- Source: 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
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.
Cofiltered categories
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.
Ind-objects and pro-objects
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}}
.
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
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.