Adjointness of Suspension and Shape Path Functors

TL;DR

This paper explores the adjoint relationship between suspension and shape path functors within a specific subcategory of shape theory, establishing natural bijections and isomorphisms for topological spaces and their shape homotopy groups.

Contribution

It introduces a subcategory of shape spaces and proves an adjointness property between suspension and shape path functors, linking shape homotopy groups.

Findings

Established a natural bijection between shape morphisms involving suspension and shape path functors.

Proved isomorphisms between shape homotopy groups of spaces and their shape path spaces.

Identified a subcategory where these adjointness properties hold.

Abstract

In this paper, we introduce a subcategory $\widetilde{Sh}_*$ of Sh$_*$ and obtain some results in this subcategory. First we show that there is a natural bijection $Sh (\Sigma (X, x), (Y,y))\cong Sh((X,x),Sh((I, \dot{I}),(Y,y)))$, for every $(Y,y)\in \widetilde{Sh}_*$ and $(X,x)\in Sh_*$. By this fact, we prove that for any pointed topological space $(X,x)$ in $\widetilde{Sh}_*$, $\check{\pi}_n^{top}(X,x)\cong \check{\pi}_{n-k}^{top}(Sh((S^k, *),(X,x)), e_x)$, for all $1\leq k \leq n-1$.

Equations

$${\check{\pi }}_1(X\times Y,(x,y))\cong {\check{\pi }}_1(X,x)\times {\check{\pi }}_1(Y,y).$$

$$q_{\mu {\mu }^{\prime }}{f}_{{\mu }^{\prime }}{p}_{f({\mu }^{\prime })\lambda }\simeq f_{\mu }{p}_{f(\mu )\lambda }.$$

$$f_{\mu }{p}_{f(\mu )\lambda }\simeq f_{\mu }^{\prime }{p}_{f^{\prime }(\mu )\lambda }.$$

$$\mathbf{gf}=\mathbf{h}=[(h,h_{\nu })].$$

$$\begin{array}{ccc}\mathbf{X}& \stackrel{\mathbf{i}}{\to }& {\mathbf{X}}^{\prime }\\ \downarrow \mathbf{f}& & {\mathbf{f}}^{\prime }\downarrow \\ \mathbf{Y}& \stackrel{\mathbf{j}}{\to }& {\mathbf{Y}}^{\prime }.\end{array}$$

$$\begin{array}{ccc}\mathbf{X}& \underset{\mathbf{p}}{\leftarrow }& X\\ \downarrow \mathbf{f}& & f\downarrow \\ \mathbf{Y}& \underset{\mathbf{q}}{\leftarrow }& Y.\end{array}$$

$$\begin{array}{ccc}{D}^2& \underset{f}{\leftarrow }& D^2\\ \downarrow \varphi & & \varphi \downarrow \\ \mathbb{R}{P}^2& \underset{\bar{f}}{\leftarrow }& \mathbb{R}{P}^2,\end{array}$$

$$\mathbb{R}{P}^2\stackrel{\bar{f}}{⟵}\mathbb{R}{P}^2\stackrel{\bar{f}}{⟵}\cdots .$$

$$f_{\mu }\circ \Sigma p_{f(\mu )\lambda }\simeq q_{\mu {\mu }^{\prime }}\circ f_{{\mu }^{\prime }}\circ \Sigma p_{f({\mu }^{\prime })\lambda }(rel\{\Sigma x_{\lambda }\}).$$

$$f_{\mu }^{♯}(p_{f(\mu )\lambda }(x))=\mathcal{S}(l_{p_{f(\mu )\lambda }(x)\mu }),$$

$$l_{p_{f(\mu )\lambda }(x)\mu }(t)=f_{\mu }([p_{f(\mu )\lambda }(x),t])=f_{\mu }\circ \Sigma p_{f(\mu )\lambda }([x,t])$$

$$(q_{\mu {\mu }^{\prime }}{)}_{\ast }\circ l_{p_{f({\mu }^{\prime })\lambda }(x){\mu }^{\prime }}(t)=q_{\mu {\mu }^{\prime }}\circ f_{{\mu }^{\prime }}([p_{f({\mu }^{\prime })\lambda }(x),t])=q_{\mu {\mu }^{\prime }}\circ f_{{\mu }^{\prime }}\circ \Sigma p_{f({\mu }^{\prime })\lambda }([x,t]).$$

