Torsion theories and radicals in normal categories
M. M. Clementino∗, D. Dikranjan and W. Tholen†
Abstract
We introduce a relativized notion of (semi)normalcy for categories that come equipped
with a proper stable factorization system, and we use radicals and normal closure operators
in order to study torsion theories in such categories. Our results generalize and complement
recent studies in the realm of semi-abelian and, in part, homological categories. In particular, we characterize both, torsion-free and torsion classes, in terms of their closure under
extensions. We pay particular attention to the homological and, for our purposes more importantly, normal categories of topological algebra, such as the category of topological groups.
But our applications go far beyond the realm of these types of categories, as they include,
for example, the normal, but non-homological category of pointed topological spaces, which
is in fact a rich supplier for radicals of topological groups.
1
Introduction
Ever since the concept was defined in [22], semi-abelian categories have been investigated intensively by various authors. Their main building block, Bourn protomodularity [8], in conjunction
with Barr exactness [3], and the presence of a zero object and of finite limits and colimits provide
all the tools necessary for pursuing many themes of general algebra and of homology theory of
not necessarily commutative structures. The monograph [5] gives a comprehensive account of
these developments.
It has been observed in [6] and [9] that the category TopGrp of topological groups satisfies all conditions of a semi-abelian category, except for Barr-exactness, i.e., equivalence relations are not necessarily effective. But TopGrp is still regular, that is: it has a pullback-stable
(RegEpi, Mono)-factorization system. The term homological was used in [5, 6] for finitely complete, regular and protomodular categories with a zero object. Hence, TopGrp is the prototype
of a homological category, which still allows for the establishment of the essential lemmata of
homology theory, but fails to be semi-abelian.
Let us examine TopGrp’s failure to be semi-abelian a bit more closely. Recall that a category
C is semi-abelian if, and only if, (see [22]):
(1) C is finitely complete and has a zero object;
(2) C has a pullback-stable (RegEpi, Mono)-factorization system;
∗
†
Partial financial assistance by CMUC/FCT is gratefully acknowledged.
Partial financial assistance by NSERC is gratefully acknowledged.
1
(3) for every commutative diagram
//
//
p
1
//
2
//
in C with a regular epimorphism p, if 1 and 1 2 are pullback diagrams, 2 is also one;
(4) equivalence relations in C are effective, i.e., are kernel pairs;
(5) C has finite coproducts.
A semi-abelian category has, in fact, all finite colimits. Conditions (1)-(3) say precisely that C
is homological. In a homological category, RegEpi = N ormEpi = the class of normal epimorphisms, i.e., of cokernels, and Mono = 0-Ker = the class of morphisms with zero kernel (see
[8, 5]). Now, in the presence of the other conditions, (4) may be rephrased in old-fashioned terms
by:
(4′ ) images of normal monomorphisms (=kernels) under normal epimorphisms are normal
monomorphisms.
Here “image” is to be understood with respect to the (RegEpi, Mono) = (N ormEpi, 0-Ker)factorization system of C. But in TopGrp the natural image (as a subobject of the codomain) is
not formed via the (RegEpi, Mono)- but the (Epi, RegM ono)-factorization system. (Consider,
for example, the normal epimorphism q : R ,2 R/Z and the cyclic subgroup H of R gener√
ated by 2; q(H) is, as a subgroup of R/Z, dense, whereas the quotient topology of H would
make it discrete.) Hence, the only condition that prevents TopGrp (and even TopAbGrp, the
category of topological abelian groups) from being semi-abelian could be rescued if we would
interpret “image” naturally, i.e., switch to the “correct” factorization system. The example
HausGrp of Hausdorff topological groups shows that, in general, the “correct” factorization system (namely: (surjective, embeddings)) may be given neither as the (RegEpi, Mono)- nor as
the (Epi, RegM ono)-system. This is why we are proposing relativized notions of homologicity
and of semi-abelianess in this paper, for categories that come equipped with a proper and stable
(E, M)-factorization system, which we call (E, M)-seminormal and (E, M)-normal, respectively.
(We are aware of the fact that the term normal for categories has been used in the older literature, particularly in [28, 29], but not with any lasting impact. Hence, we hope that our
terminology does not lead to any confusion.) These relativized notions reach categories such
as Set∗ and Top∗ (pointed sets and pointed topological spaces), which are very far from being
homological. Nevertheless, our axioms are strong enough to establish the key characterization
theorems for torsion and torsion-free classes, in terms of their closure property under extensions.
In fact, most of the results in this paper remain true for categories that satisfy only a fraction of
the conditions for (E, M)-(semi)normalcy. We therefore refer always directly to the conditions
used at each stage, rather than working with the blanket assumption of (E, M)-normalcy.
The paper rests on two (well-known, in principle) correspondences, between torsion theories
and radicals, and between radicals and closure operators, the latter of which is adapted from [18],
and the composition of which has been the subject of a recent paper by Bourn and Gran [9]. We
generalize and extend their results from the context of homological (or semi-abelian categories)
2
to that one of seminormal (or normal) categories, and beyond, by adapting to the present context
many results on (pre)radicals presented in [18]. This adaptation follows the lead of [9] where
the notion of closure operator for arbitrary subobjects (as used in [17, 18]) is restricted to one
for normal subobjects. We believe that the factorization of the correspondence between torsion
theories and closure operators through radicals greatly clarifies matters, and it also connects the
results better to existing work, especially to that of Barr [4] and Lambek [23]. (The Lambek
paper contains many references to the literature of the time, which was predominantly concerned
with torsion theories for R-modules. Later papers present general categorical approaches, for
example [30, 27, 11].)
Consequently, after specifying our setting in Section 2, we start off by presenting (pre)radicals
in Section 3. Their easy relation with torsion theories follows in Section 4, while the more involved
relation with closure operators is presented in Section 5. We give a summary of results in Section
6 and present examples in Section 7.
2
The setting
Throughout this paper we consider a pointed, finitely complete category C, which has cokernels
of kernels. We also assume that C comes with a fixed (orthogonal) factorization system (E, M),
which is proper and stable. Hence, E ⊆ Epi and M ⊆ Mono are pullback-stable classes of
morphisms such that C = M · E, and the so called (E, M)-diagonalization property holds. As a
consequence, for every morphism f : X → Y one has the adjunction
f (−)
M/X oo
⊥
f −1 (−)
//
M/Y
between inverse image and image under f of M-subobjects, given by pullback and (E, M)factorization, respectively. Note that, because of pullback stability, one has f ∈ E if, and only
if,
f (f −1 (N )) = N for all N ∈ M/Y.
We also note that, because the system is proper, every regular monomorphism represents an
M-subobject, and every regular epimorphism lies in E; in particular, every normal monomorphism (=kernel of some morphism) is in M, and every normal epimorphism (=cokernel of some
morphism) is in E.
Denoting by 0-Ker the class of morphisms with trivial kernel (which contains all monomorphisms), first we note that C automatically has a second factorization system given by
(N ormEpi, 0-Ker) which, however, generally fails to be proper or stable, but which coincides
with (RegEpi, Mono) when C is homological.
Proposition 2.1 Every morphism factors into a normal epimorphism followed by a morphism
with a trivial kernel, and this constitutes an orthogonal factorization system.
p
h
Proof. For every f , in the factorization f = (X → X/K −→ Y ) with K = ker f , the morphism
k
p
// X/K factors through M = ker h, by a morphism p′ : K → M , which is epic
// X
K ,2
since, with p ∈ E, also p′ ∈ E. Since pk = 0, we have M = 0.
3
To show the diagonalization property, let tp = ms with a normal epimorphism p and ker m =
0, and let m′ be the pullback of m along t. Then ker m′ = ker m = 0, and for the arrow v induced
by the pullback property one has vk = 0, with k : ker p → X.
X LLL
LLLv
LLL
L&&
p
P
rrr
r
r
rr ′
r
xx rr t
s
U
//
r88 Y
m′rrrr
rr
rrr
t
// V
m
The induced arrow Y → P , together with t′ : P → U , gives the desired fill-in morphism.
