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       On the relative CR category
by
Shigeru Takeuchi
Department of Mathematics, Faculty of Education, Gifu University
Yanagido 1-1, Gifu, Japanemail: takeuchi@gifu-u.ac.jp(received January 8th and revised January 30th 2009)
Abstract
The purpose of the present note is to consider the relative CR category and to provide
the set of morphisms with the functorial CR structures at least in an abelian CR category.
For other CR categories, e.g., complex one the relative notion is well known, but there seemnot so many investigations in this direction to exist.

1

Introduction

Let us consider an arbitrary category A. We prepare some basic categorical notions necessary
for the later arguments. First for surjectivity and injectivity, we are concerned in the categoryof sets. So it would be relevant to introduce the two diﬀerent notions, set theoretic one andcategorical one. We are indebted to [11] for the following deﬁnition:
Deﬁnition 1.1 (epic and monic)(1) A morphism f : M → N is categorically surjective or epic, if for any object X andany morphisms g = h : N → X we have f ∗ (g) := g ◦ f = h ◦ f =: f ∗ (h). Namely the(contravariantly) induced morphismf ∗ ; Hom(N, X) → Hom(M, X)
is (set theoretically) injective.
(2) A morphism f : M → N is categorically injective or monic, if for any object X and any
morphisms g = h : X → M we have f∗ (g) := f ◦ g = f ◦ h =: f∗ (h). Namely the (covariantly)
induced morphismf∗ ; Hom(X, M ) → Hom(X, N )is (set theoretically) injective.
It would be irrelevant to discuss what kind of implication holds concerning the two notions
without any speciﬁc conditions on which it depends. In the following sections we will discuss
this problem for speciﬁc categories. Let M be an object. Then a subobject L of M is amorphism i : L → M which is monic. As the dual notion we have a quotient object N of M
deﬁned by a morphism p : M → N which is epic. For speciﬁc categories of various geometricspaces, these notions diﬀer from the geometric notions of subspaces or quotient ones of theoriginal space. From a category A we can derive a new category A in the following way. An
object in A is a triplet (M, f, N ), where f : M → N is a morphism of an object M to another
one N in A. Then for two objetcs f1 : M1 → N1 , f2 : M2 → N2 , a morphism (g1 , h1 ) of
(M1 , f1 , N1 ) → (M2 , f2 , N2 ), is a pair of morphism g1 : M1 → M2 and h1 : N1 → N2 , such thatthe following diagram commutes:
f1

M1 → N1
↓ g1
↓ h1
M2

f2

→ N2 .
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For later references we name this category A the derived category of A, and denote it always
with the prime symbol ’ attached to the original symbol, e.g., A. An object in this category is
named derived object and a morphism is named derived morphism. Then it is clear that thederived category is well deﬁned since the composed morphism of the morphism (g1 , h1 ) withanother one (g2 , h2 ) of f2 : M2 → N2 to f3 : M3 → N3 is deﬁned in the usual fashion; thefollowing diagram is commutative.
f1

M1 → N1
↓ g1
↓ h1
f2

M2 → N2
↓ g2
↓ h2
M3

f3

→ N3 .

We denote the object M f N of a derived category sometimes byf : M → N following theexpression in the original category. For a pair of objects M f N and M  f  N  , we denote the setof morphisms of M f N to M  f  N by Hom(M f N, M  f  N  ).

2

Linear CR category revision

We review brieﬂy some of the main results in [5] for the readers’ convenience. First a CR
vector space is a triplet M = (U, RU , JU ) of a real vector space U and a subspace RU of U
with the complex structure JU on R. If the dimensions are concerned we have m = dim M ,2rU = dim RU and we refer sM := n − 2rU as CR-codimension of the object M . The pair
(RU , JU ) is referred to the CR-structure of the object M and (rU , sM ) is sometimes called thetype of the CR-structure of M . For simplicity of notation the suﬃxes U ,M are omitted if there
is no fear of confusion. The CR structure is totally real iﬀ r = 0 and totally complex iﬀ s = 0.
Let N = (V, RV , JV ) be another CR vector space with the CR structure of type (rV , sN ).
Then a (CR)-morphism f : M → N is an R-linear mapping compatible with the respective
CR structures:f (RU ) ⊂ RV
f (JU (u)) = JV (f (u)),

