On products in the coarse shape and strong coarse shape categories

0.22 MB·English

by  Tayyebe Nasri

On products in the coarse shape and strong coarse shape
categories
∗
Tayyebe Nasri, Behrooz Mashayekhy , Hanieh Mirebrahimi
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures,
Ferdowsi University of Mashhad,
3 P.O.Box 1159-91775, Mashhad, Iran.
1
0
2

n
a
J
Abstract
8

The paper is devoted to introduce a new category, called strong coarse shape cat-
]
T egory for topological spaces, SSh∗(Top), that is coarser than strong shape cate-
A
gory SSh(Top). If the Cartesian product of two spaces X and Y admits a strong
. ∗ ∗ ∗
h HPol -expansion (HPol -expansion), which is the Cartesian product of strong HPol -
t ∗
a expansions (HPol -expansions) of these spaces, then X×Y is a product in the strong
m
coarse shape category (coarse shape category). As a consequence, the Cartesian
[ product of two compact Hausdorff spaces is a product in the coarse shape and strong

1 coarse shape categories.
v
1 Keywords: Coarse shape category, Shape category, Inverse limit.
4
2010 MSC: 55P55, 54C56, 54B10, 55Q07.
6
1
.
1
0 1. Introduction
3
1 F.W. Bauer[1]wasthefirst todefineandstudy strongshapeforarbitraryspaces.
:
v Recently, N. Ugleic and N. Bilan [3] have extended the shape theory by constructing
i ∗
X a coarse shape category, denoted by Sh , whose objects are all topological spaces. Its
r isomorphisms classify topological spaces strictly coarser than the shape does. The
a
∗
shape category Sh can be considered as a subcategory of Sh . The shape and coarse
shape coincide on the class of spaces having homotopy type of polyhedra.
In this paper, we take inspiration from these two categories and introduce in
∗
Section 2 the strong coarse shape category SSh (Top), that is coarser than strong
∗
Corresponding author
Email addresses: [email protected](Tayyebe Nasri), [email protected]
(Behrooz Mashayekhy), h−[email protected](Hanieh Mirebrahimi)
Preprint submitted to January 9, 2013
shape category SSh(Top).
Keesling in 1974 [5] proved that if X and Y are compact Hausdorff spaces, then
X × Y is a product in the ordinary shape category. Also, Mardeˇsi´c in 2004 [6]
investigated on the product in the strong shape and showed that if X ×Y admits a
strong expansion, then X×Y is a product in the strong shape category. In particular
the Cartesian product of two compact Hausdorff spaces is a product in strong shape.
In Section 3, we study the existence of product in the coarse shape and strong
coarse shape categories. We prove that if the Cartesian product of two spaces X
∗ ∗
and Y admits a strong HPol -expansion (HPol -expansion), which is the Cartesian
∗ ∗
product of strong HPol -expansions (HPol -expansions) of these spaces, then X ×Y
is a product in the strong coarse shape category (coarse shape category). As a
consequence we show that the Cartesian product of two compact Hausdorff spaces
is a product in the coarse shape and strong coarse shape categories. Moreover, we
show that the shape groups and the coarse shape groups commute with the product.
By the fact that every inverse system is isomorphic to a cofinite inverse system [8,
Remark 1.1.5], in this paper every inverse system is assumed to be cofinite inverse
system.
2. The strong coarse shape category
In this section we define the strong coarse shape category for topological spaces,
∗
denoted by SSh (Top) that is coarser than the strong shape category SSh(Top). The
objects of this category are topological spaces. To define its morphisms similar to
strong shape morphisms, one needs the coarse coherent homotopy category CCH
which define similar to the coherent homotopy category CH. For the sake of com-
pleteness, let us briefly recall the well known notions and main facts concerning the
coherent homotopy category and the strong shape category (see [7]).
Let HTop be the homotopy category of topological spaces, then the objects of the
coherent homotopy category are inverse systems in HTop. To define its morphisms,
for a pre-ordered set M, we consider the sets M of all increasing (m+1)-tuples
m
µ¯ = (µ ,...,µ ), m ≥ 0, where µ ∈ M andµ ≤ ... ≤ µ . Wecall µ¯isa multiindex
0 m i 0 m
of length m. We also define face operators djm : Mm → Mm−1, j = 0,...,m, m ≥ 1
and degeneracy operators sj : M → M , j = 0,...,m, m ≥ 0 by putting
m m m+1
djm(µ0,...,µm) = (µ0,...,µj−1,µj+1,...,µm),
sj (µ ,...,µ ) = (µ ,...,µ ,µ ,µ ,...,µ ).
m 0 m 0 j j j+1 m
2
Also we consider face operators dm : ∆m−1 → ∆m, j = 0,...,m, m ≥ 1 and degen-
j
eracy operators sm : ∆m+1 → ∆m, j = 0,...,m, m ≥ 0 by
j
dmj (t0,...,tm−1) = (t0,...,tj−1,0,tj,...,tm−1),
smj (t0,...,tm+1) = (t0,...,tj−1,tj +tj+1,tj+2,...,tm+1).
A coherent mapping f : X = (Xλ,pλλ′,Λ) → Y = (Yµ,qµµ′,M) is a map which
consists of an increasing function f : M → Λ and of mappings f : X ×∆m →
µ¯ f(µm)
Y , where µ¯ = (µ ,µ ,...,µ ) ∈ M such that
µ0 0 1 m m
qµ0µ1fd0µ¯(x,t) j = 0,
f (x,d t) = f (x,t) 0 < j < m,
µ¯ j  djµ¯
fdmµ¯(pf(µm−1)f(µm)(x),t) j = m