✷
The additional conditions used in this paper, which will give the notion of (E, M)-(semi)normalcy, are essentially about the interaction of the two factorization systems of C, (E, M)
and (N ormEpi, 0-Ker), and are best motivated by the interaction of the quotient and subspace
topology in TopGrp (where quotient map means normal epimorphism, and where we would choose
for M the class of embeddings). To this end we first consider a commutative square
X
f
(∗)
// Y
p
q
Z
g
// W
in C and show:
Proposition 2.2 Let f ∈ M and p ∈ E. Then ker p = ker q if, and only if, ker g = 0 and
ker q ≤ X (so that ker q ,2 // Y factors through f ). These equivalent conditions hold true when
(*) is a pullback diagram, and they imply that (*) is a pushout diagram in case q is a normal
epimorphism.
Proof. One always has ker p ≤ ker q. If (*) is a pullback one obtains h with f h = k =
( ker q ,2 // Y ) and ph = 0. The latter identity actually follows from the former if ker g = 0,
since g(ph) = qk = 0. Now ph = 0 makes h factor through ker p, whence ker p = ker q. Conversely, ker p = ker q trivially implies ker q ≤ X, and
ker g = p(p−1 (ker g)) = p(f −1 (ker q)) = p(ker p) = 0.
Furthermore, any morphisms a, b with ap = bf satisfy
b(ker q) = a(p(ker p)) = a(0),
so that b factors uniquely as b = cq when q is a normal epimorphism. Since p ∈ E is epic, cg = a
follows.
✷
Definition 2.3 (1) A commutative diagram (*) is a basic image square (bis) if p ∈ E, f ∈ M,
q ∈ N ormEpi, and ker p = ker q.
4
(2) C is (E, M)-seminormal when, for every bis (*),
g ∈ M if, and only if, p ∈ N ormEpi,
and when, in either case, the bis is a pullback diagram.
Remark 2.4 There are, implicitly, three conditions that make up (E, M)-seminormalcy, which
we may state independently from each other, as follows.
(CT) Every bis (*) with g ∈ M is a pullback diagram.
Equivalently: for every normal epimorphism q : Y → W and for every M-subobject X of Y with
ker q ≤ X, one has q −1 (q(X)) = X. Since we already know that q ∈ E satisfies q(q −1 (Z)) = Z
for all M-subobjects Z of W , condition (CT) in fact means that the Correspondence Theorem
of algebra holds true, i.e., M-subobjects of Y above the kernel of q correspond bijectively to
M-subobjects of W :
q(−)
∼
ker q\(M/Y ) oo
q −1 (−)
//
M/W
Here is an equivalent formulation of (CT) that we shall use frequently:
(CT′ ) For every normal epimorphism q : Y → W and all M-subobjects X1 , X2 of Y with ker q ≤
X2 and q(X1 ) ≤ q(X2 ), one has X1 ≤ X2 .
The next ingredient to (E, M)-seminormalcy is:
(PN) For every bis (*) that is a pullback, g ∈ M implies p ∈ N ormEpi.
(Of course, in the presence of (CT), there is no need to mention the pullback provision.) Since
E and M are stable under pullback, and in light of Proposition 2.2, (PN) just means pullback
stability of normal epimorphisms along M-morphisms, that is:
(PN′ ) For every pullback diagram (*) with g ∈ M and q ∈ N ormEpi, also p ∈ N ormEpi.
The third ingredient to (E, M)-seminormalcy is:
(QN) For every bis (*), p ∈ N ormEpi implies g ∈ M.
Since a bis is a pushout diagram (by Prop. 2.2), (QN) requests M to be stable under certain
pushouts along normal epimorphisms, briefly referred to as stability under normal quotients:
(QN′ ) For every M-morphism f : X → Y and every normal subobject N ,2 // Y with N ≤ X,
the induced morphism g : X/N → Y /N lies also in M.
We shall now consider two further conditions which, in conjunction with (CT), (PN) and (QN),
will make C (E, M)-normal.
(IN) For every commutative square (*) with p, q ∈ E and f, g ∈ M, if f is a normal monomorphism, g is also one.
Hence, (IN) stipulates that M-images of normal monomorphisms along an E-morphism be normal monomorphisms:
5
(IN′ ) For every E-morphism q : Y → W and every normal subobject N ,2 // Y , the M-subobject
q(N ) → W is also normal.
The last condition is just a mild existence condition for particular colimits:
(JN) Every two normal subobjects have a join, that is: for any two normal subobjects N ,2 // X
and K ,2 // X , there is a least normal subobject N ∨ K ,2 // X containing N and K.
Since we may switch back and forth between kernels and cokernels, (JN) means equivalently:
(JN′ ) The pushout of any two normal epimorphisms (with common domain) exists.
Definition 2.5 C is (E, M)-normal if C is (E, M)-seminormal and (IN) and (JN) hold; equivalently, if C satisfies (CT), (PN), (QN), (IN) and (JN).
Examples 2.6 (1) Every homological category C is (RegEpi, Mono)-seminormal. In fact, (PN)
is just a particular instance of RegEpi’s pullback stability, and (QN) is (vacuously) satisfied,
since Mono = 0-Ker (by Prop. 2.2). To prove (CT) in a homological category, one can just
apply property (3) of the Introduction to the diagram
ker q
// X
f
// Y
p
0
// Z
q
g
// W
In other words, (CT) is just a very special instance of the protomodularity condition of a homological category.
(2) A homological category C with binary coproducts is (RegEpi, Mono)-normal if, and only if,
it is semi-abelian. In fact, (IN) coincides with (4′ ) of the Introduction in this case.
(3) The category TopGrp (and, more generally, the category of models in Top for any semi-abelian
theory, see [6]) is homological and (E, M)-normal, with (E, M)=(surjections, embeddings), but
(generally) not semi-abelian. Similarly for HausGrp, the category of Hausdorff groups, etc.
(4) The categories Set∗ and Top∗ of pointed sets and topological spaces, respectively, are (E, M)normal, with (E, M)= (surjections, embeddings), but certainly not homological (since Mono 6=
0-Ker).
3
Normal preradicals
Definition 3.1 A (normal) preradical of the category C (as in Section 2) is a normal subfunctor
r ,2 // Id C of the identity functor of C, i.e., for all X we have a normal monomorphism rX ,2 // X ,
so that every morphism f : X → Y restricts to rX → rY . It is
• idempotent if r(rX) = rX for all objects X;
• a radical, if r(X/r(X)) = 0 for all objects X;
• M-hereditary if f −1 (rY ) = rX for every M-morphism f : X → Y ;
• E-cohereditary if f (rX) = rY for every E-morphism f : X → Y .
6
Let us note immediately that M-heredity implies idempotency (consider f : rX ,2 // X ), and Ecoheredity forces r to be a radical (consider f : X → X/rX). Both the least preradical 0 (given
by 0 → X) and the largest preradical 1 (given by X → X) satisfy these additional properties.
Remark 3.2 With every preradical r of C one associates the full subcategory Fr of r-torsion-free
objects X, which must satisfy rX = 0. Denoting by
̺X : X → X/rX = RX
the canonical projection, one obtains an endofunctor R, pointed by a natural transformation
̺ : Id C → R, which is pointwise a normal epimorphism and has Fr as its fixed subcategory:
Fr = Fix (R, ̺) = {X : ̺X iso }.
Conversely, any endofunctor S, pointed by a pointwise normal epimorphism σ : Id C → S, gives
the preradical ker σ. We thus have an adjunction
{ preradicals } oo
//
⊥
{ normal-pointed endofunctors }.
Under this adjunction, radicals correspond bijectively to those (S, σ) with Sσ iso, i.e., to full
and replete normal-epireflective subcategories,
{ radicals } oo
//
∼
{ normal-epireflective subcategories }op .
For a preradical r of C, Tr denotes the full subcategory of all r-torsion objects X, defined by
the condition rX = X. Preradicals are nothing but normal copointed endofunctors, and Tr is
just the fixed subcategory of that copointed endofunctor of C. By restriction of the principal
adjunction one obtains the bijective correspondence
{ idempotent preradicals } oo
//
∼
{ normal-monocoreflective subcategories }.
Proposition 3.3 Let r be a preradical of C. Then:
(1) r is M-hereditary if, and only if, r is idempotent and Tr closed under M-subobjects in C.
(2) r is E-cohereditary if, and only if, r is a radical and Fr is closed under E-images in C,
provided that C satisfies that (CT) and (IN).
