The Grothendieck construction is a construction used in the mathematical field of category theory.
Let
F
:
C
→
C
a
t
be a functor from any small category to the category of small categories. The Grothendieck construction for
F
is the category
Γ
(
F
)
(also written
C
∫
F
or
F
⋊
C
), with
objects being pairs
(
c
,
x
)
, where
c
∈
obj
(
C
)
and
x
∈
obj
(
F
(
c
)
)
; and
morphisms in
hom
Γ
(
F
)
(
(
c
1
,
x
1
)
,
(
c
2
,
x
2
)
)
being pairs
(
f
,
g
)
such that
f
:
c
1
→
c
2
in
C
, and
g
:
F
(
f
)
(
x
1
)
→
x
2
in
F
(
c
2
)
.
Composition of morphisms is defined by
(
f
,
g
)
∘
(
f
′
,
g
′
)
=
(
f
∘
f
′
,
g
∘
F
(
f
)
(
g
′
)
)
.