f (x,s t) =f (x,t) 0 ≤ j ≤ m.
µ¯ j sjµ¯
Acoherent homotopyfromXtoY isacoarsecoherent mappingF = (F,F ) : X×
µ¯
I → Y where X×I = (Xλ×I,pλλ′×1,Λ). F connects coherent mapping f = (f,fµ¯)
to f′ = (f′,f′), denoted by f ≃ f′, provided F ≥ f,f′ (i.e. F(µ) ≥ f(µ),f′(µ) for all
µ¯
µ ∈ M) and for all x ∈ X ,t ∈ ∆m,
F(µm)
F (x,0,t) = f (p (x),t)
µ¯ µ¯ f(µm)F(µm)
′
Fµ¯(x,1,t) = fµ¯(pf′(µm)F(µm)(x),t).
A morphism in the coherent homotopy category is the homotopy classes of co-
herent mapping f which is denoted by [f].
Inordertodefinethecompositionh = gf = (h,h ) : X → Zofcoherentmappings
ν¯
f = (f,fµ¯) : X → Y and g = (g,gν¯) : Y → Z = (Zν,rνν′,N), one decomposes ∆m
into m+1 subpolyhedra Pm = ∆i ×∆m−i, 0 ≤ i ≤ m where t = (t ,...,t ) ∈ Pm,
i 0 m i
provided i−1 t ≤ 1 ≤ i t .
j=0 j 2 j=0 j
Also there exist affine homemorphisms cm : Pm → ∆m−i×∆i given by mappings
P P i i
am : Pm → ∆m−i and bm : Pm → ∆i by am(t) = (1−2(t +...+t ),2t ,...,2t ),
i i i i i i+1 m i+1 m
bmi (t) = (2t0,...,2ti−1,1−2(t0+...+ti−1)). We now put h = fg : N → Λ and define
h : X ×∆m → Z for ν¯= (ν ,...,ν ) ∈ N and t ∈ Pm by putting
ν¯ h(νm) ν0 0 m m i
h (x,t) = g (f (x,am(t)),bm(t)).
ν¯ ν0...νi g(νi)...g(νm) i i
The composition of homotopy classes of coherent mappings is defined by compos-
ing their representatives, [g][f] = [gf].
3
Now, we define the coherent functor C : pro−Top → CH as follows:
Put C(X) = X if f = (f,f ) : X → Y is a morphism in pro-Top, put C(f) = [(f,f )]
µ µ¯
where f : X ×∆m → Y is defined by
µ¯ f(µm) µ0
f (x,t) = f p (x).
µ¯ µ0 f(µ0)f(µm)
Also, we define the forgetful functor E : CH → pro−HTop. Put E(X) = X and
if f = (f,f ) : X → Y is a coherent map, then E(f) to be the homotopy class of the
µ¯
(f,f ) : X → Y.
µ0
Now, let us to recall some of the main notions concerning the coarse shape cat-
egory and the pro∗-HTop (see [3]). Let X = (Xλ,pλλ′,Λ) and Y = (Yµ,qµµ′,M) be
two inverse systems in HTop. An S∗-morphism of inverse systems, (f,fn) : X → Y,
µ
consists of an index function f : M → Λ and of a set of mappings fn : X → Y ,
µ f(µ) µ
n ∈ N and µ ∈ M such that for every related pair µ ≤ µ′ in M there exists λ ∈ Λ,
λ ≥ f(µ),f(µ′), and there exists an n ∈ N so that, for every n′ ≥ n,
qµµ′fµn′′pf(µ′)λ ≃ fµn′pf(µ)λ.
The composition of S∗-morphisms (f,fn) : X → Y and (g,gn) : Y → Z =
µ ν
(Zν,rνν′,N) is a S∗-morphism (h,hnν) = (g,gνn)(f,fµn) : X → Z, where h = fg and