Proof. (1) Let f : X → Y be in M. If rY = Y , then f −1 (rY ) = X, and rX = X follows
when r is hereditary. Conversely, assuming closure of Tr under M-subobjects, since rY ∈ Tr
by idempotency of r, we conclude f −1 (rY ) ∈ Tr . Trivially, f −1 (rY ) ≤ X, hence f −1 (rY ) =
r(f −1 (rY )) ≤ rX, and rX ≤ f −1 (rY ) is always true.
(2) Let f : X → Y be in E. If rX = 0, then f (rX) = 0, and rY = 0 follows when r is
cohereditary. Conversely, we now assume Fr to be closed under E-images. From (IN) we have
that f (rX) is normal in Y , and we can form the commutative diagram
rX
// X
̺X
// X/rX
f
f (rX)
// Y
7
f
q
// Y /f (rX)
Since f ̺X = qf ∈ E, also f ∈ E, so that from X/rX ∈ Fr (for the radical r) one can derive
Z := Y /f (rX) ∈ Fr , by hypothesis. Since trivially f (rX) ≤ rY , in the presence of (CT) we
can deduce equality of these two M-subobjects from q(rY ) = 0; in fact, rY ,2 // Y
through rZ = 0.
q
// Z factors
✷
Remarks 3.4 (1) Of course, for any epimorphism f : X → Y in C and any preradical r, when
X ∈ Tr also Y ∈ Tr . Likewise, if f is a monomorphism, Y ∈ Fr implies X ∈ Fr . These assertions
are immediate consequences of the functoriality of r, as an inspection of the naturality diagram
rX
// rY
// Y
X
f
reveals.
(2) For an idempotent radical r, both Tr and Fr are closed under extensions, that is: if in the
short exact sequence
k
// Y ,2
0
// X
p
2,
Z
// 0
both Y and Z lie in Tr (Fr ), X also lies in Tr (Fr ). More precisely, Tr (Fr ) is closed under
extensions if r is a radical (idempotent preradical, respectively). In fact, for a radical r consider
the commutative diagram
// rX
// rZ
rY
// X
k
Y
̺Y
p
// Z
̺X
̺Z
// RX
RY
// RZ
If Y, Z ∈ Tr , so that rY = Y , rZ = Z, ̺X must factor through p by a (normal) epimorphism
Z → RX, so that with Z also RX ∈ Tr , by (1). But RX ∈ Fr since r is a radical, so that
RX ∈ Tr ∩ Fr must be 0, which means X ∈ Tr . Symmetrically, if r is an idempotent preradical
and Y, Z ∈ Fr , so that rY = 0 = rZ, rX → X must factor through k by a monomorphism
rX → Y , so that with Y also rX ∈ Fr , by (1). But rX ∈ Tr , since r is idempotent, so that
rX ∈ Tr ∩ Fr , must be 0, which means X ∈ Fr .
(3) M-heredity of a preradical r means, by definition, that every M-morphism f : X → Y yields
a pullback diagram
// rY
rX
X
f
// Y
in C. Next we show that E-coheredity fully deserves the dual name also under this perspective.
8
Corollary 3.5 A preradical r is E-cohereditary if, and only if, for every E-morphism f : X → Y ,
X
f
// Y
̺X
̺Y
X/rX
Rf
// Y /rY
is a pushout diagram in C.
Proof. Assume E-coheredity and consider g : X/rX → Z, h : Y → Z with g̺X = hf . (E, M)factoring and exploiting the (E, M)-diagonalization property we see that we may assume g, h ∈ E,
without loss of generality. Then X/rX ∈ Fr implies Z ∈ Fr with 3.3(2), so that h factors through
the Fr -reflection ̺Y by a morphism t : Y /rY → Z, which also satisfies tRf = g since ̺X is epic.
Conversely, exploiting the pushout diagram for f = ̺X we obtain ̺Y iso with Y = X/rX, hence
rY = 0. For general f ∈ E we just need to show that X ∈ Fr implies Y ∈ Fr ; but that is trivially
true since ̺X iso implies ̺Y iso.
✷
Relativizing the notion given in [21], let us call a full replete subcategory B of C to be EBirkhoff when B is normal-epireflective and closed under E-images in C. Dually, B is called
M-co-Birkhoff if B is normal-monocoreflective and closed under M-subobjects in C.
Restricting the correspondences of 3.2 one obtains from 3.3:
Theorem 3.6 (1) M-hereditary preradicals of C correspond bijectively to M-co-Birkhoff subcategories of C.
(2) If C satisfies (CT) and (IN), E-cohereditary (pre)radicals of C correspond bijectively to
E-Birkhoff subcategories of C.
A useful observation is:
Lemma 3.7 For any preradical r and every morphism f : X → Y one has f −1 (rY ) = rX if,
and only if, ker (Rf ) = 0, with Rf as in 3.5.
Proof. First assume f −1 (rY ) = rX, i.e., that the naturality diagram 3.4 is a pullback, and form
the pullback diagram
k′
̺−1
X (ker (Rf ))
// X
̺X
e
k
ker (Rf )
// X/rX
Since ̺Y f k ′ = (Rf )ke = 0, f k ′ must factor through rY , and then k ′ must factor through rX.
This yields ̺X k ′ = ke = 0, which forces k = 0 since e is epic.
Conversely, assume ker (Rf ) = 0 and consider morphisms g, h with f g = mh, with m =
(rY ,2 // Y ) . Hence g factors through rX, which is a factorization also for h since m is monic.
✷
Corollary 3.8 A radical r is M-hereditary if, and only if, the reflector R of Fr takes every
M-morphism to a morphism with trivial kernel.
9
We note that the corollary remains valid if r is just a preradical, since the endofunctor R is
still available in that case.
Examples 3.9 (1) The only preradicals in Set∗ (see 2.6(4)) are 0 and 1; likewise in the category
of vector spaces over a field.
(2) The category Top∗ (see 2.6(4)) has a very large supply of preradicals. Here is a first general
scheme for obtaining radicals, that actually works in the abstract category C of our setting,
provided that the needed limits exist. Let B be a class of objects in C, and for X ∈ C let qB X be
the intersection of the kernels of all morphisms X → B, B ∈ B. Of course, when B is reflective,
qB X = ker ̺X , as described in 3.2. Then qB is a radical, which we call the B-radical in C, and
just B-radical when B = {B}. In Top∗ now, taking for B the Sierpinski dyad S with closed
base point, then qS (X, x) = {x} is the closure of x in X. If S ∗ denotes the Sierpinski dyad with
open base point, qS ∗ (X, x) is the intersection of all open neighbourhoods of x in X. Both, qS
and qS ∗ are (M-)hereditary (with M = {embeddings}) and therefore idempotent. Applying
the same procedure to the 2-point discrete space D we obtain for qD (X, x) the quasi-component
of x in X, a non-idempotent radical. Likewise, for the real line R pointed by 0, the radical qR
fails to be idempotent. (qR (X, x) contains all points in X that cannot be separated from x by
a continuous R-valued function; non-idempotency is therefore witnessed by an infinite pointed
regular T1 -space (X, x) such that qR (X, x) 6= {x} is finite.)
(3) Let A be is a class of objects of Top∗ , and for X in Top∗ let pA be the union of the images of
all maps A → X, A ∈ A. When A is normal-monocoreflective, only one such morphism suffices.
Then pA is an idempotent preradical of Top∗ . Certainly, one can replace A by its normalmonocoreflective hull. In the particular case when the class A is singleton {A}, we just put pA .
It is easy to see that pS = qS ∗ , and pS ∗ = qS , while pR coincides with the arc-component of
the base point (so it differs from qR ). (In general, pA = qB may occur only when A and B have
at most the zero objects in common.)
(4) If A is a connectedness in Top and if B is the corresponding disconnectedness (in the sense
of [1]), then pA = qB . In particular, we denote by p the idempotent radical of Top∗ obtained by
the connected component of the base point (see also [11]).
(5) Every preradical r of the semi-abelian category Grp of groups may be naturally lifted to a
preradical of the (E, M)-normal category TopGrp (see 2.6), by just regarding rG as a subspace of
the topological space G, for a given topological group G. Surprisingly, essentially every preradical
r of Top∗ may also be lifted to TopGrp, that is: for a topological group G one can expect rG
to be not just a subspace of G but a subgroup. In fact, this is true for any of the examples
mentioned in (2), (3) and (4). (A more precise discussion of this phenomenon follows in 5.11
below.) Specifically, the connected component pG of the neutral element of the topological
group G defines an idempotent radical of TopGrp. Likewise, qG = qD (G, eG ) defines a radical
q of TopGrp that, also in this category, fails to be idempotent: for every ordinal α one can find
a topological group Gα such that the transfinite iterations qGα , q(qGα ), etc., form a chain of
subgroups of Gα of length α ([13]).
