D IAGRAMMES

J. ROSICKÝ
Abstract tangent functors
Diagrammes, tome 12 (1984), exp. no 3, p. JR1-JR11
<http://www.numdam.org/item?id=DIA_1984__12__A3_0>

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Diagrammes, Vol, 12 , 1984.

A.M.S. Sub. Class.
18 F 15 , 55 R 65.

Abstract tangent functors
J. Rosicktf

Our aim is to axiomatize properties of the tangent functor
T : M -> M on the category of smooth manifolds. The resulting
abstract tangent functor

T : C •* C

on a category

C

has the

property that there is atfellr-behavedbracket opération of its
sections

A - TA

on any object

A e C. Representable abstract

tangent functors are closely connected with rings of line type
in the sensé of Kock - Lawvere.
1. Natural group bundles
Let
p : G - F

C

be a category, F,G : C -» C

functors and

a natural transformation, Let © dénote products

in the category of functors over

F. It means that

P
is a pullback. We say that

(G,p)

is a natural group bundle

over F if it is a group in the category of functors over
means that

G

is equipped with natural transformations

+ : G ® G - G ,
which are over

F, It

F

- : G - G ,

0 :F - G

(i.e. p.+ = p.p, , p.- = p

and

p.O = 1) and

satisfy the group axioms. A natural group bundle is a natural
group bundle over the identity functor on

C.

JR 2
Let

(G,p)

be a natural group bundle over

a natural group bundle over

F

and

(H,q)

E. A homomorphlsm (f,g):(G,p) - (H,q)

is a couple of natural transformations

f : G - H

and

g : F - E

such that the following diagrams oommute
f©f

f

G

G©G

*H

G© G

(G,p)

be a natural group bundle. If the pullbacks

and

G©G©G

are pointwise then

G

with respect to

2
(Gp,p) : (G ,pG) - (G,p)

G

*H©H

Let

bundle over

G©G

»

and
and

(G2,pG)

+G, -G, OG

is a natural group
and

is a homomorphism. If

G

préserves

then (G2,Gp) is a natural group bundle over
2
(pG,p) : (G ,Gp) - (G,p) is a homomorphism.
G©G©G

2. Tangent functors
We say that

T : C - c

is a tangent functor if there are

p, i and m such that
(Tl) (T,p) is a natural commutative group bundle such that
pullbacks T©T and T©T©T are pointwise and preserved by T.
(T2) (i,l):(T ,pT) - (T2,Tp) is a homomorphism and i 2 = 1.
2
(T3) (m,0):(T,p) -* (T ,pT) is a homomorphism such that i.m=m
and the following diagram is a pointwise equalizer

"f?L_~

-HL_*T2.
0. p. pT
(T4) The following diagrams commute

JR 3
iT

3
,J
J. •

3
* T3

Ti

3
^T3

iT

Ti J

m

•*• T -

-••T*

iT

Ti

mT
Thèse axioms are satisfied by the tangent functor
T : M - M on the category of smooth manifolds. Since
= &
of

=

, the following example also indicates the description
i

and

m

in local coordinates.
Ab

Example 1 : Let
and

T(Rn)

T = (-)2. Then

be the category of abelian groups

p A (a f b) = a, (a,b) + A (a,c) = (a,b+c),

- A (a,b) = (a,-b), 0 A (a) = (a,0), iA(a,b,c,d) = (a,c,b,d) and
m A (a,b) = (a,0,0,b)
indicate that

make from

T®T = (-)3 and

= (a,b,c+c',d+d')

and

Example 2 :
Let

a tangent functor. We

(afb,c,d) +_.

+TA

and

T+A

dénote the addition

(T 2 A,p TA ) and (T2A,T(p) )
Let

(a,bfc',d') =

(a,b,c,d) T + A (a,b',c,d') =

= (a,b+b',c,d+d'). Hère
in group bundles

T

R

TA = A[x]\x . Then

over

TA.

be the category of commutative rings,
TA

consists of polynomials

a + bx,

(T®T)A = A[x,y]\x 2 ,y 2 ,xy of polynomials a+bx+cy and
2
2 2
T A = A[x,y]\x ,y
of polynomials a+bx+cy+dxy. Then T
a tangent functor and its structure is given by the same
formulas as in the preceding example.

is

JR 4
Example 3 : Let

R

be a ring of the line type in a
2
E

cartesian closed category
C

[2]). Let

and

D = {deR\d
E

be a full subcategory of

= 0} ( see Kock

which consists of

ail infinitesimaly linear objects having the property
T = (-)D

T©T = (-) D ( 2 ) where

C. Hère

is a tangent functor on

D(2) = {(d 1 / d 2 ) e D x D X d ^ d j = 0},. + : T©T -> T
by the diagonal
the symmetry

D - D(2), T 2 = (-) D x D ,

s : DxD -> DxD

and

m

W. Then

i

is represented

is represented by

by the multiplication

. : DxD -* D.
E

A concrète example is the category
the category

of functors from

R

of finitely presented commutative rings to
2
the category of sets. R is R (Z,-)
and D is i? (Z[x]\x , - ) .
o
o

We took

RQ

to avoid set theoretical difficulties with functor

catégories. It is évident that Example 2 works for
Hence the tangent functor

(-)

, too.

is derived from

R . It is a gênerai phenomenon.

the tangent functor on

C

Proposition 1 : Let

be a small category and

C. Let

a tangent functor on

C C E

on

R

B

T : C - C

be the full subcategory of the

Q

functor category
pullbacks

Set

T©T, T©T©T

consisting of ail functors which préserve
and the equalizer from

T*(V) = V.T

,

yields the tangent functor

(T3). Then

T*(o) = aT

T* : B -> B .