hn = gnfn . The identity S∗-morphism on X is a S∗-morphism (1 ,1n ) : X → X,
wνhere 1ν gi(sν)the identity function and 1n = 1 for all n ∈ N and λΛ ∈XΛλ.
A S∗Λ-morphism (f,fn) : X → YXiλs saidXλto be equivalent to a S∗-morphism
µ
(f′,f′n) : X → Y denoted by (f,fn) ∼ (f′,f′n) provided every µ ∈ M admits λ ∈ Λ
µ µ µ
and n ∈ N such that λ ≥ f(µ),f′(µ) and for every n′ ≥ n,
fµn′pf(µ)λ ≃ fµ′n′pf′(µ)λ.
∗
The relation ∼ is an equivalence relation. The category pro -HTop has as objects all
inverse systems X in HTop and as morphisms all equivalence classes f∗ = [(f,fn)].
µ
∗
The composition in pro -HTop is well defined by putting
g∗f∗ = h∗ = [(h,hn)].
ν
In particular if (X) and (Y) are two rudimentary inverse systems of HTop, then
every set of mappings fn : X → Y, n ∈ N, induces a map f∗ : (X) → (Y) in
∗
pro -HTop.
Let p : X → X and p′ : X → X′ be two HPol-expansions of the same space
X in HTop, and let q : Y → Y and q′ : Y → Y′ be two HPol-expansions of the
same space Y in HTop. Then there exist two natural isomorphisms i : X → X′ and
4
j : Y → Y′ in pro-HTop. Consequently i∗ = J(i) : X → X′ and j∗ = J(j) : Y → Y′
∗ ∗
are isomorphisms in pro -HTop ( where J : pro−HTop → pro −HTop is a functor
that J(X) = X and J([(f,f )]) = [(f,fn)], where fn = f for all n ∈ N). A
µ µ µ µ
morphism f∗ : X → Y is said to be equivalent to a morphism f′∗ : X′ → Y′, denoted
by f∗ ∼ f′∗, provided the following diagram in pro∗-HTop commutes:
X −−i−∗→ X′
f∗ f′∗ (1)
Y −−j−∗→ Y′,
y y
∗ ∗
Now, the coarse shape category Sh is defined as follows: The objects of Sh are
topological spaces. A morphism F∗ : X → Y is the equivalence class < f∗ > of a
mapping f∗ : X → Y in pro∗-HTop.
The composition of F∗ =< f∗ >: X → Y and G∗ =< g∗ >: Y → Z is defined
by the representatives, i.e., G∗F∗ =< g∗f∗ >: X → Z. The identity coarse shape
∗ ∗
morphism on a space X, 1 : X → X, is the equivalence class < 1 > of the identity
X X
∗ ∗
morphism 1 in pro -HTop.
X
In the following we introduce the coarse coherent mapping.
Definition 2.1. Let X = (Xλ,pλλ′,Λ) and Y = (Yµ,qµµ′,M) be inverse systems in
HTop. A coarse coherent mapping f∗ = (f,fn) : X → Y consists of an increasing
µ¯
function f : M → Λ, called the index function and of a set of mappings fn : X ×
µ¯ f(µm)
∆m → Y , where n ∈ N and µ¯ = (µ ,µ ,...,µ ) ∈ M , such that there exists n ∈ N
µ0 0 1 m m