$$f_{\mu }^{♯}\circ p_{f(\mu )\lambda }(x)=\mathcal{S}(l_{p_{f(\mu )\lambda }(x)\mu })=\mathcal{S}((q_{\mu {\mu }^{\prime }}{)}_{\ast }\circ l_{p_{f({\mu }^{\prime })\lambda }(x){\mu }^{\prime }})=(q_{\mu {\mu }^{\prime }}{)}_{\ast }\circ f_{{\mu }^{\prime }}^{♯}(p_{f({\mu }^{\prime })\lambda }(x)).$$

$$(g_{\mu {\mu }^{\prime }}{)}_{\ast }\circ g_{{\mu }^{\prime }}\circ p_{g({\mu }^{\prime })\lambda }\simeq g_{\mu }\circ p_{g(\mu )\lambda }(rel\{x_{\lambda }\}).$$

$$(g_{\mu {\mu }^{\prime }}{)}_{\ast }\circ g_{{\mu }^{\prime }}\circ p_{g({\mu }^{\prime })\lambda }=g_{\mu }\circ p_{g(\mu )\lambda }.$$

$$g_{\mu }^{♭}\circ \Sigma p_{g(\mu )\lambda }([x,t])=g_{\mu }^{♭}([p_{g(\mu )\lambda }(x),t])=g_{\mu p_{g(\mu )\lambda }(x)}^{\prime }(t)$$

$$q_{\mu {\mu }^{\prime }}\circ g_{{\mu }^{\prime }}^{♭}\circ \Sigma p_{g({\mu }^{\prime })\lambda }([x,t])=q_{\mu {\mu }^{\prime }}\circ g_{{\mu }^{\prime }}^{♭}([p_{g({\mu }^{\prime })\lambda }(x),t])=q_{\mu {\mu }^{\prime }}\circ g_{{\mu }^{\prime }{p}_{g({\mu }^{\prime })\lambda }(x)}^{\prime }(t).$$

$$\mathcal{S}(g_{\mu p_{g(\mu )\lambda }(x)}^{\prime })=g_{\mu }(p_{g(\mu )\lambda }(x))$$

$$\mathcal{S}(q_{\mu {\mu }^{\prime }}\circ g_{{\mu }^{\prime }{p}_{g(\mu )\lambda }(x)}^{\prime })=q_{\mu {\mu }^{\prime }}\circ g_{{\mu }^{\prime }}(p_{g({\mu }^{\prime })\lambda }(x)).$$

$$g_{\mu p_{g(\mu )\lambda }(x)}^{\prime }\simeq q_{\mu {\mu }^{\prime }}\circ g_{{\mu }^{\prime }{p}_{g(\mu )\lambda }(x)}^{\prime }(rel\{\dot{I}\})$$

$$g_{\mu }^{♭}\circ \Sigma p_{g(\mu )\lambda }\simeq q_{\mu {\mu }^{\prime }}\circ g_{{\mu }^{\prime }}^{♭}\circ \Sigma p_{g({\mu }^{\prime })\lambda }(rel\{\Sigma x_{\lambda }\}).$$

$$Sh(\Sigma (X,x),(Y,y))\cong Sh((X,x),(Sh((I,\dot{I}),(Y,y)),e_y)).$$

$${\tau }_{XY}:Sh(\Sigma (X,x),(Y,y))\to Sh((X,x),(Sh((I,\dot{I}),(Y,y)),e_y))$$

$${\theta }_{XY}:Sh((X,x),(Sh((I,\dot{I}),(Y,y)),e_y))\to Sh(\Sigma (X,x),(Y,y))$$

$$Sh((I,\dot{I}),(X\times Y,(x,y)))=Sh((I,\dot{I}),(X,x))\times Sh((I,\dot{I}),(Y,y))$$

$${\check{\pi }}_1(X\times Y,(x,y))={\check{\pi }}_1(X,x)\times {\check{\pi }}_1(Y,y).$$

$$g_{\mu }^{♯}\circ p_{g(\mu )}(x)=\mathcal{S}(l_{p_{g(\mu )}(x)\mu }),$$

$$l_{p_{g(\mu )}(x)\mu }(t)=g_{\mu }([p_{g(\mu )}(x),t])=g_{\mu }\circ \Sigma p_{g(\mu )}([x,t]).$$

$$f_{\mu }^{♭}\circ \Sigma p_{f(\mu )}([x,t])$$

$$=f_{\mu p_{f(\mu )}(x)}^{\prime }(t),$$

$$g_{\mu }^{♯}\circ p_{g(\mu )}(x)=\mathcal{S}(l_{p_{g(\mu )}(x)\mu })=\mathcal{S}(f_{\mu p_{f(\mu )}(x)}^{\prime })=f_{\mu }(p_{f(\mu )}(x)).$$