It is interesting to note that, when G is a compact abelian Hausdorff group, the topologically
defined subgroup pG coincides with the algebraically defined maximally divisible subgroup dG
(cf. [12, Example 4.1]).
10
(6) Let us now apply the principle described in (2) to subclasses B of TopGrp directly. For
example (see [14]), for Z the class of zero-dimensional groups (so that they have a base of clopen
neighbourhoods), we obtain the radical z = qZ which, like q = qD , is not idempotent (for the
same reason as q). Another non-idempotent radical is obtained by considering the class D of
discrete groups. Then oG = qD G is the intersection of all open normal subgroups of G. There
is a chain of inclusions
pG ⊆ qG ⊆ zG ⊆ oG,
each of which may be proper. The inequality p 6= q follows from the fact that p is idempotent
while q is not. The highly non-trivial fact that q 6= z was established by Megrelishvili [25].
Answering a question of Arhangel′ skij, he gave an example of a totally disconnected group G
(so, qG = {e}) that admits no coarser Hausdorff zero-dimensional group topology (so, zG 6= {e}).
Finally, the properness of the last inclusion is witnessed by the subgroup G = Q/Z of all torsion
elements of the circle group (it is obviously zero-dimensional, but has no proper clopen subgroup,
so zG = {e}, while oG = G).
(7) Here is an important classical example which, again, arises by the scheme of (6). The class
CompGrp of compact Hausdorff groups is a reflective subcategory of TopGrp. For every topological
group G the reflection ̺G : G → bG has dense-image and is the Bohr compactification of G. The
CompGrp-radical is known as von Neumann’s kernel and usually denoted by n. According to
von Neumann [26], the groups of Fn are called maximally almost periodic (briefly, MAP) while
the groups of Tn are called minimally almost periodic (briefly, MinAP). The radical n is neither
idempotent ([24]), nor cohereditary (there exists a MAP group G with a non-trivial Hausdorff
quotient G/N that is MinAP [2]). More about this radical can be found in [16].
4
Torsion theories
Definition 4.1 [9] A torsion theory of C is a pair (T, F) of full replete subcategories of C such
that:
(1) for all Y ∈ T and Z ∈ F, every morphism f : Y → Z is zero;
(2) for every object X ∈ C there exists a short exact sequence
0
// Y ,2
k
// X
p
2,
Z
// 0
(so that k = ker p and p = coker k) with Y ∈ T and Z ∈ F.
T is the torsion part of the theory, and F is its torsion-free part. Any full replete subcategory of
C is called torsion (torsion-free) if it is the torsion (torsion-free, respectively) part of a torsion
theory.
One proves easily:
Theorem 4.2 (T, F) is a torsion theory of C if, and only if, there exists a (uniquely determined)
idempotent radical r of C with T = Tr and F = Fr .
Proof. For an idempotent radical r and any object X, we have the short exact sequence
0
// rX ,2
// X
2,
X/rX = RX
11
// 0
with rX ∈ Tr and RX ∈ Fr . Furthermore, because of the commutativity of
rY
// rZ
// Z
Y
h
any morphism h must be 0 when Y ∈ Tr and Z ∈ Fr , i.e., when rY = Y and rZ = 0.
Conversely, given a torsion theory (T, F), let us first point out that Y, Z of 4.1(2) depend
functorially on X. Indeed, given any morphism f : X → X ′ , consider
0
// Y ,2
0
// Y ′ ,2
// X
2,
Z
// 0
2, ′
Z
// 0
f
// X ′
with both rows short exact, Y ′ ∈ T and Z ′ ∈ F. Then, since any arrow Y → Z ′ is 0 we
have the fill-in arrows Y → Y ′ and Z → Z ′ . In particular, given X we can put rX = Y and
RX = X/rX ∼
= Z to obtain a preradical r. Clearly, T = Tr since, for X ∈ T, because RX ∈ F,
the morphism ̺X is 0, so that rX = ker 0 = X; conversely, when X = rX one has X ∈ T since
rX ∈ T. Analogously, F = Fr . These identities tell us also that r is an idempotent radical.
Finally, as the coreflector of T, r is uniquely determined by T.
✷
Remark 4.3 For every preradical r of C, the pair (Tr , Fr ) satisfies (1) of 4.1, but need not be a
torsion theory. However, if it is, then r is necessarily an idempotent radical, by the uniqueness
part of Theorem 4.2.
With the initial considerations of Section 3 one obtains immediately:
Corollary 4.4 In a torsion theory, the torsion part and the torsion-free part determine each
other uniquely. A full replete subcategory is torsion if, and only if, it is normal-monocoreflective
such that the coreflector is a radical; it is torsion-free if, and only if, it is normal-epireflective
such that the (pre)radical given by the kernels of its reflections is idempotent.
Corollary 4.5 The following conditions are equivalent for a pair (T, F) of full replete subcategories of C satisfying condition 4.1(2):
(i) (T, F) is a torsion theory;
(ii) T ∩ F = {0}, T is closed under E-images, and F is closed under M-subobjects.
(iii) T ∩ F = {0}, and for all morphisms f : X → Y one has
(f normal epi & X ∈ T ⇒ Y ∈ T),
(ker f = 0 & Y ∈ F ⇒ X ∈ F).
Proof. Assuming (i), we have (T, F) = (Tr , Fr ) for an idempotent radical r. Clearly, Tr ∩Fr = {0},
and (ii) follows from 3.4(1). Now only the last assertion of (iii) needs proof; but the diagram
// rY
rX
X
f
12
// Y
shows that, when rY = 0, rX factors through ker f , hence rX = 0 when ker f = 0.
Conversely, assuming (ii) or (iii), in order to show that any f : X → Y with X ∈ T and
Y ∈ F is 0, factor f (E, M) or (N ormEpi, 0-Ker) (see 2.1), respectively. Then the factoring
object must be 0 by hypothesis, in each of the two cases.
✷
We now prove that closure under extensions (see 3.4(2)) is characteristic for both torsion-free
and torsion subcategories.
Theorem 4.6 (1) Let C satisfy (QN) and (IN). Then a normal-epireflective subcategory F of
C is torsion-free if, and only if, it is closed under extensions and, for its induced radical r,
rrX is normal in X, for all objects X.
(2) Let C satisfy (PN). Then a normal-monocoreflective subcategory T of C is torsion if, and
only if, it is closed under extensions.
Proof. (1) It suffices to show that a radical r is idempotent when Fr is closed under extensions
and rrX is normal in X, for all X ∈ C. The left square of
f
rX 2,
p
_
_
g
RrX = rX/rrX
1
// X
q
h
// X/rrX
// X
_
̺X
// X/rX = RX
is a bis for all objects X; in fact, ker p = rrX = ker q. By (QN), with f ∈ M we have g ∈ M,
and by (IN) we know that g is even a normal monomorphism. Since hq = ̺X is a normal
epimorphism, h is also one, with kernel g:
ker h = h−1 (0) = q(q −1 (h−1 (0))) = q(̺−1
X (0)) = q(rX) = g.
Consequently, with RrX and RX ∈ Fr , also X/rrX ∈ Fr , that is: r(X/rrX) = 0. The
commutative diagram
// r(X/rrX) = 0
rX
_
_
q
X
// X/rrX
now shows rX ≤ ker q = rrX, as desired.
(2) It suffices to show that an idempotent preradical r is a radical when Tr is closed under
extensions. With RX = X/rX and ̺X : X → RX the projection, we form the pullback
2,
̺−1
X (rRX)
// X
̺X
p
rRX 2,
// RX
r
(In the terminology of 5.3 below, ̺−1
X (rRX) = maxX (rX).) By (PN), p is a normal epimorphism,
and by Prop. 2.2,
ker p = ker ̺X = rX.
