Table Of ContentThe dual fibration in elementary terms
5
1
0 Anders Kock
2
n
a
J
We give an elementary construction of the dual fibration of a fibration. It
8
does not use the non-elementary notion of (pseudo-) functor into the category of
] categories. In fact, it is clear that the construction we present makes sense for
T
C internalcategoriesand fibrationsin anyexact category.
. ThedualfibrationofafibrationX →BoverBisdescribedine.g.[Borceux]
h
t II.8.3 via a pseudofunctor F : Bop →Cat (the category of categories), by com-
a
m posingF withthe(covarariant!) dualizationfunctorCat →Cat;choosingsuchan
[ F is tantamount to choosing a cleavage for the fibration. In the present section,
we give an alternative description of the dual fibration, which is elementary and
1
v choice-free.
7
4
9
1 Fibrations
1
0
.
1 Werecall here someclassicalnotions.
0
Letp :X →B beanyfunctor. Fora :A→BinB,andforobjectsX,Y ∈X
5
1 withp (X)=Aandp (Y)=B,lethoma (X,Y)bethesetofarrowsh:X →Y inX
v: with p (h)=a . For any arrow x :C→A, and any object Z ∈X with p (Z)=C,
i
X post-compositionwithhdefines a map
r
a
h∗ :homx (Z,X)→homx .a (Z,Y).
(we compose from left to right). Recall that h is called Cartesian if this map is a
bijection,forall suchx and Z.
If h is Cartesian, the injectivity of h implies the cancellation property that h
∗
is “monic w.r. to p ”, meaning that for parallel arrows k,k′ in X with codomain
X,and withp (k)=p (k′), wehavethat k.h=k′.h impliesk=k′.
Forlateruse,we recall abasicfact:
Lemma 1.1 If k=k′.h isCartesian,andh isCartesianthen k′ is Cartesian.
1
The functor p :X →B is called a fibrationif for every a :A→B in B and
anyY ∈X withp (Y)=B,thereexistsaCartesianarrowovera withcodomainY.
ThefibreoverA∈B isthecategorywhoseobjectsaretheX ∈X withp (X)=A,
andwhosearrowsarearrowsinX whichbyp mapto1 ;sucharrowsarecalled
A
vertical(overA).
All this is standard, dating back essentially to early French category theory
(Grothendieck, Chevalley, Giraud, Bénabou,...). For a modern account, see [1]
II.8.1, [2] B.1.3, or [3]. Note that these notions are elementary (they make sense
for category objects in any left exact category), and they do not depend on the
non-elementarynotionsofcleavage, orCat-valuedpseudofunctor.
2 The “factorization system” for a fibration
Inthediagramsbelow,wetrytomakedisplayverticalarrowsvertically,andCarte-
sianarrows horizontally.
Recall from the literature that if p : X → B is a fibration, then every arrow
z in X may be written as a compositeof a vertical arrow followedby a cartesian
arrow. And, crucially, thisdecompositionof z is uniquemoduloa uniquevertical
isomorphism. Or,equivalently,moduloanarrowwhichisatthesametimevertical
and cartesian. (Recall that for vertical arrows, cartesian is equivalent to isomor-
phism (= invertible).) This means that every arrow z in X may be represented
by a pair (v,h) of arrows with v vertical and h cartesian, with z = v.h. Thus the
codomainofvis thedomainofh. Wecall such apair a“vh compositionpair”, to
maketheanalogywithvhspans,tobeconsideredbelow,moreexplicit. Twosuch
pairs(v,h)and(v′,h′)representthesamearrowiffthereexistsaverticalcartesian
(necessarily unique,and necessarily invertible)i suchthat
v.i=v′ andi.h′ =h. (1)
We say that (v,h) and (v′,h′) are equivalent if this holds. The composition of
arrows in X can be described in terms of representative vh composition pairs,
as follows. If z is represented by (v ,h ) for j = 1,2, then z .z is represented
j j j 1 2
by (v .w,k.h ), where k is cartesian over p (h ) and w is vertical, and the square
1 2 1
2
displayedcommutes:
·
v
1
❄ h
· 1 ✲ ·
w v
2
❄ ❄
· ✲ · ✲ ·
k h
2
Such k and w exists (uniquely, up to unique vertical cartesian arrows): construct
first k as a cartesian lift of p (h ), then use the universal property of cartesian
1
arrowsto constructw.