Full text

1 Introduction and Motivation

Morón et al. [11] gave a complete, non-Archimedean metric (or ultrametric) on the set of shape morphisms between two unpointed compacta (compact metric spaces) $X$ and $Y$, $Sh(X,Y)$. They mentioned that this construction can be translated to the pointed case. Consequently, as a particular case, they obtained a complete ultrametric induces a norm on the shape groups of a compactum Y and then presented some results on these topological groups [12]. Also, Cuchillo-Ibanez et al. [5] constructed several generalized ultrametrics in the set of shape morphisms between topological spaces and obtained semivaluations and valuations on the groups of shape equivalences and $k$th shape groups. On the other hand, Cuchillo-Ibanez et al. [6] introduced a topology on the set $Sh(X,Y)$, where $X$ and $Y$ are arbitrary topological spaces, in such a way that it extended topologically the construction given in [11]. Also, Moszyńska [10] showed that the $k$th shape group $\check{\pi}_{k}(X,x)$, $k\in\mathbb{N}$, is isomorphic to the set $Sh((S^{k},*),(X,x))$ consists of all shape morphisms $(S^{k},*)\rightarrow(X,x)$ with a group operation, for all compact Hausdorff space $(X,x)$. Note that, Bilan [1] mentioned that this fact is true for all topological spaces. The authors [13] applied this topology on the set of shape morphisms between pointed spaces and proved that the $k$th shape group $\check{\pi}_{k}(X,x)$, $k\in\mathbb{N}$, with the above topology is a Hausdorff topological group, denoted by $\check{\pi}_{k}^{top}(X,x)$. In this paper, we introduce a subcategory $\widetilde{Sh}_{*}$ of Sh∗ and obtain some results in this subcategory. It is well-known that the pair $(\Sigma,\Omega)$ is an adjoint pair of functors on hTop∗ and therefore, there is a natural bijection $Hom(\Sigma(X,x),(Y,y))\cong Hom((X,x),\Omega(Y,y))$, for every pointed topological spaces $(X,x)$ and $(Y,y)$. In this paper, we show that there is a natural bijection $Sh(\Sigma(X,x),(Y,y))\cong Sh((X,x),(Sh((I,\dot{I}),(Y,y)),e_{y}))$, for every $(Y,y)\in\widetilde{Sh}_{*}$ and $(X,x)\in Sh_{*}$. By this fact we conclude that the functor $Sh((I,\dot{I}),-)$ preserves inverse limits such as products, pullbacks, kernels, nested intersections and completions, provided inverse limit exists in the subcategory $\widetilde{Sh}_{*}$. Also, the functor $\Sigma$ preserves direct limits of connected spaces in this subcategory. As a consequence, if $(X\times Y,(x,y))$ is a product of pointed spaces $(X,x)$ and $(Y,y)$ in the subcategory $\widetilde{Sh}_{*}$, then

[TABLE]

It is well-known that for any pointed space $(X,x)$ and for all $1\leq k\leq n-1$, $\pi_{n}(X,x)\cong\pi_{n-k}(\Omega(X,x),e_{x})$. In this paper, we show that for any pointed topological space $(X,x)$ in $\widetilde{Sh}_{*}$, $\check{\pi}_{n}(X,x)\cong\check{\pi}_{n-k}(Sh((S^{k},*),(X,x)),e_{x})$, for all $1\leq k\leq n-1$. We then exhibit an example in which this result dose not hold in the category Sh∗.

Endowed with the quotient topology induced by the natural surjective map $q:\Omega^{n}(X,x)\rightarrow\pi_{n}(X,x)$, where $\Omega^{n}(X,x)$ is the $n$th loop space of $(X,x)$ with the compact-open topology, the familiar homotopy group $\pi_{n}(X,x)$ becomes a quasitopological group which is called the quasitopological $n$th homotopy group of the pointed space $(X,x)$, denoted by $\pi_{n}^{qtop}(X,x)$ (see [2, 3, 4, 8]). Nasri et al. [14], showed that for any pointed topological space $(X,x)$, $\pi_{n}^{qtop}(X,x)\cong\pi_{n-k}^{qtop}(\Omega^{k}(X,x),e_{x})$, for all $1\leq k\leq n-1$. In this paper, we prove that for any pointed topological space $(X,x)$ in $\widetilde{Sh}_{*}$, $\check{\pi}_{n}^{top}(X,x)\cong\check{\pi}_{n-k}^{top}(Sh((S^{k},*),(X,x)),e_{x})$, for all $1\leq k\leq n-1$.