u ∈ RU

Before proceeding further, letting M = (U, RU , JU ) be a CR vector space, we have to prepare
the following
Deﬁnition 2.1 (subspace) A CR space L = (T, RT , JT ) is a (geometric) CR subspace of M
if(i) L is an R-subspace of U ,(ii) RT is an JU invariant subspace of RU , and JT is the restriction of JU to RT .
Thus the natural inclusion ι : L → M , that is injective, is monic ( see the Theorem 2.1 below), and hence deﬁnes a subobject in the sense of the 1st section. There exists a more strong
subspace notion, which is the induced subspace as was mentioned in [5]:Deﬁnition 2.2 (induced CR structure) A CR subspace L = (T, RT , JT ) of M is a CR subspace with the induced CR structure (RT , JT ), or an induced CR subspace for short, if(iii) (RT , JT ) is the CR structure induced from RU ; RT = (T ∩ RU ) ∩ JU (T ∩ RU )or equivalently(iii)’ RT is the maximal JU invariant R-subspace of T .
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The induced CR structure is characterized by the following universality condition.
Theorem 2.1 (cf. [5]) Let T be an R-subspace of a CR vector space M . Then the natural
inclusion ι : L → M is a morphism, where L = (T, RT , JT ) is the induced CR subspace of M .
The CR structure (RT , JT ) is maximal in the set of CR structures on T with this property.
Proof. Let (R, J) be any other CR structure on T such that ι is a morphism. Then we haveby deﬁnition R = ι(R) ⊂ RU . Since R is a (J invariant) subset of T we have R ⊂ T ∩ RU and
Q.E.D.
J(R) ⊂ J(T ∩ RU ), which deduce R ⊂ (T ∩ RU ) ∩ J(T ∩ RU ) .
Deﬁnition 2.3 (generic)An R-subspace G of U is generic in M = (U, R, J), if
U = G + J(R ∩ G).

Then we have
R = R ∩ G + J(R ∩ G),
which implies G and J generate the CR vector space U = (U, R, J).
Deﬁnition 2.4 (generically surjective) An R-linear morphism f : M → N is genericallysurjective if the set theoretic image f (M ) is generic in the target space N .
As applications of the above deﬁnition, we have the following deﬁnition and theorems.
Deﬁnition 2.5 (complexiﬁcation [5]) Let U be a CR vector space. Then a complexiﬁcationof U is a pair (f, V ) of a monomorphism f : U → V of U into a totally complex CR vector
space V satisfying the following universality condition:(uc) For any totally complex object W and a morphism g : U → W in LCR, there exists
uniquely a morphism h : V → W , such that the following diagram commutes:
U

f

→

V
g ↓h
W

Then a complexiﬁcation of U , denoted (f, U C ), does exist uniquely in the sense as follows :Theorem 2.2
The complexiﬁcation (f, U C ) exists uniquely up to isomorphisms; if (f  , V  )is another complexiﬁcation of U , then we have an isomorphism ι : V → V  such that thefollowing diagram commutes.
f
V
U →

f ↓ι
V
Furthermore the complexiﬁcation f is realized as a generic (CR) imbedding f : U → U C of U
into a totally complex CR vector space U C (see [5] for the deﬁnition and more details of thegeneric embeddings).
Proof. The deﬁnition above is a little bit diﬀerent from the original one in [5], and the theorem
itself is formulated in a diﬀerent way, but we may follow the main idea of proof there. The
uniqueness follows from the (uc). We may drop the monic condition of f , since it seems
superﬂuous, but yet to be proved without this assumption.
Q.E.D.
Theorem 2.3
In the LCR a morphism g is(1) epic iﬀ it is generically surjective,(2) monic iﬀ it is set-theoretically injective.

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Homomorphisms in the linear CR category