Thearrows z and z may be inserted, completingthe diagramwith two com-
1 2
mutative triangles, since z = v .h . But if we refrain from doing so, we have a
j j j
blueprint for a succinct and choice-free description of the fibrewise dual X ∗ of
thefibration X →B.
Note that a vh factorization of an arrow in X is much reminiscent of the
factorizationforanE-M factorizationsystem,asin[Borceux]I.5.5,say,(withthe
class of vertical arrows playing the role of E, and the class of cartesian arrows
playingtheroleofM;however,notethatnoteveryisomorphisminX isvertical.
3 The dual fibration X ∗
The construction presented in this Section is still elementary, but requires more
than just left exactness in the category where it is performed, namely exactness;
this implies that good quotients exist for equivalence relations, and that maps on
such a quotientcan be defined by assigningvalueson representativeelementsfor
the equivalence classes. – We present the construction in the exact category of
sets,forsimplicity.
Given a fibration p :X →B. We describe another category X ∗ over B, as
follows: The objects of X ∗ are the same as those of X ; the arrows X →Y are
represented byvh spans,in thefollowingsense:
3
Definition3.1 Avh span inX fromX toY isa diagraminX of theform
h
· ✲ Y
v (2)
❄
X
withvvertical andh cartesian.
The set of arrows in X ∗ from X to Y are equivalence classes of vh spans from
X to Y, for the equivalence relation ≡ given by (v,h) ≡ (v′,h′) if there exists a
verticalisomorphismi (necessarily unique)inX so that
i.v.=v′ andi.h=h′. (3)
We denote the equivalence class of the vh span (v,h) by {(v,h)}. They are the
arrows of X ∗; the direction of a the arrow {(v,h)} is determined by its cartesian
part h.
Composition has to be described in terms of representative pairs; it is in fact
the standard composite of spans, but let us be explicit: If z is represented by
j
(v ,h ) for j =1,2, then z .z is represented by (w,k), where k is cartesian over
j j 1 2
p (h )and wis vertical,andthesquaredisplayedcommutes:
1
k h
· ✲ · 2 ✲ ·
w v
2
❄ ❄
· ✲ · (4)
h
1
v
1
❄
·
Such k and w exists (uniquely, up to unique vertical cartesian arrows): construct
first k as a cartesian lift of p (h ), then use the universal property of cartesian
1
4
arrowstoconstructw. (Thesquaredisplayedwillthenactuallybeapull-backdia-
gram,thusthecompositiondescribedwillbethestandardcompositionofspans.)
Compositionofvhspansdoesnotgiveadefinitevhspan,butratheran equiv-
alence class of vh spans. So referring to (4), the compositeof {(v ,h )} with the
1 1
of{(v ,h )}is defined by
2 2
{(v ,h )}.{(v ,h )}:={(w.v ,k.h )}.
1 1 2 2 1 2
There is a functor p ∗ from X ∗ to B; on objects, it agrees with p :X →B;
and p ∗({(v,h)}) = p (h). Note that if v : X′ → X is vertical, the vh span (v,1)
represents amorphismX →X′ inX ∗.
Clearly, a vertical arrow in X ∗ has a unique representative span of the form
(v,1). So the fibres of p ∗ : X ∗ →B are canonically isomorphic to the duals of
thefibres ofp :X →B, i.e. (X∗) ∼=(X )op;soX ∗ is“fibrewisedual”toX
A A
(butisnotingeneraldualtoX ,sincethefunctorp ∗:X ∗→Bisstillacovariant
functor). The arrows in X ∗, we call comorphisms; it is ususally harmless to use
thename“comorphism”alsoforarepresenting vhspan (v,h).
There are twospecial classes ofcomorphisms: thefirst class consistsofthose
comorphismsthat can berepresented by a pair(v,1)where 1 is the relevantiden-
tity arrow. They are precisely the vertical arrows for X ∗ → B. – The second
classconsistsofthosecomorphismsthatcanberepresentedbyapair(1,h)where
1 is the relevant identity arrow. We shall see that these are precisely the cartesian
morphismsinX ∗. ’
Wefirst notethatif(v,h)represents an arbitrary arrow inX ∗,then
(v,h)∈{(v,1)}.{(1,h)}; (5)
thisiswitnessedbythediagram
1 h
· ✲ · ✲ ·
1 1
❄ ❄
· ✲ ·
1
v
❄
·
5
sincetheupperleft squareis oftheform consideredin (4).