2 Preliminaries

In this section, we recall some of the main notions concerning the shape category and the pro-HTop (see [9]). Let $\mathbf{X}=(X_{\lambda},p_{\lambda\lambda^{\prime}},\Lambda)$ and $\mathbf{Y}=(Y_{\mu},q_{\mu\mu^{\prime}},M)$ be two inverse systems in HTop. A pro-morphism of inverse systems, $(f,f_{\mu}):\mathbf{X}\rightarrow\mathbf{Y}$, consists of an index function $f:M\rightarrow\Lambda$ and of mappings $f_{\mu}:X_{f(\mu)}\rightarrow Y_{\mu}$, $\mu\in M$, such that for every related pair $\mu\leq\mu^{\prime}$ in $M$, there exists a $\lambda\in\Lambda$, $\lambda\geq f(\mu),f(\mu^{\prime})$ so that

[TABLE]

The composition of two pro-morphisms $(f,f_{\mu}):\mathbf{X}\rightarrow\mathbf{Y}$ and $(g,g_{\nu}):\mathbf{Y}\rightarrow\mathbf{Z}=(Z_{\nu},r_{\nu\nu^{\prime}},N)$ is also a pro-morphism $(h,h_{\nu})=(g,g_{\nu})(f,f_{\mu}):\mathbf{X}\rightarrow\mathbf{Z}$, where $h=fg$ and $h_{\nu}=g_{\nu}f_{g(\nu)}$. The identity pro-morphism on $\mathbf{X}$ is pro-morphism $(1_{\Lambda},1_{X_{\lambda}}):\mathbf{X}\rightarrow\mathbf{X}$, where $1_{\Lambda}$ is the identity function. A pro-morphism $(f,f_{\mu}):\mathbf{X}\rightarrow\mathbf{Y}$ is said to be equivalent to a pro-morphism $(f^{\prime},f^{\prime}_{\mu}):\mathbf{X}\rightarrow\mathbf{Y}$, denoted by $(f,f_{\mu})\sim(f^{\prime},f^{\prime}_{\mu})$, provided every $\mu\in M$ admits a $\lambda\in\Lambda$ such that $\lambda\geq f(\mu),f^{\prime}(\mu)$ and

[TABLE]

The relation $\sim$ is an equivalence relation. The category pro-HTop has as objects, all inverse systems $\mathbf{X}$ in HTop and as morphisms, all equivalence classes $\mathbf{f}=[(f,f_{\mu})]$. The composition of $\mathbf{f}=[(f,f_{\mu})]$ and $\mathbf{g}=[(g,g_{\nu})]$ in pro-HTop is well defined by putting

[TABLE]

An HPol-expansion of a topological space $X$ is a morphism $\mathbf{p}:X\rightarrow\mathbf{X}$ in pro-HTop, where $\mathbf{X}$ belongs to pro-HPol characterised by the following two properties:

(E1) For every $P\in HPol$ and every map $h:X\rightarrow P$ in HTop, there is a $\lambda\in\Lambda$ and a map $f:X_{\lambda}\rightarrow P$ in HPol such that $fp_{\lambda}\simeq h$.

(E2) If $f_{0},f_{1}:X_{\lambda}\rightarrow P$ satisfy $f_{0}p_{\lambda}\simeq f_{1}p_{\lambda}$, then there exists a $\lambda^{\prime}\geq\lambda$ such that $f_{0}p_{\lambda\lambda^{\prime}}\simeq f_{1}p_{\lambda\lambda^{\prime}}$.

Let $\mathbf{p}:X\rightarrow\mathbf{X}$ and $\mathbf{p^{\prime}}:X\rightarrow\mathbf{X^{\prime}}$ be two HPol-expansions of an space $X$ in HTop, and let $\mathbf{q}:Y\rightarrow\mathbf{Y}$ and $\mathbf{q^{\prime}}:Y\rightarrow\mathbf{Y^{\prime}}$ be two HPol-expansions of an space $Y$ in HTop. Then there exist two natural isomorphisms $\mathbf{i}:\mathbf{X}\rightarrow\mathbf{X}^{\prime}$ and $\mathbf{j}:\mathbf{Y}\rightarrow\mathbf{Y}^{\prime}$ in pro-HTop. A morphism $\mathbf{f}:\mathbf{X}\rightarrow\mathbf{Y}$ is said to be equivalent to a morphism $\mathbf{f^{\prime}}:\mathbf{X^{\prime}}\rightarrow\mathbf{Y^{\prime}}$, denoted by $\mathbf{f}\sim\mathbf{f^{\prime}}$, provided the following diagram in pro-HTop commutes:

[TABLE]

Now, the shape category Sh is defined as follows: The objects of Sh are topological spaces. A morphism $F:X\rightarrow Y$ is the equivalence class $<\mathbf{f}>$ of a mapping $\mathbf{f}:\mathbf{X}\rightarrow\mathbf{Y}$ in pro-HTop. The composition of $F=<\mathbf{f}>:X\rightarrow Y$ and $G=<\mathbf{g}>:Y\rightarrow Z$ is defined by the representatives, i.e., $GF=<\mathbf{g}\mathbf{f}>:X\rightarrow Z$. The identity shape morphism on a space $X$, $1_{X}:X\rightarrow X$, is the equivalence class $<1_{\mathbf{X}}>$ of the identity morphism $1_{\mathbf{X}}$ in pro-HTop.

Let $\mathbf{p}:X\rightarrow\mathbf{X}$ and $\mathbf{q}:Y\rightarrow\mathbf{Y}$ be HPol-expansions of $X$ and $Y$, respectively. Then for every morphism $f:X\rightarrow Y$ in HTop, there is a unique morphism $\mathbf{f}:\mathbf{X}\rightarrow\mathbf{Y}$ in pro-HTop such that the following diagram commutes in pro-HTop.

[TABLE]

If we take other HPol-expansions $\mathbf{p^{\prime}}:X\rightarrow\mathbf{X^{\prime}}$ and $\mathbf{q^{\prime}}:Y\rightarrow\mathbf{Y^{\prime}}$, we obtain another morphism $\mathbf{f^{\prime}}:\mathbf{X^{\prime}}\rightarrow\mathbf{Y^{\prime}}$ in pro-HTop such that $\mathbf{f^{\prime}}\mathbf{p^{\prime*}}=\mathbf{q^{\prime}}f$ and so we have $\mathbf{f}\sim\mathbf{f^{\prime}}$. Hence every morphism $f\in HTop(X,Y)$ yields an equivalence class $<[\mathbf{f}]>$, i.e., a shape morphism $F:X\rightarrow Y$ which is denoted by $\mathcal{S}(f)$. If we put $\mathcal{S}(X)=X$ for every topological space $X$, then we obtain a functor $\mathcal{S}:HTop\rightarrow Sh$, called the shape functor. Also if $Y\in$HPol, then every shape morphism $F:X\rightarrow Y$ admits a unique morphism $f:X\rightarrow Y$ in HTop such that $F=\mathcal{S}(f)$ [9, Theorem 1.2.4].

Similarly, we can define the categories pro-HTop∗ and Sh∗ on pointed topological spaces (see [9]).

3 Main Results

In this section, we introduce a subcategory $\widetilde{Sh}_{*}$ of Sh∗ consists of all pointed topological spaces having bi-expansions. Then we consider the well-known suspension functor $\Sigma:Sh_{*}\rightarrow Sh_{*}$ (see [9]) and $Sh((I,\dot{I}),-):Sh_{*}\rightarrow Sh_{*}$ and show that there is a natural bijection $Sh(\Sigma(X,x),(Y,y))\cong Sh((X,x),(Sh((I,\dot{I}),(Y,y)),e_{y}))$, for every $(Y,y)\in\widetilde{Sh}_{*}$ and $(X,x)\in Sh_{*}$. Then using this bijection we conclude some results in subcategory $\widetilde{Sh}_{*}$.

Definition 3.1**.**

We say that a pointed topological space $(X,x)$ has a bi-expansion $\mathbf{p}:(X,x)\rightarrow(\mathbf{X},\mathbf{x})$ whenever $\mathbf{p}$ is an HPol∗-expansion of $(X,x)$ such that $\mathbf{p_{*}}:Sh((I,\dot{I}),(X,x))\rightarrow\mathbf{Sh}((I,\dot{I}),(X,x))$ is an HPol∗-expansion of $Sh((I,\dot{I}),(X,x))$.

In follow, we recall some conditions on topological space $X$ under which $X$ has a bi-expansion.