Let M = (U, RU , JU ), N = (V, RV , JV ) be CR vector spaces. Now we consider the set of CRmorphisms, denoted Hom(M, N ), as CR object. It has the natural R-module structure, whichis compatible with the CR structure on the respective objects M, N . As in [5] we can equip
H := Hom(M, N ) with the canonical CR structure (RH , JH ) in the following manner.
RH := {f ∈ H|f (U ) ⊂ RV }(JH (f ))(u) := JV (f (u)),

u∈U

Then H = (H, RH , JH ), thus obtained, is a CR vector space with the maximal CR structure,in the sense it is maximal in the set of CR structures on H such that the natural operation ρof H on the source object M
ρ : H × M → N,
is CR. More exactly speaking the mapping ρ is CR bilinear and is referred to as evaluation
mapping in [5]. The maximality was not mentioned explicitly there. Here we only remark
the maximality of RH without any proof. The exact statement (with proof) of this fact isessentially given in a similar theorem in the 5th section. For the special case of M = N , weadopt the conventional notation and terminology
E := End(M )in stead of Hom(M, M ). Then E has the multiplication μ deﬁned by the composition ofmorphisms:
μ : E × E → E,
which is CR bilinear. This multiplication satisﬁes the associativity and distributivity, respectively and hence deﬁnes the (non commutative CR algebra structure on E (see [6] for thedeﬁnition of CR algebra). The invertible elements of E, denoted G := GLCR (M ), constitutethe general linear CR group of M , which is a real Lie subgroup of the general linear group
GL(U, R) of the real vector space U . The group G has the invariant CR structure, in the sensethat the restriction ρ|G of ρ to G:
ρ|G : G × M → M
is a CR morphism in the (non linear) CR category (see the next section and [12]). In this sense
we referred to this action as a CR Lie group action on a CR manifold M [12, 13, 14]. In thefollowing sections we investigate a little bit more general CR category, which is non-linear. In
this extended category the isomorphism group of the object M , denoted A := Aut(M ), contains
G := GLCR (M ) as an abstract subgroup. If A has an invariant CR structure, compatible withthat of G, then we may have the CR inclusion (which is yet to be deﬁned precisely)ι : G → A.

4

Aﬃne CR category

In this section we consider a more general CR category than that of CR vector spaces. Take asmooth m-dimensional manifold M with the tangent bundle T M . Here ”smooth” means theC ∞ diﬀerentiability. A smooth CR manifold M = (M, RM , JM ) is a triplet of smooth manifold
M and an R-subbundle RM of T M R-rank 2r, and an almost complex structure tensor JM
on RM . We refer to the pair (RM , JM ) as the CR structure of M , the pair (r, s) as the type
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and s as the CR codimension of the CR structure, respectively, where s = m − 2r. As is thelinear case, the CR structure is totally real iﬀ r = 0 and totally complex if iﬀ s = 0. Then amorphism f : M → N of M to another CR manifold N = (N, RN , JN ) is a smooth mapping
compatible with the respective CR structures of M, N :df (RM ) ⊂ RN
JN ((df )(u)) := df (JM (u)),