Proposition3.2 An arrow g is cartesian in X ∗ iff it admits a vh representative
oftheform(1,h).
Proof. Inonedirection,let(1,h)representacomorphismY →Z overb ∈B,and
let (v,k) represent a comorphism X →Z over a .b . We display these data as the
fullarrows inthefollowingdisplay(in X andB):
·
vX❄..............k...′...............✲..Yk ✲✲Z;
h
: : :
· ✲ · ✲ ·
a b
The dotted arrow k′ comes about by using the universal property of the cartesian
arrow h in X . Since k and h are Cartesian, then so is k′, by the Lemma 1.1. So
(v,k′)isacomorphismovera ,and(v,k′).(1.h)≡(v,k),andusingthecancellation
property of Cartesian arrows, (v,k′) is easily seen to be the unique comorphism
overa .b composingwith (1,h)togive(v,k).
In the other direction, let g be a cartesian arrow in X ∗. Let (w,k) be an arbi-
trary representative of g. Then by (5), g={(w,1)}.{(1,k)}. Since g is assumed
cartesian in X∗, and {(1,k)} is cartesian by what is already proved, it follows
from Lemma1.1 that {(w,1)}is cartesian. Since it is also vertical, it followsthat
itisanisomorphisminX ∗,hencewisanisomorphisminX . Sincekiscartesian
inX , w−1.k is cartesian as well,and
(w,k)≡(1,w−1.k),
sog has arepresentativeoftheclaimedform.
Proposition3.3 Thefunctorp ∗ :X ∗ →B isa fibrationover B
6
Proof. Let b :A →B be an arrow in B, and letY be an object in over B. Since
X →B is a fibration,there existsin X a cartesian arrow h overb , and then the
vhspan (1,h)represents, by theabove,acartesian arrow inX ∗ overb .
SinceX ∗ →B isafibration, wemayask foritsfibrewisedualX ∗∗:
Proposition3.4 ThereisacanonicalisomorphismoverBbetweenX andX ∗∗.
Proof. We describe an explicit functor y : X → X ∗∗. Let us denote arrows in
X ∗ by dotted arrows; they may be presented by vh spans (v,h) in X . We first
describe y on vertical and cartesian arrows separately. For a vertical v in X , say
v : X′ → X, we have the vh span (v,1) in X , which represents a vertical arrow
v:X′99KX inX ∗;thuswehaveavhspan(v,1)inX ∗,whichinturnrepresents
a vertical arrow X → X′ in X ∗∗. This arrow, we take as y(v) ∈ X ∗∗. Briefly,
y(v) =((v,1),1). – For a cartesian h : X′ →Y (over b , say), we have a vh span
(1,h) in X , which represents a horizontal arrow h : X′ 99KY in X ∗ (cartesian
overb ); thuswehaveavhspan(1,h)inX ∗,henceanarrow inX ∗∗, fromX′ to
Y whichwetakeas y(h)∈X ∗∗;briefly, y(h)=(1,(1,v)).
Then, for a general f : X →Y in X , we factor it v.h with v vertical and h
cartesian, andputy(f):=y(v).y(h). Weleavetothereadertoverify thatadiffer-
ent choice of v and h gives an equivalent vh span in X ∗, thus the same arrow in
X ∗∗.
Conversely, given an arrow g : X →Y in X ∗∗, represent it by a vh span in
X ∗, (v,h),
h
X′ ...................✲.. Y
.
.
.
.
.
.
.
.
.
v.
.
.
.
.
.
.
.❄
X
Since v is vertical, we may pick a representative of v in the form (v,1) with v :
X →X′, and since h is cartesian in X ∗, we may pick a representative of it if the
form (1,h), with h:X′ →Y in X . Then the compositev.h:X →Y makes sense
inX , and itgoes byyto thegiveng.
Example. Consider a group homomorphism p : X → B. It is a fibration iff
p is surjective. Assume this. Then the fibre (over the unique object ∗ of B) is
the kernel K of p . Every h ∈ X is Cartesian; the vertical arrows are those of
K . Then X∗ is canonically isomorphic to X . For, an element (arrow) (v,h)
7
of X ∗ may be presented by either (1,v−1.h), so may be presented in the form
(1,x). The map (v,h) 7→ v−1.h gives a canonical isomorphism J : X ∗ → X .