Remark 3.2**.**

[13, Remark 4.11]**. If $\mathbf{p}:(X,x)\rightarrow(\mathbf{X},\mathbf{x})$ is an HPol∗-expansion of $X$, then $\mathbf{p_{*}}:Sh((S^{k},*),(X,x))\rightarrow\mathbf{Sh}((S^{k},*),(X,x))$ is an inverse limit of $\mathbf{Sh}((S^{k},*),(X,x))=(Sh((S^{k},*),(X_{\lambda},x_{\lambda})),(p_{\lambda\lambda^{\prime}})_{*},\Lambda)$ (see [6, Theorem 2]). Now, if $Sh((S^{k},*),(X,x))$ is compact and $Sh((S^{k},*),(X_{\lambda},x_{\lambda}))$ is compact polyhedron for all $\lambda\in\Lambda$, then by [7, Remark 1], $\mathbf{p_{*}}$ is an HPol∗-expansion of $Sh((S^{k},*),(X,x))$.

Lemma 3.3**.**

[13, Lemma 4.12]**. Let $(X,x)$ have an HPol∗-expansion $\mathbf{p}:(X,x)\rightarrow((X_{\lambda},x_{\lambda}),p_{\lambda\lambda^{\prime}},\Lambda)$ such that $\pi_{k}(X_{\lambda},x_{\lambda})$ is finite, for every $\lambda\in\Lambda$. Then $\mathbf{p_{*}}:Sh((S^{k},*),(X,x))\rightarrow\mathbf{Sh}((S^{k},*),(X,x))$ is an HPol∗-expansion of $Sh((S^{k},*),(X,x))$, for all $k\in\mathbb{N}$.

Example 3.4**.**

[13, Example 4.13]** (see also [9]). Let $\mathbb{R}P^{2}$ be the real projective plane. Consider the map $\bar{f}:\mathbb{R}P^{2}\rightarrow\mathbb{R}P^{2}$ induced by the following commutative diagram:

[TABLE]

where $D^{2}=\{z\in\mathbb{C}\ \ |\ \ |z|\leq 1\}$ is the unit 2-cell, $f(z)=z^{3}$ and $\phi:D^{2}\rightarrow\mathbb{R}P^{2}$ is the quotient map identifies pairs of points $\{z,-z\}$ of $S^{1}$. We consider $X$ as the inverse sequence

[TABLE]

Since $\mathbb{R}P^{2}$ is compact polyhedron, by [7, Remark 1] $X$ is compact and $\mathbf{p}:X\rightarrow(\mathbb{R}P^{2},\bar{f},\mathbb{N})$ is an HPol-expansion of $X$. Since $\bar{f}$ is onto and $\pi_{k}(\mathbb{R}P^{2})\cong\mathbb{Z}_{2}$ is finite, $\mathbf{p_{*}}:Sh((S^{k},*),(X,x))\rightarrow\mathbf{Sh}((S^{k},*),(X,x))$ is an HPol∗-expansion of$Sh((S^{k},*),(X,x))$, for all $k\in\mathbb{N}$.

The well-known suspension functor $\Sigma:HTop_{*}\rightarrow HTop_{*}$ is extended to a suspension functor $\Sigma:Sh_{*}\rightarrow Sh_{*}$ (see [9]). Note that, if $(X,x)$ is a pointed topological space, then $\Sigma(X,x)=(\Sigma X,\Sigma x)$ is also a pointed topological space. Therefore, whenever $\mathbf{p}:(X,x)\rightarrow(\mathbf{X},\mathbf{x})$ is an HPol∗-expansion of $(X,x)$, then $\mathbf{\Sigma p}:\Sigma(X,x)\rightarrow\Sigma(\mathbf{X},\mathbf{x})=(\Sigma(X_{\lambda},x_{\lambda}),\Sigma p_{\lambda\lambda^{\prime}},\Lambda)$ is an HPol∗-expansion of $\Sigma(X,x)$.

Remark 3.5**.**

Let $(X,x)$ be a connected topological space and $\mathbf{p}:(X,x)\rightarrow(\mathbf{X},\mathbf{x})=((X_{\lambda},x_{\lambda}),p_{\lambda\lambda^{\prime}},\Lambda)$ be an HPol∗-expansion of $(X,x)$. Since $X$ is connected, one can assume that all $X_{\lambda}$ are connected, by [9, Remark 4.1.1] and so $\pi_{1}(\Sigma(X_{\lambda},x_{\lambda}))=0$, for all $\lambda\in\Lambda$ (by Van Kampen Theorem). Therefore, the HPol∗-expansion $\mathbf{\Sigma p}:\Sigma(X,x)\rightarrow\Sigma(\mathbf{X},\mathbf{x})$ satisfies in the conditions of Lemma 3.3 and so $\Sigma(X,x)\in\widetilde{Sh}_{*}$.