u ∈ RM ,where df denotes the diﬀerential of f . Thus the category of CR manifolds is deﬁned and denoted
CR. Then LCR can be considered as a subcategory of CR by assigning to each CR vector
space U = (U, RU , JU ) the associated smooth CR-manifold denoted M = (M, RM , JM ), where
M denotes the smooth manifold deﬁned by the real vector space U with the natural smooth
structure U = Rm under an arbitrary base {e1 , e2 , . . . , em } and RM denotes the subbundle ofT M = U × Rm deﬁned at each point p ∈ M = U by Rp M = R2r , which is identiﬁed with
RU irrespective of the point p involved. A morphism f : U → V in LCR naturally deﬁnes
a smooth (linear) CR mapping of M to N = (N, RN , JN ), where N is deﬁned by CR vector
space V = (V, RV , JV ) as in the case of M , since its diﬀerential df : T M → T N is represented
by a constant matrix, which is CR at each point p ∈ M (see [13] for matrix representationsof linear CR mappings). Usually we impose an additional assumption integrability on the CR
structure (RM, JM ) deﬁned by[JX, JY ] + J[X, JY ] + J[JX, Y ] = [X, Y ], X, Y ∈ Γ(RM ),where Γ(RM ) denotes the (germs of local) smooth sections of RM . This is equivalent to thecomplexiﬁed condition [1, 3, 5]:[Γ(AM ), Γ(AM )] ⊂ Γ(AM )the so-called involutiveness of AM , where AM denotes the anti-holomorphic part of the comC := R ⊗ C of R
plexiﬁed bundle RM
M
M and Γ(AM ) denotes its sections. As is well known the
integrability condition is satisﬁed if the CR structure is induced from an imbedding of M into
a totally complex CR manifold. For our linear case the CR vector space U = (U, RU , JU ) admits a complexiﬁcation U C ([5]), which is a totally complex aﬃne CR space by deﬁnition. We
can verify the CR structure of U coincides with the induced structure and hence integrable.
We can deduce the integrability directly by calculating the bracket relation above, since thetangent bundle T U is trivial.
Remark 4.1 We remark that LCR is not a full subcategory of CR, since the smooth CR
mappings are far more abundant than the linear ones. One of the most important reasons why
we have to consider more general CR category than the linear one is that a CR vector space
V is not an abelian CR group in this category, since the addition is not a CR morphism evenfactor-wise separately. This problem can be solved if we consider the general CR category,
which is however too general to obtain more speciﬁc results. So now it’s time to consider atractable and moderately general subcategory which contains LCR.
Deﬁnition 4.1 (Aﬃne CR space) An aﬃne CR space is a triplet (A, RA , JA ) of an aﬃne
R-space A (of dim A = n), a rank 2r subbundle of the tangent bundle T A of A and the almost
complex structure tensor JA of RA . We refer to the pair (RA , JA ) as the CR structure of theaﬃne CR space, and (r, s) as the type of the CR structure. Let B = (B, RB , JB ) be anotheraﬃne space. Then an aﬃne mapping f : A → B is an aﬃne CR morphism, if its diﬀerential
df : TA → TB is compatible with the respective CR structures:
df (RA ) ⊂ RB
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df (JA (u)) = JB (df (u),u ∈ RA ,as a bundle mapping.
Thus the category of aﬃne CR spaces, denoted CRA, is deﬁned.
Remark 4.2 It is more appropriate to deﬁne CRA not relying on the tangent bundle structure, since it is trivial as in the linear case. But for the present moment we adopt the abovedeﬁnition so as to imbed CRA in CR, although it is not a full subcategory as is the case in
the linear one.
We can easily observe this category is closed under direct product with the self-evident product
CR structure. Namely we have the following
Deﬁnition 4.2 (Direct product) For objects A, B in CRA, we have the product aﬃne space,denoted A × B, deﬁned by(A × B, RA ⊕ RB , JA ⊕ JB ),where A × B denotes the set theoretic direct product of R- aﬃne spaces A, B and RA ⊕ RB the
direct sum of the respective tangent subbundles and JA ⊕ JB the direct sum of the respective
tensor ﬁelds. The CR structure (RA ⊕ RB , JA ⊕ JB ) is referred to as the product CR structure.
Exactly speaking we need a proof of the above theorem, which is, however, left to the readers
since it is not diﬃcult but is only tedious. Then in the situation as above we have the following
Theorem 4.1
The respective projections pA : A × B → A, pB : A × B → B are CR are
morphisms in CRA. Furthermore the product CR structure is characterized as the maximal
CR structure such that they are morphisms.
We have already pointed out that a CR vector space U can be considered a CR aﬃne space
A. If there is no fear of confusion we may use the same notation U in stead of A. Anyway U
has an additive group structure. Then we may introduce the notion of an aﬃne CR group as
follows:
Theorem 4.2 A CR vector spaceU , considered canonically an CR aﬃne space, is an aﬃne
CR group in the sense that the group (additive) multiplication or addition
μ:U ×U →U
deﬁned by μ(u, v) = u + v, uv ∈ U is an aﬃne CR morphism with respect to the product aﬃnestructurein U × V .
Proof. We have given the main points of the theorem in the process of deﬁning CRA. Q.E.D.