13
Hence, we may apply the hypothesis to the short exact sequence
0
// rX ,2
// ̺−1 (rRX)
X
2,
rRX
// 0
and obtain ̺−1
X (rRX) ∈ Tr . Consequently, we have the commutative diagram
2,
̺−1
X (rRX)
// rX
2,
̺−1
X (rRX)
// X
−1
that is: ̺−1
X (rRX) ≤ rX, and trivially rX ≤ ̺X (rRX) (from the pullback property). This fact
easily implies RX ∈ Fr (as desired), as we shall show explicitly, and more generally, in 5.3 below:
since rX is maxr -closed, RX = X/rX ∈ Fr .
✷
Remark 4.7 (1) The additional condition in 4.6(1) that rrX be normal in X (i.e., that the
composite
// rX ,2
// X
rrX ,2
of normal monomorphisms be normal again) is, of course, satisfied in categories of modules
where each subobject is normal. But it also holds in Grp (since, generally, rrX is a fully
invariant subobject of X), and therefore in TopGrp. But we do not know whether the condition
is redundant in a more general context, in all semi-abelian categories, in all categories of models
in Top for a semi-abelian theory?
(2) One may wonder why no extra condition is needed in 4.6(2): simply because normal epimorphisms are, unlike normal monomorphisms, always closed under composition in C. The proof
refers to this property implicitly since ̺−1
X (rRX) is the kernel of the composite morphism
X
2,
RX
,2
RRX,
which makes the intrinsic duality of the two proofs more apparent. Also note that, instead of
the conjunction of (QN) and (IN), in 4.6(1) it would suffice to require the precise categorical
dual of (PN′ ), namely that normal monomorphisms be stable under pushout along E-morphisms
(or just along normal epimorphisms).
(3) Without any additional hypotheses on C one can easily show that a normal-epireflective
subcategory F of C is torsion-free if, and only if, it is closed under extensions, and the full image
in C of its induced radical functor r is closed under (normal) epimorphisms. For C homological,
this criterion was proved in [9].
We say that a full reflective subcategory B of C has stable reflections if, for all objects Y and
all f : X → RY with X ∈ B, the morphism e in the (N ormEpi, 0-Ker)-factorization m · e of the
pullback of the B-reflection ̺Y : Y → RY is also a B-reflection (of X ×RX Y ). Under condition
(CT), this property turns out to be characteristic for torsion-free subcategories.
Theorem 4.8 Let C satisfy condition (CT). Then a normal-epireflective subcategory is torsionfree if, and only if, it has stable reflections.
14
Proof. Let us first assume that F is torsion-free, hence F = Fr for an idempotent radical r. We
consider a pullback diagram
f′
P
// Y
̺Y
p
f
X
// RY
p
with X ∈ F. Since (rP → P −→ X) factors through rX = 0, there is a morphism v : RP → X
with v̺P = p. By the pullback property, there is a unique k making the diagram
k
rY
f′
// P
// Y
̺Y
p
// X
0
f
// RY
commutative, with f ′ k = (rY ,2 // Y ) . Since the right and the whole rectangle are pullbacks,
so is the left, that is: rY is the kernel of p. The morphism v gives rP ≤ rY , and in order
to show equality, by (CT) it suffices to show ̺Y (rP ) = 0; but that follows trivially from the
commutativity of
rY
k
// P
̺P
̺rY
0 = RrY
Rk
// RP
(the identity 0 = RrY is due to the idempotency of r). Since k = ker p = rY = rP , ̺P coincides
with the normal-epi component in the (N ormEpi, 0-Ker)-factorization of p. This proves that F
has stable reflections.
Conversely, let us consider a normal-epireflective subcategory F with stable reflection ̺; we
may assume F = Fr for a radical r and must show idempotency of r. For every object X, since
rX = ker ̺X , we have the pullback diagram
rX
// X
̺X
// RX
0
Since 0 ∈ Fr and rX → 0 is a normal epimorphism, the hypothesis of stability gives RrX = 0,
so that r(rX) = ker (rX → RrX) = ker (rX → 0) = rX.
✷
Following [9] we say that a full replete subcategory B of C is a fibred reflection if for all
f : X → RY with X ∈ B and ̺Y : Y → RY the B-reflection of some C-object Y , the pullback of
̺Y along f is also a B-reflection. Clearly, this property is stronger than having stable reflections,
but if N ormEpi is pullback-stable (in particular, in homological categories), these properties
coincide. As an immediate corollary of the above theorem we obtain the following fact, proved
in [9, Theorem 4.11] for homological categories.
Corollary 4.9 Let C satisfy condition (CT), and let normal epimorphisms be stable under pullback in C. Then a normal-epireflective subcategory is torsion-free if, and only if, it is a fibred
reflection.
15
The following example shows that the above corollary fails to be true without the assumption
of pullback-stability, and it justifies our definition of having stable reflections as used in Theorem
4.8. Consider the idempotent radical p of Top∗ defined in Example 3.9(4). Then the torsionfree category Fc fails to be a fibred reflection. For a counter-example take Y = ([0, 1], 0) and
X = (Q, 0), then cY = Y , so RY = 0 and f : X → 0. Consequently, the pullback of ̺Y : Y → 0
is nothing else but the projection X × Y → X. Obviously, it is not a reflection map.
Definition 4.10 A torsion theory (T, F) is M-hereditary if T is closed under M-subobjects,
and it is E-cohereditary if F is closed under E-images.
From 3.3 and 4.2 we have:
Corollary 4.11 A torsion theory (T, F) in C is M-hereditary (E-cohereditary) if, and only if,
T is an M-co-Birkhoff (F is an E-Birkhoff ) subcategory of C, and (if C satisfies (CT) and (IN))
it is therefore equivalently described by an M-hereditary (idempotent E-cohereditary) radical of
C.
5
Normal closure operators
Definition 5.1 A normal closure operator c = (cX )X∈C of C assigns to every normal subobject
N ,2 // X a normal subobject cX (N ) of X such that, for every object X,
• cX is extensive: N ≤ cX (N ) for all N ;
• cX is monotone: N ≤ K ⇒ cX (N ) ≤ cX (K);
• the continuity condition is satisfied: cX (f −1 (L)) ≤ f −1 (cY (L)) for all f : X → Y and
normal subobjects L of Y .
The normal subobject N of X is c-closed in X if cX (N ) = N , and c-dense in X if cX (N ) = X.
One calls c
• idempotent, if cX (N ) is c-closed in X for all N ,2 // X ;
• weakly hereditary, if N is c-dense in cX (N ) for all N ,2 // X .
Remarks 5.2 (1) A closure operator c of C can be defined just like a normal closure operator,
except that cX acts on all M-subobjects of X (see [17, 18]). However, such a closure operator
very often maps normal subobjects to normal subobjects, i.e., yields a normal closure operator.
For example this property was observed in [18, Exercise 4V] for every closure operator of the
category of groups; in fact, it holds for every closure operator of a semi-abelian variety of universal
algebras (cp. [6, 7]).
If, in our category C, every M-subobject M → X has a normal closure νX (M ) in X, i.e.,
a least normal subobject containing M , then every normal closure operator c is the restriction
to normal subobjects of a closure operator d ≥ ν which maps normal subobjects to normal
subobjects: simply put d := cν, i.e., dX (M ) := cX (νX (M )). The existence of a normal closure of
every M-subobject is guaranteed if every M-subobject has a cokernel; then νX (M ) is simply the
kernel of coker (M → X). In particular, the normal closure exists when C is finitely cocomplete.
16
(2) Every normal closure operator c of C induces a normal preradical of C, namely
rad c X := cX (0),
defining in fact a functor of preordered classes:
rad : {normal closure operators} −→ {(normal) preradicals}.
Given a preradical r, there is a largest normal closure operator c with rad c = r, namely c = maxr ,
defined by
max rX (N ) = p−1 (r(X/N )) for all N ,2 // X ;
here p : X → X/N is the projection (see [17, 18]).
If (JN) is satisfied in C (see Section 2), then for every preradical r there is also a least normal
closure operator c with rad c = r, namely c = minr , defined by
minrX (N ) = rX ∨ N for all N ,2 // X .
One calls maxr and minr the maximal and minimal (normal) closure operator induced by r,
respectively, and easily verifies that for any normal closure operator c,
(c ≤ maxr ⇔ rad c ≤ r) and (min r ≤ c ⇔ r ≤ rad c ).