′
so that, for every n ≥ n,
q fn′ (x,t) j = 0,
µ0µ1 d0µ¯
fn′(x,d t) = fn′ (x,t) 0 < j < m, (2)
µ¯ j  djµ¯
fn′ (p (x),t) j = m
dmµ¯ f(µm−1)f(µm)
fµ¯n′(x,sjt) =fsnj′µ¯(x,t) 0 ≤ j ≤ m.
If X and Y are inverse systems in HTop over the same index set Λ and f = 1 ,
Λ
then (1 ,fn) is said to be a level coarse coherent mapping.
Λ λ¯
Lemma 2.2. With the notation and assumptions of the first of this section, let
f∗ = (f,fn) : X → Y and g∗ = (g,gn) : Y → Z be coarse coherent mappings
µ¯ ν¯
of inverse systems. Then h∗ = (h,hn) : X → Z, where h = fg : N → Λ and
ν¯
hn : X ×∆m → Z , n ∈ N, ν¯ ∈ N is given by
ν¯ h(νm) ν0 m
hn(x,t) = gn (fn (x,am(t)),bm(t)),
ν¯ ν0...νi g(νi)...g(νm) i i
is a coarse coherent mapping.
5
Proof. Taking into account the construction of hn and coarse coherent properties of
ν¯
fn and gn, this lemma is obvious.
µ¯ ν¯
The above lemma enables us to define the composition of coarse coherent map-
pings of inverse systems. If f∗ = (f,fn) : X → Y and g∗ = (g,gn) : Y → Z are
µ¯ ν¯
two coarse coherent mappings of inverse systems, then g∗f∗ = h∗ = (h,hn) : X → Z,
ν¯
which is define in Lemma 2.2. This composition is associative.
Definition 2.3. A coarse coherent homotopy from X to Y is a coarse coherent
mapping F∗ = (F,Fµ¯n) : X×I → Y where X×I = (Xλ ×I,pλλ′ ×1,Λ). F∗ con-
nects coarse coherent mappings f∗ = (f,fn) and f′∗ = (f′,f′n), denoted by f∗ ≃ f′∗
µ¯ µ¯
provided F ≥ f,f′ and there exists n ∈ N such that for every n′ ≥ n and for all
x ∈ X ,t ∈ ∆m
F(µm)
Fn′(x,0,t) = fn′(p (x),t)
µ¯ µ¯ f(µm)F(µm)
Fµ¯n′(x,1,t) = fµ¯′n′(pf′(µm)F(µm)(x),t).
A morphism in the coarse coherent homotopy category is the homotopy classes
of coarse coherent mapping f∗ which is denoted by [f∗].
Lemma 2.4. Homotopy of coarse coherent mappings is an equivalence relation.
Proof. Reflexivity and symmetry are obvious. To prove transitivity, assume that
F′∗ = (F′,F′n) connects f∗ to f′∗ and F′′∗ = (F′′,F′′n) connects f′∗ to f′′∗. Let
µ¯ µ¯
F : M → Λ be an increasing function such that F ≥ F′,F′′. We define Fn :
µ¯
X ×I ×∆m → Y , µ¯ ∈ M , by
F(µn) µ0 m
Fn(x,s,t) = Fµ¯′n(pF′(µm)F(µm)(x),2s,t) 0 ≤ s ≤ 1/2,
µ¯ (Fµ¯′′n(pF′′(µm)F(µm)(x),2s−1,t) 1/2 ≤ s ≤ 1.
Then F∗ = (F,Fn) is a coarse coherent homotopy, which connects f∗ to f′′∗.
µ¯
The following lemma has a similar argument to the proof of [7, Theorem 2.4].
Lemma 2.5. Let f∗,f′∗ : X → Y and g∗,g′∗ : Y → Z be coarse coherent mappings.