Let $F:\Sigma(X,x)\rightarrow(Y,y)$ be a shape morphism represented by $\mathbf{f}:\Sigma(\mathbf{X},\mathbf{x})\rightarrow(\mathbf{Y},\mathbf{y})$ consists of $f:M\rightarrow\Lambda$ and $f_{\mu}:\Sigma(X_{f(\mu)},x_{f(\mu)})\rightarrow(Y_{\mu},y_{\mu})$. If $(Y,y)$ has a bi-expansion $\mathbf{q}:(Y,y)\rightarrow(\mathbf{Y},\mathbf{y})$, then $F$ determines a map $F^{\sharp}:(X,x)\rightarrow(Sh((I,\dot{I}),(Y,y)),e_{y})$ represented by $\mathbf{f^{\sharp}}:(\mathbf{X},\mathbf{x})\rightarrow(\mathbf{Sh}((I,\dot{I}),(Y,y)),\mathbf{e_{y}})$ consists of $f:M\rightarrow\Lambda$ and $f_{\mu}^{\sharp}:(X_{f(\mu)},x_{f(\mu)})\rightarrow(Sh((I,\dot{I}),(Y_{\mu},y_{\mu})),e_{y_{\mu}})$ which is defined as $f_{\mu}^{\sharp}(x)=\mathcal{S}(l_{x\mu})$, where $l_{x\mu}:(I,\dot{I})\rightarrow(Y_{\mu},y_{\mu})$ is a map in HTop∗ such that $l_{x\mu}(t)=f_{\mu}([x,t])$.

In the following lemma we show that $F^{\sharp}$ is a shape morphism.

Lemma 3.6**.**

The map $F^{\sharp}$ defined in the above is a shape morphism.

Proof.

With the above notation, first we show that $f_{\mu}^{\sharp}:X_{f(\mu)}\rightarrow Sh((I,\dot{I}),(Y_{\mu},y_{\mu}))$ is continuous. Since $Y_{\mu}$ is a polyhedron, the space $Sh((I,\dot{I}),(Y_{\mu},y_{\mu}))$ is discrete by [6, Corollary 1]. Therefore, it is sufficient to show that $f_{\mu}^{\sharp}$ is locally constant. Let $x\in X_{f(\mu)}$. Since $X_{f(\mu)}$ is polyhedron, there is an open neighborhood $V_{x}$ of $x$ that is contractible to $x$ in $X_{f(\mu)}$. We will show that $f_{\mu}^{\sharp}$ is constant on $V_{x}$. Let $x^{\prime}\in V_{x}$, then by path connectedness of $V_{x}$, there exists a path $\alpha:I\rightarrow X_{f(\mu)}$ such that $\alpha(0)=x$ and $\alpha(1)=x^{\prime}$. We define the map $H:I\times I\rightarrow Y_{\mu}$ by $H(t,s)=f_{\mu}([\alpha(s),t])$. Since $f_{\mu}$ and $\alpha$ are continuous and $V_{x}$ is contractible to $x$ in $X_{f(\mu)}$, the map $H$ is well-defined and continuous. Moreover, $H$ is a relative homotopy between $f_{\mu}([x,-])$ and $f_{\mu}([x^{\prime},-])$. Hence $l_{x\mu}\simeq l_{x^{\prime}\mu}\ (rel\{\dot{I}\})$ and so $\mathcal{S}(l_{x\mu})=\mathcal{S}(l_{x^{\prime}\mu})$. Therefore $f_{\mu}^{\sharp}(x)=f_{\mu}^{\sharp}(x^{\prime})$ and so $f_{\mu}^{\sharp}$ is constant on $V_{x}$. Finally, we conclude that $f_{\mu}^{\sharp}$ is continuous.

Now, let $\mathbf{p}:(X,x)\rightarrow(\mathbf{X},\mathbf{x})$ be an HPol∗-expansion of $(X,x)$ and $\mathbf{q}:(Y,y)\rightarrow(\mathbf{Y},\mathbf{y})$ be a bi-expansion of $(Y,y)$. The map $\mathbf{f}^{\sharp}$ is a morphism in pro-HTop∗. Indeed, for any pair $\mu^{\prime}\geq\mu$, there is a $\lambda\geq f(\mu),f(\mu^{\prime})$ such that

[TABLE]

Also, for every $x\in X_{\lambda}$,

[TABLE]

and for every $t\in I$,

[TABLE]

[TABLE]