Another gênerai construction of new tangent functors is
the following one. Let

T

be a tangent functor on a category
B = C\A

Consider the comma category
*(X,f) = (TX,f.px)
yields the tangent functor

,

where

A e C. Then

T(h) = T(h)

T : B - S. Let

C.

JR 5

V

T(f)
X

* TX

TX

be an equalizer. If

rzî TA
0 A .p A .T(f)

T

préserves this equalizer then

T(X,f) = (TX,f.p .v )
*x x
T : B •* B.

provides the tangent functor
If

T : M- M

is the tangent functor on the category of

smooth manifolds then

T

gives the vertical bundle on the

category of fibered manifolds.
The following property of tangent functors is very important,
Lemma 1 : Let
composition
e : T©T

T

be a tangent functor and consider the

.
m /~\
<m.p2,T(o).pi>
» (T©T)T

+T

*T

'Then the diagram
pT
e

T^T

2

»T '

»T
O.p.pT

is an equalizer.
o

Since

T(p).e = p 1 , it says that if

natural transformation egualizing
a unique natural transformation
(1)

T

A morphism

Hence

r

and

O.p.pT

g : S - T

is a
then there is

such that

f = T(p.O).f + T m.g.

3

Let

pT

f : S -* T

* Bracket opération
C

be a tangent functor on a category
r : A •* TA

is a

is called a section of

T-coalgebra

T

and
if

A e C.
p, .r = 1.

^A
in the terminology of K e l l y [ 1 ] .

JR 6

If

T : M - M is the usual tangent functor, the sections are

vector fields. We want to define the bracket
r,s : A -* TA

[r,s] of sections

of any tangent functor. For this purpose, we would

need the description of the bracket of vector fields in terms of
T

only and not using functions on manifolds. This description

was given by Koléf [3] and we will follow it. Very similar
description is stated in White [8].
Let

r,s : A -> TA

be sections. Since

T(p) .T(s).r = r =

= T(p)A.iA.T(r).s, it is defined the différence
v = (T(s).r)T-A (iA.T(i).s) .
It is easy to see that
a unique morphism

PTA.v = 0 A . Following lemma 1, there is

v : A - (T®T)A

[r,s] : A

v

-(T©T)A

such that

e .v = v. Put

(pa).
^-^. TA .
2

In example 1, sections
morphisms

r : A -> A

correspond to endo-

r : A - A. The bracket is the usual bracket of endo-

morphisms
[r,s] = s.r - r.s .
2
In example 2, sections

r : A - A[x]\x

dérivations

and the bracket is the usual bracket of

r :A - A

coincide with

dérivations. In example 3, the bracket is the bracket from
the synthetic differential geometry (see [2]).It follows from
the fact that
»A- [r ' sls (+ -k>TA-T2( + .k)A.T(i)TA.-T3A.T(-)T2A.T3(s).T2(r).T(8).r
and that this formula corresponds to the commutator of infini2
tesimal transformations. Hère k : T - T©T is given by
PA.k = pT and p2«k = Tp.

JR 7

Theorem 1 : Let

T

be a tangent functor. Then the

bracket opération has properties
(Bl)

[r+s,t] = [r,t] +

(B2)

[s,r] = - [r,s]

(B3)

[r,[s,t]] + [s,[t,r]] + [t,[r,s]]

[s,t]

It is not difficult to calculate (Bl) - (B3) for manifolds
without using functions (see Van2urova" [7]). In the gênerai case,
we are

facing the cohérence problem for tangent functors. Remark

that (Bl) and (B2) hold on the basis of axioms (TI) - (T3)

only.

The proofs from synthetic differential geometry (see Reyes, Wraith
[6], Lavendhomme [5] and Kock [2]) do not work in the gênerai
case. Our proof is "additive" and consists in calculations with
T, T 2 and

natural group bundles
by

p,pT, Tp, pT 2 , pTp

and

T3

over

1, T

and

T2

given

T 2 p.

We did not need the /?-linear structure on

T : M - M.

However, it is présent in the gênerai case, too. We say that
h : T - T

is an endomorphism of a tangent functor

T : C - C

if

the following diagrams commute
h® h

T

T©T

-* T

m

It means that

h

is an endomorphism of the natural group bundle

JR 8

T

and préserves the tangent structure given by

The set

R

of ail endomorphisms of

T

i

and

m.

is a ring with the

composition as the multiplication and the addition
<g,h>

+

g + h : T
The morphisms
and

T

» T©T

h A : TA - TA

put an

*T .
R-module

structure on

becomes a natural R-module bundle. If

r,s : A -* TA

are sections of

T

h e R

TA

and

then it holds

[hA.r,s] = 1^ .[rfs] .
T : M - M then natural transforma-

Kolâf [4] proved that if
tions

h : T - T

with

p.h = p

are precisely homotheties given

by multiplying with

x <= R.