If f∗ ≃ f′∗ and g∗ ≃ g′∗, then g∗f∗ ≃ g′∗f′∗.
By Lemmas 2.4 and 2.5 we can define the composition of homotopy classes of
coarse coherent mappings by composing their representatives, [g∗][f∗] = [g∗f∗].
6
∗ ∗ ∗
Let us define functors C : inv − Top → CCH, C : pro − Top → CCH, C :
pro∗ −Top → CCH, called the coarse coherent functors. Put C∗(X) = X, for every
inverse system X in Top. If f is a map, put C∗(f) = [(f,fn)], where
µ¯
fn(x,t) = fn p (x) = f p (x).
µ¯ µ0 f(µ0)f(µm) µ0 f(µ0)f(µm)
Therefore if f : X → Y is an isomorphism in pro-Top, then C∗(f) is an isomor-
∗ ∗
phism in CCH. Also, we define the forgetful functor E : CCH → pro − HTop as
follows: Put E∗(X) = X, for every inverse system X in HTop. If [f∗] = [(f,fn)] ∈
µ¯
CCH(X,Y), put E∗([f∗]) = [(f,fn )].
µ0
Also, we define the functor J∗ : CH → CCH by J∗(X) = X, for every inverse
system X in HTop. If [f] = [(f,f )] ∈ CH(X,Y), put J∗([f]) = [f∗] = [(f,fn)] ∈
µ¯ µ¯
CCH(X,Y), where for every n ∈ N, fn = f for all µ¯ ∈ M .
µ¯ µ¯ m
Now we are ready to introduce the strong coarse shape category. First, we Recall
that an equivalent definition of a strong HPol-expansion of X. A map p : X → X is
a strong HPol-expansion of X provided, for any inverse system Y in HPol and any
morphism [h] : X → Y of CH, there exists a unique morphism [f] : X → Y of CH
such that [h] = [f]C(p) (see [7]). Now, let p : X → X and p′ : X → X′ be two
cofinite strong HPol-expansions of the same space X in HTop, and let q : Y → Y
and q′ : Y → Y′ be two cofinite strong HPol-expansions of the same space Y in
HTop. We know that the maps [i] : X → X′ and [j] : Y → Y′ are isomorphisms
in CH, consequently [i∗] = J∗([i]) : X → X′ and [j∗] = J∗([j]) : Y → Y′ are
isomorphisms in CCH. A morphism [f∗] : X → Y is said to be equivalent to a
morphism [f′∗] : X′ → Y′, denoted by [f∗] ∼ [f′∗], provided the following diagram in
CCH commutes:
X −−[i−∗→] X′
[f∗] [f′∗] (3)
Y −−[j−∗→] Y′,
y y
∗
Now, we define the strong coarse shape category SSh as follows: The objects
∗ ∗
of SSh are topological spaces. A morphism F : X → Y is the equivalence class
< [f∗] > of a coarse coherent mapping [f∗] : X → Y with respect to any choice of a
pair of strong HPol-expansions p : X → X and q : Y → Y. In other words, a strong
∗
coarse shape morphism F : X → Y is given by a diagram
X ←−−− X
p
[f∗] (4)