This isomorphism preserves p ; note that the p for X ∗ takes (v,h) to p (h). Let
us for clarity denote it p ′, so p ′{(v,h)}=p (h). The kernel K ′ for p ′ consists of
elementswhichmayberepresentedintheform(v,1)withv∈K ,soK ′may,asa
set,beidentifiedwithK byidentifying(v,1)∈K ′⊆X ∗ withv∈K ⊆X . But
this identification is an anti-isomorphism, since (v,1) by J goes to v−1.1 = v−1.
So K ′ is identified as a group with K op. Thus we have a diagram of group
homomorphisms
(−)−1
K op ✲ K
∼
=
i ⊆
❄ J ❄
X ∗ ✲ X
∼
=
p ′ p
❄ ❄
B ✲ B
id
wherei(v)={(v,1)}. IncasewhereB=1,andX isthegroupG,thefourmaps
ofthetop squareare moreexplicitly‘thefourgroup isomorphisms
(−)−1
Gop i ✲ G
v7→{(v,1)} =
❄ J ❄
G∗ ✲ G
{(v,h)}7→v−1.h
where the inverse of J is given by h7→{(1,h)}. If we denote the inverse of J by
j, wecan writetheinformationinthisdiagrammoresymmetrically:
i j
Gop ✲ G∗ ✛ G
withi(v):={(v,1)}and j(h):={(1,h)}.
8
4 The case of a (pseudo-) functor Bop →Cat
ItiswellknownthatapseudofunctorF :Bop→Cat,givesrisetoafibrationover
B. Itisdescribedin,say,[2]B.1.3,orin[1]II.8.3. Thisfibrationisknownasthe
GrothendieckconstructionforF. Wedescibeitbrieflyintermsofthefactorization
systemalluded toinSection 1.
Given a functor (or just a pseudo-functor) F : Bop →Cat. Then we have
a category X whose objects are pairs (X,A) with A an object of B and X an
object in F(A). Arrows (X,A) → (Y,B) are pairs (v,a ), where a : A → B and
v:X →a ∗(Y)inF(A)(andwherea ∗ denotesthefunctorF(a ):F(B)→F(A)).
Thefunctorp :X →B takes thisarrowto a .
Let usdenotethearrow (1a ∗(Y),a )by a ⊳Y, thus
a ⊳Y
a ∗(Y) ✲ Y
This is a Cartesian arrow over a in X , and every Cartesian arrow is of this form
modulo unique vertical isomorphisms. There is then a canonical factorization of
general arrows in X , namely, the arrow given by a pair (v,a ), as above, factors
as
(X,A)
(v,1 ) .
A
❄
(a ∗(Y),A) ✲ (Y,B)
a ⊳Y
LetF′beF followedbythedualizationfunctorCat→Cat. Thenamorphismover
a inthefibrationcorrespondingtoF′,from(X,A)to(Y,B),isgivensimilarly,but
nowwithv:a ∗(Y)→X,whichintermsofthecategoryF(A)ratherthan(F(A))op
maybedisplayedinterms ofthevh span
a ⊳Y
(a ∗(Y),A) ✲ (Y,B)
(v,1 ) ,
A
❄
(X,A)
and from this, it is clear that the fibration corresponding to F′ is isomorphic to
X ∗ as described inthepreviousSections.
9
One motivation for the present note is to extract the pure category theory behind
“fibrewise contravariant functors" (like fibrewise duality for vector bundles), and “star-
bundlefunctors”, asin[Kolaretal,1993]41.2. Thisisstillanongoing project.
Icannotimaginethattheconstructions inthepresentnotearenotknown,butIdonot
presently knowofanyavailable account.
References
[1] F.Borceux,HandbookofCategoricalAlgebra,vols.1-3,EncyclopediaofMathemat-
icsanditsApplications 50-52(CambridgeUniversityPress1994).
[2] P.T. Johnstone, Sketches of an Elephant: a Topos Theory Compendium, vols. 1-2,
OxfordLogicGuides44and45(OxfordUniversityPress2002).
[3] A.Kock,FibrationsasEilenberg-Moore Algebras, arXiv:1312.1608 v1,2013.
[4] I. Kolar, P.W. Michor, and J. Slovak, Natural Operations in Differential Geometry,
SpringerVerlag1993.
UniversityofAarhus,January 2015
mail: kock(at)math.au.dk
10
