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the remaining data are determined by these. As a matter of convenience, we
usually omit the indexes unless they are necessary for comprehension. If D =
hD0 ; D1 ; u; s; t; ci is another category object, then an internal functor f : D Â¡
!
C is given by homomorphisms f0 : D0 Â¡ C0 and f1 : D1 Â¡ C1 such that the
! !
following diagrams commute, where f2 : D2 Â¡ C2 is the unique arrow for which
!
pi Â± f2 = f1 Â± pi ; i = 1; 2.
f1  f1 
D1 C1 D1 C1
s s t t
? ? ? ?
 C0  C0
D0 D0
f0 f0
76 Toposes
(5.3)
f2  f0 
D2 C2 D0 C0
c c u u
? ? ? ?
 C1  C1
D1 D1
f1 f1
5.7.2 External functors The notion of a functor from a category into the
category of sets can be extended to describe a functor from a category in to
itself. Such a functor is called an external functor. This construction is based
on the construction in Section ES 4.2 for Set. Theorem ES 4.3.7 gives an equiva
lence of categories between external functors and the split opÂ¯brations which are
produced by the Grothendieck construction, and so justiÂ¯es the representation
by split opÂ¯brations of external functors deÂ¯ned on a category object.
Essentially, what we will do is use objects and arrows representing sets and
functions involved in the Grothendieck construction, Â¯nd enough cones and com
mutative diagrams to characterize it, and take that as the deÂ¯nition of external
functor in an arbitrary category.
5.7.3 Let be a category, and let
C = (C0 ; C1 ; source; target; unit; comp)
. An external functor C Â¡
!
be a category object in consists of data
(D0; D1 ; D2; d 0 ; d 1 ; u; Â¼0 ; Â¼1; p1 ; p2 ; c)
for which the Di are objects of and the arrows have sources and targets as
indicated:
di D1 Â¡ D0 ; i = 1; 2
!
:
D 0 Â¡ D1
!
:
u
Di Â¡ Ci ; i = 0; 1
!
:
Â¼i
D2 Â¡ D1 ; i = 1; 2
!
:
pi
D 2 Â¡ D1 :
!
:
c
D0 is the object corresponding to the disjoint union of the values of the functor.
Note that we are not given a functor F as we were in Section ES 4.2 { we are being
guided by Theorem ES 4.3.7 and the details of the Grothendieck construction to
deÂ¯ne an external functor, and Â¼0 represents the projection (taking (x; C) to C
in the case of the Grothendieck construction). D1 is the object corresponding to
the arrows of the category G(C; F ) as deÂ¯ned in Section ES 4.2, and Â¼1 takes
5.7 External functors 77
(x; f ) to f . Thus Â¼0 and Â¼1 together describe the functor G(F ) in the case of the
Grothendieck construction.
d 0 and d 1 are the source and target maps of that category, c is the composition
and u picks out the identities.
The data are subject to the requirements E{1 through E{4 below.
E{1 All three diagrams below must commute and (a) must be a pullback:
  
Â¼1 Â¼1 Â¼o
D1 C1 D1 C1 D1 C1
6 6
source d 1 target u
d0 unit
(5.4)
? ? ? ?
 C0  C0 
D 0 Â¼0 D0 Â¼0 D0 Â¼1 C0
(a) (b) (c)
That (a) is a pullback says in the case of the Grothendieck construction that,
up to unique isomorphism, D1 consists of elements of the form (x; f) with f an
arrow of and x 2 F (C) where C is the source of f. This is part of GS{2 (see
Section ES 4.2). The commutation of (b) says that d1(x; f ) 2 F (C 0 ) where C 0 is
the target of F ; that follows from GS{1 and GS{2 (there, x0 must be in F (C 0)).
That of (c) says that u(x; C) must be (idC ). In the case of the Grothendieck
construction that follows from the fact that F is given as a functor.
E{2 The following diagram is a pullback:
p2 
D2 D1
p1 d1
? ?

D1 D0
d0
(d)
In the case of the Grothendieck construction this forces D2 to be the set of
composable pairs of arrows of G(C; F ).
E{3 The following diagram must commute:
78 Toposes
Â¼1 
D1 C1
6 6
p1 p1
 C2
D2 Â¼2
p2 p2
? ?
 C1
D1 Â¼1
(e)
Â¼2 is deÂ¯ned by Diagram (ES 5.3) (called f2 there). In the case of the Grothen
dieck construction this follows from GS{3: (x; g 0) composes with (x; f ) only if g
composes with f (but not conversely!).
E{4 The following diagrams must commute.
 p1
 p2

Â¼2
D2 C2 D2 D1 D2 D1
comp
c c c
d0 d1
(5.5)
? ? ? ? ? ?
 C1  
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