 F f(x)  (F g ± F f)(x)
x
 
Ff Fg
(4.4)

x0 F g(x0 )
 g 
f
C0 C 00
C
represents the elements of F (C). The arrows from x to F f (x) and from x0
of
to F g(x0 ) are there informally to illustrate what the set functions F f and F g do.
These informal arrows become actual arrows in G0 ( ; F ).
4.2.3 De¯nition G0 ( ; F ) is the category de¯ned as follows.
GS{1 An object of G0( ; F ) is a pair (x; C) where C is an object of and x is
an element of F (C) (as observed, the C occurs in the pair (x; C) because
we want the disjoint union of the values of F ).
GS{2 An arrow is a pair of the form (x; f) : (x; C) ¡ (x0 ; C 0 ) where f is an arrow
!
f : C ¡ C 0 of for which F f (x) = x0 .
!
GS{3 If (x; f ) : (x; C) ¡ (x0; C 0 ) and (x0; g) : (x0 ; C 0) ¡ (x00 ; C 00), then (x0 ; g) ±
! !
(x; f) : (x; C) ¡ (x00 ; C 00 ) is de¯ned by
!
(x0 ; g) ± (x; f) = (x; g ± f )
Note in GS{3 that indeed F (g ± f )(x) = x00 as required by GS{2.
The reason that we use the notation (x; f ) is the requirement that an arrow
must determine its source and target. The source of (x; f ) is (x; C), where C is
the source of f and x is explicit, while its target is (x0; C 0), where C 0 is the target
of f and x0 = F f(x). In the literature, (x; f ) is often denoted simply f , so that
4.2 The Grothendieck construction 45
the same name f may refer to many di®erent arrows { one for each element of
F (C). We used lacunary notation of this sort in de¯ning slice categories in 2.6.10.
Projection on the second coordinate de¯nes a functor
G0 (F ) : G0 ( ; F ) ¡
!
G0 ( ; F ) together with G0 (F ) is called the split discrete op¯bration induced
by F , and is the base category of the op¯bration.
If C is an object of , the inverse image under G0 (F ) of C is simply the set
F (C), although its elements are written as pairs so as to form a disjoint union.
This discrete op¯bration is indeed an op¯bration, in fact a split op¯bration. If
f : C ¡ C 0 in and (x; C) is an object of G0 ( ; F ), then an opcartesian arrow
!
is (x; f ) : (x; C) ¡ (F f (x); C 0 ) (Exercise ES 2). The word `discrete' refers to the
!
fact that the ¯bers are categories in which the only arrows are identity arrows;
such categories are essentially the same as sets.
4.2.4 Semidirect products We now describe a more general version of the
Grothendieck construction that has the semidirect product of monoids as a special
case. We ¯rst de¯ne the semidirect product of monoids: it is constructed from
two monoids, one of which acts on the other.
4.2.5 De¯nition If M and T are monoids, an action of M on T is a function
® : M £ T ¡ T for which
!
®(m; 1T ) = 1T for all m 2 M .
MA{1
®(m; tu) = ®(m; t)®(m; u) for all m 2 M and t; u 2 T .
MA{2
®(1M ; t) = t for all t 2 T .
MA{3
®((mn); t) = ®(m; ®(n; t)) for all m; n 2 M and t 2 T .
MA{4
If we curry ® as in 6.1.2, we get a family of functions Á(m) : T ¡ T with the
!
properties listed in MA0 {1 through MA0{4 below.
MA0{1 Á(m)(1T ) = 1T for all m 2 M .
MA0{2 Á(m)(tu) = Á(m)(t)Á(m)(u) for all m 2 M and t; u 2 T .
MA0{3 Á(1M )(t) = t for all t 2 T .
MA0{4 Á(mn)(t) = Á(m)[Á(n)(t)] for all m; n 2 M and t 2 T .
Thus we see that an alternative formulation of monoid action is that it is a
monoid homomorphism Á : M ¡ End(T ) (End(T ) being the monoid of endo
!
morphisms of T ). MA {1 and MA0 {2 say that each function Á(m) is an endo
0
morphism of T , and MA0{3 and MA0{4 say that Á is a monoid homomorphism.
46 Fibrations
4.2.6 De¯nition The semidirect product of M and T with the given action
® as just de¯ned is the monoid with underlying set T £ M and multiplication
de¯ned by
(t; m)(t0; m0) = (t®(m; t0 ); mm0 )
To see the connection with the categorical version below you may wish to
write this de¯nition using the curried version of ®.
4.2.7 The categorical construction corresponding to a monoid acting on a mon
oid is a functor which takes values in Cat rather than in Set. A functor F :
¡ Cat can be regarded as an action of on a variable category which plays the
!
role of T in the de¯nition just given.
In the case of a monoid action de¯ned by MA{1 through MA{4, the variable
category is actually not varying: it is the category C(T ) determined by the monoid
T . The functor F in that case takes the single object of M to the single object of
T , and, given an element m 2 M , F (m) is the endomorphism of T which takes
t 2 T to mt: in other words, F (m)(t) = mt. Thus F on the arrows is the curried
form of the action ®.
A setvalued functor is a special case of a categoryvalued functor, since a set can
be regarded as a category with only identity arrows. Note that this is di®erent from
the monoid case: an action by a monoid on a set is not in general a special case of
an action by the monoid on a monoid. It is, however, a special case of the action of
a monoid on a category { a discrete category.
¡ Cat, the Grothendieck construction in this
!
4.2.8 Given a functor F :
more general setting constructs the op¯bration induced by F , a category G( ; F )
de¯ned as follows:
GC{1 An object of G( ; F ) is a pair (x; C) where C is an object of and x is
an object of F (C).
GC{2 An arrow (u; f ) : (x; C) ¡ (x0 ; C 0 ) has f : C ¡ C 0 an arrow of
! ! and
0 0
u : F f(x) ¡ x an arrow of F (C ) (note that by de¯nition F f(x) is an
!
object of F (C 0)).
GC{3 If (u; f ) : (x; C) ¡ (x0; C 0 ) and (v; g) : (x0 ; C 0) ¡ (x00; C 00 ), then (v; g) ±
! !
00 00
(u; f ) : (x; C) ¡ (x ; C ) is de¯ned by
!
(v; g) ± (u; f ) = (v ± F g(u); g ± f )
4.2.9 Theorem Given a functor F : ¡ Cat, G( ; F ) is a category and the
!
second projection is a functor P : G( ; F ) ¡ ! which is a split op¯bration with
splitting
·(f; X) = (idF fx ; f ) : (x; C) ¡ (F fx; C 0)
!
for any arrow f : C ¡ C 0 of
! and object (x; C) of G( ; F ).
4.2 The Grothendieck construction 47
We omit the proof of this theorem. G( ; F ) is called the crossed product
£ F by some authors.