ñòð. 17 
structed from a setvalued functor. In the case that F is setvalued, the Â¯rst
component u of an arrow (u; f ) : (x; C) Â¡ (x0 ; C 0) has to be an identity arrow
!
and it has to be idF f(x) . Thus the only arrows are of the form (idF f(x) ; f ) : (x; C)
Â¡ (x0 ; C). Such an arrow is denoted (x; f) in GS{1 through GS{3.
!
To visualize the Catvalued Grothendieck construction, we can modify the
picture in Diagram (ES 4.4) to get Diagram (ES 4.5). The arrows from inside one
 F f(x)  (F g Â± F f)(x)
x
 
Ff Fg
u F g(u)
? ?

x0 F g(x0 )
(4.5)
v
?
x00
 g 
f
C0 C 00
C
box to inside another, such as the arrow from x to F f (x), are parts of arrows of
G(f), which are now (in contrast to the discrete case) allowed to miss the target
and be rescued by an internal arrow of the codomain category.
Thus in the picture above there is an arrow from x to F f(x) and F f (x)
is not necessarily x0 ; the gap is Â¯lled by the arrow u : F f (x) Â¡ x0 of F (C 0 ).
!
0 0
The arrow (u; f) : (x; C) Â¡ (x ; C ) of G( ; F ) may be pictured as the arrow
!
from x to F f (x) followed by u. Observe that the deÂ¯nition of composition says
that the square in the picture with corners F f (x), (F g Â± F f )(x), x0 and F g(x0)
`commutes'.
As before, one writes (x; C) for x and (x0 ; C 0 ) for x0 only to ensure that the
union of all the categories of the form F (C) is a disjoint union.
4.2.10 An analogous construction, also called the Grothendieck construction (in
fact this is the original one), produces a split Â¯bration F( ; G) given a functor
G : op Â¡ Cat.
!
FC{1 An object of F( ; G) is a pair (C; x) where C is an object of and x is
an object of G(C).
FC{2 An arrow (f; u) : (C; x) Â¡ (C 0 ; x0 ) has f : C Â¡ C 0 an arrow of
! ! and u : x
Â¡ Gf (x0 ) an arrow of G(C).
!
48 Fibrations
FC{3 If (f; u) : (C; x) Â¡ (C 0 ; x0) and (g; v) : (C 0; x0) Â¡ (C 00 ; x00), then (g; v)
! ! Â±
(f; u) : (C; x) Â¡ (C 00; x00 ) is deÂ¯ned by
!
(g; v) Â± (f; u) = (g Â± f; Gf (v) Â± u)
4.2.11 In line with the concept that a category is a mathematical workspace,
one could ask to construct objects in a suitably rich category which themselves
are categories. The Grothendieck construction provides a way to describe func
tors from such a category object to the ambient category which is worked out
in ES 5.7.2.
4.2.12 Exercises
1. Verify that GS{1 through GS{3 deÂ¯ne a category.
Â¡ Set, G0 (F ) : G0( ; F ) Â¡
! !
2. Show that for any functor F : is a split
opÂ¯bration.
3. Verify that GC{1 through GC{3 deÂ¯ne a category.
Â¡ Cat, G(F ) : G( ; F ) Â¡
! !
4. Show that for any functor F : is a split
opÂ¯bration.
5. Verify that the deÂ¯nition of the semidirect product in ES 4.2.6 makes T Â£ M
a monoid.
6. Let F : Â¡ Cat be a functor. Show that for each object C of , the arrows
!
of the form (u; idC ) : (x; C) Â¡ (y; C) (for all arrows u : x Â¡ y of F (C)) (and
! !
their sources and targets) form a subcategory of the opÂ¯bration G( ; F ) which
is isomorphic to F (C).
4.3 An equivalence of categories
In this section, we describe how the construction of a functor from an opÂ¯bration
given in Proposition ES 4.1.8 (in one direction) produces an equivalence of cate
gories (with the Grothendieck construction as pseudoinverse) between a category
of functors and a suitably deÂ¯ned category of split opÂ¯brations.
4.3.1 Catvalued functors For a category , Func( ; Cat) is the category
whose objects are functors from to the category of categories, and whose arrows
are natural transformations between them.
Â¡ Cat is such a functor and f : C Â¡ D is an arrow of , then
! !
If F :
F (C) and F (D) are categories and F f : F (C) Â¡ F (D) is a functor. If also G :
!
Â¡ Cat and Â® : F Â¡ G is a natural transformation, then for each object of C,
! !
4.3 An equivalence of categories 49
Â®C : F (C) Â¡ G(C) is a functor and the following diagram is a commutative
!
diagram of categories and functors:

F (C) Â®C G(C)
Ff Gf (4.6)
? ?
 G(D)
F (D)
Â®D
! and P 0 : 0
Let P : Â¡
4.3.2 The category of split opÂ¯brations of
with splittings Â· and Â·0
Â¡! be two split opÂ¯brations of the same category
respectively. A homomorphism of split opÂ¯brations is a functor Â³ : Â¡ 0 !
for which
HSO{1 The diagram

Â³ 0
@ Â¡
P@ Â¡ P0 (4.7)
@ Â¡
@
R Âª
Â¡
commutes.
HSO{2 For any arrow f : C Â¡ D in
! and object X of such that P (X) = C,
Â³(Â·(f; X)) = Â·0 (f; Â³(X))
Thus a homomorphism of split Â¯brations `takes Â¯bers to Â¯bers' and `preserves
the splitting'.
4.3.3 DeÂ¯nition Split opÂ¯brations of and homomorphisms between them
form a category SO( ).
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