such that u w = v w {\displaystyle uw=vw} .

A cofiltered limit is a limit of a functor F : J → C {\displaystyle F:J\to C} where J {\displaystyle J} is a cofiltered category.

Given a small category C {\displaystyle C} , a presheaf of sets C o p → S e t {\displaystyle C^{op}\to Set} that is a small filtered colimit of representable presheaves, is called an ind-object of the category C {\displaystyle C} . Ind-objects of a category C {\displaystyle C} form a full subcategory I n d ( C ) {\displaystyle Ind(C)} in the category of functors (presheaves) C o p → S e t {\displaystyle C^{op}\to Set} . The category P r o ( C ) = I n d ( C o p ) o p {\displaystyle Pro(C)=Ind(C^{op})^{op}} of pro-objects in C {\displaystyle C} is the opposite of the category of ind-objects in the opposite category C o p {\displaystyle C^{op}} .

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 J {\displaystyle J} of the form {     } → J {\displaystyle \{\ \ \}\rightarrow J} , { j       j ′ } → J {\displaystyle \{j\ \ \ j'\}\rightarrow J} , or { i ⇉ j } → J {\displaystyle \{i\rightrightarrows j\}\rightarrow J} . 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 J {\displaystyle J} is filtered (according to the above definition) if and only if there is a cocone over any finite diagram d : D → J {\displaystyle d:D\to J} .

Extending this, given a regular cardinal κ, a category J {\displaystyle J} is defined to be κ-filtered if there is a cocone over every diagram d {\displaystyle d} in J {\displaystyle J} 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 F : J → C {\displaystyle F:J\to C} where J {\displaystyle J} is a κ-filtered category.