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.

Filtered categories

A category ${\displaystyle J}$ is filtered when

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

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

Cofiltered categories

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

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

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

Ind-objects and pro-objects

Given a small category ${\displaystyle C}$, a presheaf of sets ${\displaystyle C^{op}\to Set}$ that is a small filtered colimit of representable presheaves, is called an ind-object of the category ${\displaystyle C}$. Ind-objects of a category ${\displaystyle C}$ form a full subcategory ${\displaystyle Ind(C)}$ in the category of functors (presheaves) ${\displaystyle C^{op}\to Set}$. The category ${\displaystyle Pro(C)=Ind(C^{op})^{op}}$ of pro-objects in ${\displaystyle C}$ is the opposite of the category of ind-objects in the opposite category ${\displaystyle C^{op}}$.

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

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

References

• 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.