Any homothety is an endomorphism

of

R = R

in this case. R = X in ex. 1 and 2.

T

and therefore

In synthetic differential geometry, any morphism
h : D - D

with

(-)

- * ( - ) . The ring of endomorphism of (-)

: (-)

of

O

ring

h(0) = O

gives an endomorphism

preserving morphisms
R

is the ring

D - D. However, it is the starting
to

of the line type. To see it G ne has

realize that

the line type property implies that

O

h : D - D

-.x, x e R. Hence the line

R

coincide with morphisms

preserving morphism

is determined by its infinitésimal segment

D.

4. Representable tangent functors
Let
T = (-)

C

be a cartesian closed category and

: C •* C

is a tangent functor. Let

be represented by

0:1-D,

- D2

. : D2 - D .

t : D

and

Dec

P/+,-/0,i

such that
and

m

ô : D - * D * D , - : D - * D , D-*l,

JR 9

Hère,
-*- D
ÏÏI

-*• D*D
TÏ2

is a pushout. There is a unique morphism
+

+ : D*D -* D

such that

. T l i = 4 - . 7 l 2 = l .

Proposition 2 : (D,.)
2
such that d = o for any

is a semigroup with the zéro O
d e D.

Categorical logic justifies the set theoretical terminology.
The last assertion means that

-» D

commutes where
that

A

A

is the diagonal. It follows from the fact

is the composition
6
D

M

-vD*D

0

D*

and that
D*D-

-+ D '

-* D

commutes. The morphism
x.ïï^d) = (0,d)

and

w

represents

x.ïï2(d) = (d f 0)

k : T

•• T©T, i.e.

for any

x(D*D) C D ( 2 ) = {(d 1 # d 2 ) e D 2 \ d x .d 2 = 0} .

d e D. Hence

JR 10

From now on, assume that
that

u

because

is the symmetry

T

is represented in such a way

s. Then

(D,.)

is commutative

i.m = m.

Let an endomorphism of
h : D - D.

Then

T

be represented by a morphism

h(0) = 0 ,

ô.h = (h*h).ô

and

h(d 1 .d 2 ) = h(d 1 ).d 2
for any

d ,d2 e D. Let

R

be the ring of thèse endomorphisms

h : D - D. The exponential transpose /> : D -* D
2
. : D - D yields a morphism j : D - R.
Proposition 3 : For any morphism
a unique élément

h e 5

f : D - R

of

there is

such that

f(d) = f(0) + h.j(d)
holds for any

d e D.

It is the translation of formula (1). Hence
R

R

= R

and

thus has some properties of a ring of a line type. We are

missing

the commutativity of

R

and the fact that

D = {h e R \ h 2 = 0 } .
Since

O x D = O

where

O

is an initial object, the

tangent functors from examples 1 and 2 are not representable.
The tangent functor from example 1 is not a restriction of a
representable tangent functor on a subcategory closed with
respect to a terminal object. It follows from the fact that

O

is not a unique natural transformation

1 - T. On the other hand,

there are very convenient extensions of

T : M •* M

to a represent-

able tangent functor (see Kock [2]).
This paper had a rather slow development due to difficulties
with the proof of theorem 1. The other material was completed

JR 11

in 1982. Proofs will appear elsewhere. I profited from
discussions with G. Wraith and A. Kock. However, I am
especially indebted to I. Kolàf who patiently introduced
me into basic differential geometry.

Références
[1] G. M. Kelly, A unified treatment of transfinite constructions
for free algebras, free monoids, colimits, associated sheaves,
and so on, Bull. Austral. Math. Soc. 22 (1980), 1-83.
[2] A. Kock, Synthetic Differential Geometry, Cambridge Univ.
Press 1981.
[3] I. Kolâf, On the second tangent bundle and generalized Lie
derivatives, Tensor 38 (1982), 98-102.
[4] I. Kolàï, Natural transformations of the second tangent
functor into itself, to appear in Arch. Math. (Brno) 4 (1984).
[5] R. Lavendhomme, Note sur l'algèbre de Lie d'un groupe de Lie
en géométrie différentielle synthétique, Univ. Cath. de
Louvain, Sém. de math, pure, Rapport no. 111 (1981).
[6] G. E. Reyes and G. C. Wraith, A note on tangent bundles in
a category with a ring object, Math. Scand. 42 (1978), 53-63.
[7] A. Vanèurovci, On geometry of the third tangent bundle, to
appear in Acta Univ. Olom. 82 (1985).
[8] J. E. White, The method of iterated tangents with
application in local Riemannian geometry, Pitman 1982.

Department of Mathematics
Purkynë University
Brno, Czechoslovakia