5

Relative linear CR category

In the 1st section we have considered the derived category in the most general settings, with nospeciﬁc assumptions either on the object or on the morphism. Here in this section we conﬁne
ourselves to the case where the target original object N of the derived object f : M → N
is a ﬁxed object, say, B independent of the source object M , but the morphism f varies
in accordance with the source M . In the usual categorical arguments this object is the socalled B-object and denoted simply (M, f ), and a morphism (g, h) : (M, f ) → (M  , f  ) is theso called B-morphism if h is an identity morphism of B. Henceforth we refer to the ﬁxed
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object B as base object and denote the morphism (g, idB ) simply by g following the usual
convention. The derived category thus conﬁned is named the relative category of the originalone. The argument of this section is partly based on the theory developed in [5]. Let CR
be the category of CR manifolds, or (general) CR category (may be with CR singularities).
Then the relative category, CR , or relative CR category, is deﬁned following the proceduresin the preceding section. Conﬁning ourselves to the linear case we have the category LCR of
CR vector spaces, or linear CR category. Then the relative category LCR , or relative linear
CR category, is deﬁned. Now we examine some properties of the relative linear CR category.
Before stating our main result we give
Remark 5.1 An object M in the B-object (M, f ) has a CR structure (RM , JM ) in theoriginal LCR. We use this CR structure as that of (M, f ) in CR . Thus we can call LCR therelative CR category or CR relative category either.
Now we consider two B-objects M = (M, f ), N = (N, g) in LCR where B is a ﬁxed baseobject in LCR. We denote the set of B-morphisms of M to N by HB := HomB (M, N ), thatis a subset of H = Hom(M, N ) as was mentioned in the 3rd section. Then following theprocedures in the previous section, the CR vector space H can be considered a CR manifold
H = (H, RH , JH ), whose dimension h and the type (p, q) of the CR structure (RH , JH ) aredetermined in [5] (Lemma 2.7). The tangent bundle of HB , denoted T HB , is a subbundle ofT H, since HB is a smooth submanifold of H. Then we have the induced CR structure Rh (HB )on each tangent space Th HB at h ∈ HB (see the Deﬁnition 2.1 and 2.2 in the 2nd section for
subspace and induced CR structure). Then we denote the subset ∪h∈HB Rh (HB ) of T H by
RHB . Let JHB denote the tensor ﬁeld deﬁned by (Jh (HB )(dk))(u) = J(dk(u)), where dk is atangent vector at h, with k ∈ HB and dk is a diﬀerential of k. Then we have the following
Theorem 5.1 The set of B-morphisms of M to N , denoted HB := HomB (M, N ), is an aﬃne
CR space with the canonical CR structure (RHB , JHB ) such that the natural operation ρρ : HomB (M, N ) × (M, f ) → (N, g)
is CR. Further it is maximal in the set of CR structures satisfying the above condition (see the
2nd section and [5] for details of dominance relation of CR structures).
Proof. We give only a brief sketch of the main points. Let us take an arbitrary B-morphism,
say, h : (M, f ) → (N, g). Then it satisﬁes the commutativity relation f = g ◦ h. It is easily
observed that this condition is a non homogeneous linear equation E = E(f, g; h) = 0 withrespect to the unknown variable h, where f and g are ﬁxed parameters. Since h is an elementof the CR vector space H = Hom(M, N ), the set of solutions of the non-homogeneous linear
equation E = 0 forms an aﬃne subspace of the total space H. It is proved that RHB is an Rsubbundle of the tangent bundle T HB of HB . The key point is the equidimensionality of each
CR structure Rh (HB ) independent of the morphism h. Further we can show that the almost
complex structure tensor Jh (HB ) preserves the R-subspace Rh (HB ). These phenomena are
based on the fact that each CR structure on each morphism h is invariant through translations
by any vectors in U, V .
Q.E.D.
For a detailed proof, readers are referred to [22] which may be published somewhere.

6

Concluding remarks

We have examined the structure of the set of morphisms in the relative CR category. We
can prove some complexiﬁcation theorems in CRA , which are however, not stated here. We
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may give them in explicit forms in the forthcoming [22]. We would like to know more in thecase of M = N ; EB =B (M ) has the aﬃne CR algebra structure (REB , JEB . It is non-abelianin general and so we have the (non-trivial) associated aﬃne CR Lie algebra GLCR (EB ). Our
farther target is the similar problem in a more general setting. There is a hope that someaﬃrmative answer is obtained in a relative non-linear CR category of restricted type for which
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[13] Shiga,K.,Takeuchi,S., Equivariant complexiﬁcation of homogeneous space of CR Lie group, Sci.
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[17] Takeuchi,S., Categorical images of CR morphisms, Sc. Rep. Fac. Educ., Gifu Univ.25-2(2001),1-5,[18] Takeuchi,S., On the adapted complexiﬁcation of CR manifold, Sc. Rep. Fac. Educ., Gifu Univ.281(2003),1-9.
[19] Takeuchi,S., On the categorical complexiﬁcationof CR Lie group action, Proc.11th ICFIDCA,
July27-31, 2003, Chiang Mai University, Thailand[20] Takeuchi,S., On the universal reciprocity law of the evaluation functor, Proceedings of the 12th
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Univ.30(2006),1-6.
[22] Takeuchi,S., in preparation.

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