Y ←−−− Y.

y q
7
The composition of F∗ =< [f∗] >: X → Y and G∗ =< [g∗] >: Y → Z is defined
by the representatives, i.e., G∗F∗ =< [g∗f∗] >: X → Z. The identity strong coarse
∗ ∗
shape morphism on a space X, 1 : X → X, is the equivalence class < [1 ] > of the
X X
∗
identity morphism [1 ] in CCH.
X
We say that topological spaces X and Y have the same coarse strong shape type,
∗ ∗ ∗
denoted by SSh (X) = SSh (Y), provided there exists an isomorphism F : X → Y
∗ ∗ ∗
in SSh . If there exist strong coarse shape morphisms F : X → Y and G : Y → X
∗ ∗ ∗
such that G F = 1 , then we say that the coarse strong shape of X is dominated
X
∗ ∗
by the coarse strong shape of Y, and we write SSh (X) ≤ SSh (Y).
Definition 2.6. Let X be a topological space. A strong HPol∗-expansion of X is a
morphism p∗ : X → X, where X is an inverse system in HPol with the following
property:
For any inverse system Y in HPol and any morphism [h∗] : X → Y of CCH, there
exists a unique morphism [f∗] : X → Y of CCH such that [h∗] = [f∗]C∗(p∗).
Remark 2.7. For every map f : X → Y in Top and every pair of cofinite strong
HPol∗-expansions p∗ : X → X and q∗ : Y → Y, there exists [f∗] : X → Y in CCH,
such that the following diagram in CCH commutes:
X ←−−−− X
C∗(p∗)
[f∗] f (5)
 
