In category theory, if C is a category and
F
:
C
→
S
e
t
is a set-valued functor, the category of elements of F
e
l
(
F
)
(also denoted by ∫CF) is the category defined as follows:
Objects are pairs
(
A
,
a
)
where
A
∈
O
b
(
C
)
and
a
∈
F
A
.
An arrow
(
A
,
a
)
→
(
B
,
b
)
is an arrow
f
:
A
→
B
in C such that
(
F
f
)
a
=
b
.
A more concise way to state this is that the category of elements of F is the comma category
∗
↓
F
, where
∗
is a one-point set. The category of elements of F comes with a natural projection
e
l
(
F
)
→
C
that sends an object (A,a) to A, and an arrow
(
A
,
a
)
→
(
B
,
b
)
to its underlying arrow in C.
Somewhat confusingly in some texts (e.g. Mac Lane, Moerdijk), the category of elements for a presheaf is defined differently. If
P
∈
C
^
:=
S
e
t
C
o
p
is a presheaf, the category of elements of P (again denoted by
e
l
(
P
)
, or, to make the distinction to the above definition clear, ∫C P) is the category defined as follows:
Objects are pairs
(
A
,
a
)
where
A
∈
O
b
(
C
)
and
a
∈
P
(
A
)
.
An arrow
(
A
,
a
)
→
(
B
,
b
)
is an arrow
f
:
A
→
B
in C such that
(
P
f
)
b
=
a
.
As one sees, the direction of the arrows is reversed. One can, once again, state this definition in a more concise manner: the category just defined is nothing but
(
∗
↓
P
)
o
p
. Consequentially, in the spirit of adding a "co" in front of the name for a construction to denote its opposite, one should rather call this category the category of coelements of P.
For C small, this construction can be extended into a functor ∫C from
C
^
to
C
a
t
, the category of small categories. In fact, using the Yoneda lemma one can show that ∫CP
≅
y
↓
P
, where
y
:
C
→
C
^
is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to
y
↓
−
:
C
^
→
Cat
.