In other words, the functor rad has both adjoints:
min
qq
{normal closure operators}
mm
⊥
rad
// { (normal) preradicals}
⊥
max
Obviously, in the presence of (JN), N ,2 // X is minr -closed if, and only if, rX ≤ N . Furthermore,
minr is idempotent, and it is weakly hereditary if and only if r is idempotent. The corresponding
statements for maxr are more involved:
Proposition 5.3 Let r be a preradical of C.
(1) A normal subobject N ,2 // X is maxr -closed (maxr -dense) if and only if X/N is r-torsionfree (r-torsion, respectively).
(2) For every normal subobject N ,2 // X one has the isomorphism
(X/N )/r(X/N ) ∼
= X/ max rX (N ) and, under condition (PN), r(X/N ) ∼
= max rX (N )/N.
(3) maxr is idempotent if, and only if, r is a radical, and, under condition (PN), maxr is
weakly hereditary if, and only if, r is idempotent.
Proof. In the diagram
N
// maxr (N )
X
// X
// r(X/N )
_
// X/N
0
17
the whole and the right rectangle are pullbacks (by definition), hence also the left rectangle is
one. Consequently, N = maxrX (N ) if and only if r(X/N ) = 0. Moreover, the right pullback
diagram shows immediately that maxrX (N ) = X holds if and only if r(X/N ) = X/N .
(2) Extending the above diagram to the right we obtain
maxrX (N )
// X
2, X/ maxr (N )
X
_
// X/N
_
2,
(X/N )/r(X/N )
r(X/N )
Since N ≤ maxrX (N ) there is a diagonal morphism X/N → X/ maxrX (N ) for the right rectangle,
which is easily seen to induce an inverse for the right vertical arrow.
In order to show the second isomorphism, we complete the defining pullback diagram for
Y = maxrX (N ), as follows:
// X
Y KKK
KKK
KK
)
_
u
Y /N
t
tt
4tttt
r(X/N )
GG
GGs
GG
G##
// X/N
Condition (PN) guarantees that Y → r(X/N ) is a normal epimorphism, hence, t : Y /N →
r(X/N ) is also a normal epimorphism. In order to show that t is actually an isomorphism,
it suffices to prove ker t = 0. But that is obvious since, thanks to 2.1, Y → Y /N → X/N is
// X/N , hence ker s = 0 and, consequently,
the (N ormEpi, 0-Ker)-factorization of Y ,2 // X
ker t = 0.
(3) When r is a radical, with (2) we have
r(X/ max rX (N )) ∼
= r((X/N )/r(X/N )) = 0,
which means that maxrX (N ) is maxr -closed in X, for every N ,2 // X . Hence maxr must be
idempotent. Conversely, assuming this property, we have that rX = maxrX (0) is maxr -closed,
which means r(X/rX) = 0 by (1). Hence, r must be a radical.
When maxr is weakly hereditary, 0 is maxr -dense in maxrX (0) = rX. By (1), rX ∼
= rX/0
is r-torsion, hence rrX = rX. Conversely, if r is idempotent, we must show that the normal
subobject N of X is maxr -dense in Y = maxr (N ), that is: Y /N ∈ Fr . But we have Y /N ∼
=
r(X/N ) by (2), so that r(Y /N ) = Y /N follows from the idempotency of r.
✷
Remarks 5.4 (1) We note that in 5.3 only a weak form of (PN) is involved, namely that normal
epimorphisms be stable under pullback along normal monomorphisms.
(2) For a normal closure operator c of C, a morphism f : X → Y is called c-open if f −1 (cY (N )) =
cX (f −1 (N )) for all normal subobjects N ,2 // Y (see [18]). In the presence of (IN) we call f cclosed if f (cX (K)) = cY (f (K)) for every normal subobject K ,2 // X .
Proposition 5.5 Let r = rad c be the induced preradical of a normal closure operator c of C.
Then:
18
(1) A morphism f : X → Y with trivial kernel can be c-open only if f −1 (rY ) = rX. Hence,
c-openness of M-subobjects implies M-heredity of r.
(2) c-openness of all normal epimorphisms implies c = maxr .
(3) In the presence of (IN), a morphism f : X → Y can be c-closed only if f (rX) = rY .
Hence, c-closedness of all E-morphisms implies E-coheredity of r.
Proof. (1) f −1 (rY ) = f −1 (cY (0)) = cX (f −1 (0)) = cX (0) = rX.
(2) For N ,2 // X and f : X → X/N the projection, one has
cX (N ) = cX (f −1 (0)) = f −1 (cX/N (0)) = f −1 (r(X/N )) = maxrX (N ).
(3) f (rX) = f (cX (0)) = cY (f (0)) = cY (0) = rY .
✷
Establishing converse statements takes more effort:
Theorem 5.6 Let c be a normal closure operator of C with r = rad c . Then
(1) c = maxr if and only if all normal epimorphisms are c-open.
(2) All morphisms of C are maxr -open if and only if f −1 (rY ) = rX for all morphisms f with
trivial kernel.
Proof. For every morphism f : X → Y and N ,2 // X we have the commutative diagram
r(X/f −1 (N ))
r99
rr
rr
r
rr
// r(Y /N )
r99
rrr
r
r
rrr
p′
q′
maxrX (f −1 (N ))
// maxr (N )
Y
X/f −1 (N )
99
p rrr
r
r
r
r
rrr
X
f
// Y /N
99
q rrr
rr
r
r
rr
// Y r
f
maxr -openness of f means, by definition, that the front face is a pullback, for every N ,2 // Y .
But since the two side faces are pullbacks, the front face is a pullback whenever the back face is
one.
Now, for the morphism f , let us first note that ker f = p(p−1 (ker f )) = p(f −1 (N )) = 0. To
complete the proof of (1), after 5.5(2), we can let f be a normal epimorphism. Then also f is
one, hence, in fact an isomorphism, making the back face of the above cube a trivial pullback.
To complete the proof of (2), after 5.5(1), we just point out that, since ker f = 0, the back face
of the cube becomes a pullback by hypothesis.
✷
Normal closure operators satisfying the condition of 5.6(1) were called homological in [9],
but from the perspective of preradicals, maximal seems more appropriate. Hence, c is maximal
c
precisely when c = maxrad .
19
Corollary 5.7 Let r be a preradical, and let C satisfy (JN). Then minr = maxr (so that there
is exactly one normal closure operator inducing r) if and only if f −1 (N ∨ rY ) = f −1 (N ) ∨ rX
for all normal epimorphisms f : X → Y and for all N ,2 // X .
Proof. The condition just means that all normal epimorphisms are minr -open, and the assertion
follows from 5.6(1).
✷
Corollary 5.8 Assume that C satisfies (JN) and that, for every f : X → Y in M and all
N1 , N2 ,2 // Y , f −1 (N1 ∨ N2 ) = f −1 (N1 ) ∨ f −1 (N2 ). Then all f ∈ M are minr -open if and only
if r is M-hereditary.
Proof. Apply 5.5(1), with c = minr .
✷
Unfortunately, the assumption that f −1 (−) preserve the join of normal subobjects is quite
restrictive. Again, it seems more natural to consider maxr rather than minr . In order to do so,
one calls a normal closure operator c M-hereditary if, for every f : X → Y in M and every
normal subobject N ,2 // Y with N ≤ X (so that (f −1 (N ) = N ), one has cX (N ) = f −1 (cY (N )).
Obviously, M-heredity makes c weakly hereditary (for K ,2 // X , consider f : cX (K) → X).
Theorem 5.9 Of the following statements for a preradical r, each one implies the next. Conditions (i)−(iv) are equivalent if C satisfies (QN), and all are equivalent if every morphism with
trivial kernel lies in M.
(i) Every morphism in M is maxr -open.
(ii) maxr is M-hereditary.
(iii) There is an M-hereditary normal closure operator c with radc = r.
(iv) r is M-hereditary.
Proof. (i) ⇒ (ii) ⇒ (iii) is trivial, and so is (iii) ⇒ (iv):
rX = cX (0) = f −1 (cY (0)) = f −1 (rY ).
For (iv) ⇒ (ii) consider the cube of the proof of 5.6 again. Under (QN), f ∈ M implies f ∈ M
when N ≤ X, so that the back face of that cube is a pullback diagram by hypothesis, and so is
the front face.
(ii) ⇒ (i) for M = 0-Ker was already stated and proved in 5.5(2).