Y ←−−−− Y.
 
y C∗(q∗) y
Thus every morphism f ∈ HTop(X,Y) yields an equivalence class < [f∗] > i.e.
a strong coarse shape morphism F∗ : X → Y. If we define S¯∗(X) = X for every
topological space X and S¯∗(f) = F∗ =< [f∗] >, for every map f : X → Y, then
¯∗ ∗
S : HTop → SSh becomes a functor, called the strong coarse shape functor. Also,
we can define the functor S¯∗ : HTop → SSh∗ by S¯∗ = J¯∗◦S¯where S¯ : HTop → SSh
is defined in [7] and J¯∗ : SSh → SSh∗ is induced by the functor J∗ defined in this
section.
Therefore if X and Y have the same homotopy type, then they have the same
∗
strong coarse shape type. Also, we have SSh(X) = SSh(Y) implies that SSh (X) =
∗
SSh (Y).
Now, we define the functor E¯∗ : SSh∗ → Sh∗ as follows: Put E¯∗(X) = X. If
F∗ =< [f∗] >: X → Y is a morphism in SSh∗, put E¯∗(F∗) =< E∗[f∗] >. This
∗ ∗ ∗ ∗
functor implies that if SSh (X) = SSh (Y), then Sh (X) = Sh (Y) for topological
8
spaces X and Y. Therefore the position of strong coarse shape theory is between
strong shape andcoarse shape. Fromthis fact andresults in [2] we have the following
theorems.
Theorem 2.8. Let X and Y be topological spaces and let SSh∗(X) ≤ SSh∗(Y).
Then the following assertions hold:
(i) If X is connected, then so is Y.
(ii) If the shape dimension sd(X) ≤ n, then also sd(Y) ≤ n.
(iii) If X is movable, then so is Y.
(iv) If X is n-movable, then so is Y.
(v) If X is strongly movable, then so is Y.
∗ ∗ ∗ ∗
Proof. Since SSh (X) ≤ SSh (Y) implies that Sh (X) ≤ Sh (Y), the results hold
by [2, Theorem 3].
Theorem 2.9. Let X and Y be topological spaces and let X be stable. If X and Y
have the same strong coarse shape type, then Y is also stable.
∗ ∗ ∗ ∗
Proof. Since SSh (X) = SSh (Y), so Sh (X) = Sh (Y). But X is stable, thus Y
is stable by [2, Theorem 4].
Theorem 2.10. Let X be a topological space and let P be a polyhedra. Then
∗ ∗
SSh (X) = SSh (P) if and only if SSh(X) = SSh(P).
∗ ∗ ∗ ∗
Proof. Let SSh (X) = SSh (P), then Sh (X) = Sh (P). By [2, Theorem 5],
Sh(X) = Sh(P) and so SSh(X) = SSh(P) by [7, Theorem 9.19].
The following corollary is a consequence of Theorem 2.10.
Corollary 2.11. Let X or Y be a stable space. Then X and Y have the same strong
shape type if and only if they have the same strong coarse shape type.
Thefollowingexampleshowsthat,ingeneral, theinducedfunctionE¯∗|. : SSh∗(X,Y) →
∗ ¯∗ ∗
Sh (X,Y)isnotinjectiveandtheinducedfunctionS |. : HTop(X,Y) → SSh (X,Y)
is not surjective.
Example 2.12. (i) Let X = {∗} and Y be the dyadic solenoid. As in [4, Example
17.7.2], one can shows that
∗
card(Sh (X,Y)) = 1
while
card(SSh∗(X,Y)) = c,
9
where c is the cardinal number of real numbers R. Consequently, the induced function
E¯∗|. : SSh∗(X,Y) → Sh∗(X,Y) can not be injective.
(ii) Let X = {∗} and Y = {∗}⊔{∗}. By a similar argument to [3, Example 7.4], we
can show that
card(HTop(X,Y)) = 2
while
card(SSh∗(X,Y)) = c,
where c is the cardinal number of real numbers R. Consequently, the induced function
¯∗ ∗
S |. : HTop(X,Y) → SSh (X,Y) can not be surjective.
3. Products in strong coarse shape and coarse shape
As it is shown by Keesling in 1974 [5] the ordinary shape category doesn’t have
the product, in general. He also proved that if X and Y are compact Hausdorff
spaces, then X × Y is a product in the shape category. Also, Mardeˇsi´c in 2004
[6] showed that X × Y is a product in the strong shape category when X and Y
are compact Hausdorff spaces. In this section, we intend to study the existence of
∗
productsinthecoarse shapeandstrong coarseshapecategories. Wedefine anHPol -
expansion for a topological space X and show that if the Cartesian product of two
∗ ∗
spaces X and Y admits an HPol -expansion (strong HPol -expansion), which is the
∗ ∗
Cartesian product of HPol -expansions (strong HPol -expansions) of these spaces,
then X×Y is a product in the coarse shape category (strong coarse shape category).
In particular, the Cartesian product of two compact Hausdorff spaces is a product
in these categories. Finally, we show that the k-th shape groups and the k-th coarse
shape groups commute with the product for every k ∈ N.
Definition 3.1. Let X be a topological space. An HPol∗-expansion of X is a mor-
phism p∗ : X → X in pro∗-HTop, where X is an inverse system in HPol with the
following property:
For any inverse system Y in HPol and any morphism h∗ : X → Y in pro∗-HTop,
there exists a unique morphism f∗ : X → Y in pro∗-HTop such that h∗ = f∗p∗.
Remark 3.2. Let p∗ : X → X and q∗ : Y → Y be HPol∗-expansions of X and Y
respectively, for every morphism f : X → Y in HTop, there is a unique morphism
f∗ : X → Y in pro∗-HTop such that the following diagram commutes in pro∗-HTop.
X ←−−− X
p∗
f∗ f (6)
 
Y ←−−− Y.
 
y q∗ y
10