✷
Theorem 5.9 can be “dualized”, as follows:
Theorem 5.10 For a preradical r, the following statements are equivalent when C satisfies (IN)
and (JN):
(i) r is E-cohereditary;
(ii) every E-morphism is minr -closed;
(iii) every E-morphism is c-closed, for some normal closure operator with rad c = r.
20
When, moreover, C satisfies (CT), these conditions imply minr = maxr , so that all E-morphisms
are c-closed (for the only normal closure operator c with rad c = r); furthermore, normal epimorphisms are also c-open and satisfy the condition given in 5.7.
Proof. (i) ⇒ (ii) Using (IN) and (JN), for every f : X → Y in E and K ,2 // X we have
f (minrX (K)) = f (K ∨ rX) = f (K) ∨ f (rX) = f (K) ∨ rY = min rY (f (K)).
(ii) ⇒ (iii) is trivial, and for (iii) ⇒ (i) use 5.5(3).
In order to show that (ii) implies minrX (K) = maxrX (K) for all K ,2 // X , we note that, with
f denoting the projection X → X/K, one has
f (minrX (K)) = minrX/N (f (K)) = minrX/N (0) = r(X/N ) and f (max rX (N )) ≤ r(X/N ).
Since K = ker f ≤ minrX K ≤ maxrX K in the presence of (CT), this shows minrX (K) =
maxrX (K).
The remaining claims follow from 5.6(1) and 5.7.
✷
We return to the issue of lifting preradicals from Top∗ to TopGrp (see 3.9(5)). For a preradical
r of Top∗ , we can think of r(X, x) as of the closure of the point x in the space X. In fact, writing
cX (x) = r(X, x),
one defines a fully additive [
closure operator
c = cr of Top (see 5.2(1)), with full additivity
[
cX (Mi ) for all families of subsets Mi ⊆ X. In this way
referring to the property cX ( Mi ) =
i
i
preradicals of Top∗ correspond bijectively to fully additive closure operators of Top.
When can we “lift” the preradical r from Top∗ to TopGrp? That is: when is rG = r(G, eG )
a (normal) subgroup of G, for every topological group G?
Proposition 5.11 For a preradical r of Top∗ and a topological group G, the following conditions
are equivalent:
(i) rG is a normal subgroup of G;
(ii) rG · rG ⊆ rG;
(iii) crG (crG ({eG })) = crG ({eG });
(iv) crG (crG (H)) = crG (H), for every subgroup H of G.
If these equivalent conditions are satisfied for all topological groups G, r can be considered as a
preradical of TopGrp and cr as a normal closure operator of TopGrp, which coincides with the
minimal closure operator induced by r.
Proof. Let us note first that, by definition of cr , for all H ≤ G
[
crG (H) =
crG ({a}).
a∈H
Considering the Top∗ -maps
(G, e)
(−)·a
// (G, a)
21
(−)·a−1
// (G, e)
we also see that crG ({a}) = rG · a for all a ∈ G. Hence, crG (H) = rG · H = H · rG, a formula that
shows (i) ⇒ (iv) immediately. (iv) ⇒ (iii) is trivial. (iii) ⇒ (ii):
rG · rG = crG (rG) = crG (crG ({eG })) = crG ({eG }) = rG.
(ii) ⇒ (i) Since rG is always invariant under inversion and inner automorphisms, closure under
multiplication makes rG a normal subgroup of G.
✷
Problem 5.12 Is there a preradical r of Top∗ and a topological group G such that rG is not a
subgroup of G?
Remarks 5.13 (1) Prop. 5.11 shows that the answer to 5.12 is negative if we would restrict the
search to preradicals r with cr idempotent in Top, or to preradicals r with r(G × G) = rG × rG
for every topological group G, since the latter condition implies 5.11(ii). (In fact, idempotency of
cr implies preservation of finite products in Top by cr , and therefore by r; see [18, Prop. 4.11].)
One easily checks that cr is idempotent when r is a radical of Top∗ . Consequently, all radicals
of Top∗ are liftable to TopGrp.
(2) If there is an idempotent preradical r of Top∗ as a witness to a positive answer to 5.12 (so
that there is a group G, such that rG fails to be a subgroup of G), then there is even such witness
of the form s = pD for some D ∈ Top∗ (see 3.9(3)). In fact, if rG fails to be a subgroup of G,
consider the space D = rG and let s = pD . Then, trivially, D = sD ֒→ sG, and since D ∈ Tr
(by the idempotency of r), also sG ֒→ rG. Hence, sG = rG, which still fails to be a subgroup of
G.
6
Summary
6.1 (1) The assignments
F
Fr oo
// ker (reflection),
r,
r
// max r
radc oo
c
define bijections
{normal epirefl. subcategories} oo
// {radicals} oo
// {idpt. maximal nco’s}
(see 3.2 and 5.3); here “nco” stands for “normal closure operator”, “subcategory” means “full
replete subcategory”. In what follows we will also use the abbreviations idpt. = idempotent,
wh. = weakly hereditary, hered. = hereditary and cohered. = cohereditary.
(2) Under condition (PN), the bijection (1) restricts to bijections
{torsion-free subcategories} oo
// {idpt. radicals} oo
// {wh. idpt. maximal nco’s}
(see 4.2 and 5.3). In addition, if (CT) holds, torsion-free subcategories are described as normalepireflective subcategories with stable reflections (see 4.8), and under conditions (QN) and (IN)
there is a criterion involving closure under extensions (see 4.6(1)).
22
(3) Under conditions (CT), (IN) and (JN), the bijection (1) restricts also to bijections
// {E-cohered. radicals} oo
{E-Birkhoff subcategories} oo
// {idpt. nco’s with closed E’s}
(see 3.2 and 5.10).
(4) Under conditions (CT), (IN), (JN) and (PN) (hence, in every (E, M)-normal category) one
obtains from (2), (3) the bijections
{E-cohered. torsion theories} oo
// {E-cohered. idpt. radicals} oo
// {wh. idpt. nco’s with closed E’s}
6.2 (1) The assignments
T
Tr oo
// coreflection,
r
r,
radc oo
// max r
c
define bijections
{normal monocorefl. subcategoriess} oo
// {idpt. preradicals} oo
// {wh. maximal nco’s}
(see 3.2 and 5.3).
(2) The bijection (1) restricts to bijections (under condition (PN))
{torsion subcategories} oo
// {idpt. radicals} oo
// {wh. idpt. maximal nco’s}
(see 4.2 and 5.3). In addition, if (PN) holds, torsion subcategories are characterized as the
normal-monocoreflective subcategories closed under extensions (see 4.6(1)).
(3) Under conditions (PN) and (QN), the bijection (1) restricts also to bijections
{M-co-Birkhoff subcategories} oo
// {M-hered. preradicals} oo
// {M-hered. maximal nco’s}
(see 3.2 and 5.9).
(4) Under conditions (PN) and (QN) (hence, in every (E, M)-seminormal category) one obtains
from (2), (3) the bijections
{M-hered. torsion theories} oo
// {M-hered. radicals} oo
// {M-hered. idpt. maximal nco’s}
When every morphism with trivial kernel lies in M (as in every homological category with
its (RegEpi, Mono)-factorization system), M-hereditary idempotent maximal nco’s are simply
idempotent nco’s for which every morphism is open (see 5.6).
7
Examples
Remark 7.1 In generalization of 3.2, a preradical r of the category C (as in Section 2) may be
obtained from any pointed endofunctor σ : Id C → S of C, as rX = ker σX . In the particular
case S = Id C , r is actually M-hereditary:
−1 −1
−1
f −1 (rXY ) = f −1 (σY−1 (0)) = σX
(f (0)) = σX
(0) = rX,
23
for all f : X → Y in M. Dually, any copointed endofunctor τ : T → Id C gives a preradical R of
C via RX = ker (coker τX ), granted the existence of cokernels. In the particular case T = Id C ,
if the induced morphisms X → RX lie in E, R is E-cohereditary
f (RX) = f (τX (X)) = τY (f (X)) = τY (Y ) = RY.
In an abelian category C (with (E, M) = (N ormEpi, N ormM ono)), any natural transformation
σ : Id C → Id C gives both, r and R as above, with RX = X/rX:
8B D&
:RX
DD
z
z
DD
z
z
D!!
z
z
σX
// X
// X
̺X
rX = ker σX ,2
2,
coker σX
Moreover, if C has a generator G, so that the hom-set C(G, X) is jointly epic, the morphism σX
is determined by σG , since the diagrams
G
σG
// G
x
x
X
σX
// X
commute for all x ∈ C(G, X).
Example 7.2 In the category AbGrp of abelian groups, with its generator Z, a natural transformation σ : Id → Id is determined by σZ : Z → Z, i.e., by m = σZ (1) ∈ Z: σA (x) = mx for all
x ∈ A ∈ AbGrp. One obtains with 7.1
A[m] := rA = {x ∈ A : mx = 0},
mA := RA = {mx : x ∈ A} ∼
= A/rA.
r is a hereditary preradical which, however, fails to be a radical, unless m = 0 or m = ±1, so
that r = 0 or r = 1, producing only trivial torsion theories. Likewise, R is a cohereditary radical
that is idempotent only for trivial m. In fact, AbGrp has no non-trivial cohereditary torsion
theories (see 7.3 below).
By contrast, there are precisely 2ℵ0 hereditary radicals in AbGrp. We give a brief indication of
the proof (see [20]). For every prime p let tp (A) denote the p-primary component of the subgroup
of all torsion elements of an abelian group A. It is easy to see that tp is a hereditary radical of
AbGrp. For every set P of prime numbers, the supremum tP of all tp ’s when p runs over P is
again a hereditary radical. There are no other hereditary radicals of AbGrp beyond these. Indeed,
for any hereditary radical r of AbGrp, one can completely determine r by its values on divisible
groups (as every abelian group A is a subgroup of some divisible group). Since every divisible
group is a direct sum of copies of Q and the Prüfer groups, the values of r are determined by rQ
and r(Z(p∞ )). Since rA is a fully invariant subgroup of Q (by functoriality), it is either Q or 0.
In the former case, since every Prüfer group Z(p∞ ) is a quotient of Q, also r(Z(p∞ )) = Z(p∞ ),
so that r coincides with the trivial radical 1. In case rQ = 0, we are left with determining the
values of r(Z(p∞ )). By the radical condition for r, this subgroup can only be either 0 or Z(p∞ ).
With P = {p : r(Z(p∞ )) = Z(p∞ )}, one obtains r = tP .
24
Example 7.3 For a commutative unital ring S, every ideal a of S gives (in generalization of
the radical m( ) of 7.2) a cohereditary radical ra of the category ModS of S-modules, namely:
raM = aM = {ax : a ∈ a, x ∈ M }. Any cohereditary
Mradical r of ModS arises in this way:
r = ra, with a = rS. Indeed, for a free module M =
S we certainly have rM = aM , and
α
coheredity of r gives the same for all (quotients of free) modules.
ra gives rise to a cohereditary torsion theory precisely when ra is idempotent, that is: when
the ideal a = a2 is idempotent. The existence of such ideals of S depends, of course, on the
structure of S. If a = Sa is principal, we mention two antipodal cases:
(1) If S is an integral domain, there are no non-trivial idempotent ideals.
(2) If S is regular von Neumann, every principal ideal is idempotent, each of which gives a
cohereditary torsion theory. Unless S is a field, ModS will therefore have plenty of nontrivial cohereditary torsion theories. Examples include: any Boolean ring (that is not a
field); any product of more than one field; the ring C(X) of continuous real-valued functions
on a non-trivial P -space X (so that countable intersections of open sets are still open in
X).
There is also a complete description of hereditary torsion theories in ModS , based on the
existence of an injective cogenerator of this category. We must refer to [20] for details.
Example 7.4 There is a bijective correspondence between torsion theories of Top∗ (see 2.6(4))
and pairs of classes (A, B) in Top that form a connectedness/disconnectedness in the sense of
[1] (see 3.9(4)). For each such pair one can take as the torsion class T all pointed spaces in
A, and as the torsion-free class all pointed spaces such that the A-component of the basepoint
is trivial. Top∗ has no non-trivial E-cohereditary radicals. Indeed, since every pointed space
can be regarded as the surjective image of a discrete space (with the same underlying set), any
E-cohereditary radical is determined by its values on the discrete pointed spaces, a subcategory
isomorphic to Set∗ . But Set∗ has only trivial preradicals (see 3.9(1)).
There are precisely 4 non-trivial M-hereditary preradicals of Top∗ , namely (in the notation
of 3.9(2),(4)) qS , qS ∗ , pT and pI = p{I} , where T = {0 ≤ 1 ≤ 2} has the order topology,
with basepoint 1, and where I is a 2-point indiscrete space. Indeed, as observed before Prop.
5.11, preradicals of Top∗ correspond precisely to fully additive closure operators of Top, and Mheredity gets transferred by this correspondence either way. It was shown in [19, Theorem 5.1]
that k ⊕ (= the fully additive core of the usual Kuratowski closure operator k of Top), k ∗ (= the
“inverse” Kuratowski closure operator), k ⊕ ∨ k ∗ , and the least non-discrete closure operator µ
of Top are the only non-trivial fully additive hereditary closure operators of Top. Their induced
preradicals are precisely the ones listed earlier.
qS , qS ∗ and pI are idempotent radicals and give the torsion theories (spaces with dense base
point, spaces with closed base point), (spaces X for which X is the only neighbourhood of the
base point, spaces where the base point x does not belong to the closure of any point distinct
from x), (indiscrete spaces, spaces in which the base point does not belong to any indiscrete
subspace with more than one point), but pT fails to be a radical.
Example 7.5 In 3.9(5) we already mentioned the idempotent radical p of TopGrp, lifted from
Top∗ . It gives the torsion theory (connected groups, hereditarily disconnected groups), but it is,
25
of course, neither M-hereditary nor E-cohereditary. The radicals qS , qS ∗ and pI of Top∗ , when
lifted to TopGrp, give the same hereditary torsion theory: (indiscrete groups, Hausdorff groups).
A large collection of well-studied hereditary preradicals of the category TopAbGrp of topological abelian groups arise from a natural generalization of the concept of m-torsion for discrete
groups (see 7.2). For any sequence m = (mi ) of integers, call an element x of a topological
abelian group A m-torsion if mi x converges to 0 in A. Now the subgroup tm A of all m-torsion
elements in A defines a hereditary preradical of TopAbGrp. Of particular importance is the case
mi = pi for a prime number p, giving the notion of p-torsion as studied already in the 1940s (see
[10, 31]). In this case tm A is referred to as the topological p-Sylow subgroup of A, which plays
an important role in the structure theory of topological (abelian) groups (see [12, 15, 16]). For
a general sequence m, tm fails to be a radical, unless it is eventually constant 0.
Example 7.6 Let CRng denote the pointed category of commutative, but not necessarily unital
rings, considered as a semi-abelian category. Although many classical examples of radicals, like
the Jacobson radical, fail to be functorial and therefore do not fit the setting of this paper,
there are important examples of torsion theories. For example, for a ring S, let tS be the set
of nilpotent elements x of S (so that xn = 0 for some n > 0). Then t defines an idempotent
radical that induces the torsion theory whose torsion-free part contains precisely the rings for
which xn = 0 only if x = 0. t is hereditary but fails to be cohereditary.
Here is a far-reaching generalization of the above example, where we replace the monomials
n
x (n > 0) by any set P of polynomials in m indeterminates over the integers: P ⊆ Z[x1 , . . . , xm ].
The full subcategory F of CRng containing precisely the rings S such that, for every p ∈ P , the
implication
p(a1 , . . . , am ) = 0 ⇒ a1 = · · · = am = 0
holds in S, is closed under products and subobjects, hence, it is normal-epireflective in CRng.
Since it also closed under extensions, the only obstacle that may prevent F from being torsion is
that, with r denoting its induced radical, rrS may fail to be an ideal of S (see 4.6(1)). However,
if we trade our ambient category for a subvariety of CRng, in which being an ideal is a transitive
property, such as the category of Boolean rings, then F is a torsion-free class, even though it
may be hard to characterize its torsion part and describe the radical in question.
Acknowledgements: The third author would like to thank Jiřı́ Rosický and Dominique Bourn for
useful comments received after his presentation of this work at the “StreetFest” at Macquarie
University (Sydney) in July 2005. In particular, the problem formulated in 4.6(1) was triggered
by a corresponding question asked by Rosický.
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