By (4) we have $l_{p_{f(\mu)\lambda}(x)\mu}\simeq(q_{\mu\mu^{\prime}})_{*}\circ l_{p_{f(\mu^{\prime})\lambda}(x)\mu^{\prime}}\ \ (rel\{\dot{I}\})$. Therefore

[TABLE]

∎

On the other hand, let $G:(X,x)\rightarrow(Sh((I,\dot{I}),(Y,y)),e_{y})$ be a shape morphism represented by $\mathbf{g}:(\mathbf{X},\mathbf{x})\rightarrow(\mathbf{Sh}((I,\dot{I}),(Y,y)),\mathbf{e_{y}})$ consists of $g:M\rightarrow\Lambda$ and $g_{\mu}:(X_{g(\mu)},x_{g(\mu)})\rightarrow(Sh((I,\dot{I}),(Y_{\mu},y_{\mu})),e_{y_{\mu}})$. Then we define $G^{\flat}:\Sigma(X,x)\rightarrow(Y,y)$ represented by $\mathbf{g^{\flat}}:\Sigma(\mathbf{X},\mathbf{x})\rightarrow(\mathbf{Y},\mathbf{y})$ in pro-HTop∗ consists of $g:M\rightarrow\Lambda$ and $g_{\mu}^{\flat}:\Sigma(X_{g(\mu)},x_{g(\mu)})\rightarrow(Y_{\mu},y_{\mu})$ given by $g_{\mu}^{\flat}([x,t])=g^{\prime}_{\mu x}(t)$, where $g^{\prime}_{\mu x}$ is a unique morphism in HTop∗ with $\mathcal{S}(g^{\prime}_{\mu x})=g_{\mu}(x)$ (see [9, Theorem 1.2.4]).

Lemma 3.7**.**

The map $G^{\flat}$ defined in the above is a shape morphism.

Proof.

First we show that $g_{\mu}^{\flat}$ is continuous. It is sufficient to show that $\overline{g_{\mu}^{\flat}}:(X_{g(\mu)}\times I,\{x_{g(\mu)}\}\times\dot{I})\rightarrow(Y_{\mu},y_{\mu})$ is continuous. We claim that the map $e_{\mu}:Sh((I,\dot{I}),(Y_{\mu},y_{\mu}))\times I\rightarrow Y_{\mu}$ given by $e_{\mu}(F,t)=F^{\prime}(t)$ is continuous, where $F^{\prime}$ is a unique morphism in HTop∗ with $\mathcal{S}(F^{\prime})=F$ (see [9, Theorem 1.2.4]). To prove the continuity of $e_{\mu}$, let $U$ be an open set containing an arbitrary point $e_{\mu}(F,t)=F^{\prime}(t)$. Since $F^{\prime}$ is continuous, there is an open neighbourhood $V$ of $t$ in $I$ such that $F^{\prime}(V)\subseteq U$. Hence the set $\{F\}\times V$ is an open neighbourhood of $(F,t)$ in $Sh((I,\dot{I}),(Y_{\mu},y_{\mu}))\times I$ such that $e_{\mu}(\{F\}\times V)\subseteq U$. Now, the map $\overline{g_{\mu}^{\flat}}$ is equal to the composition $e_{\mu}\circ(g_{\mu}\times id)$ and so it is continuous.

Let $\mathbf{p}:(X,x)\rightarrow(\mathbf{X},\mathbf{x})$ and $\mathbf{q}:(Y,y)\rightarrow(\mathbf{Y},\mathbf{y})$ be HPol∗-expansions of $(X,x)$ and $(Y,y)$, respectively. The map $\mathbf{g^{\flat}}:\Sigma(\mathbf{X},\mathbf{x})\rightarrow(\mathbf{Y},\mathbf{y})$ is a morphism in pro-HTop∗. To prove this, let $\mu^{\prime}\geq\mu$, then there is a $\lambda\geq g(\mu),g(\mu^{\prime})$ such that

[TABLE]

Since $Y_{\mu}$ is a polyhedron, the space $Sh((I,\dot{I}),(Y_{\mu},y_{\mu}))$ is discrete by [6, Corollary 1]. But homotopic maps in a discrete space are equal, so

[TABLE]

Also, for every $x\in X_{\lambda}$ and $t\in I$,

[TABLE]

and

[TABLE]

Also,

[TABLE]

and

[TABLE]

Hence, using (6) and [6, Theorem 1.2.4],

[TABLE]

and so

[TABLE]

∎

Let $\widetilde{Sh}_{*}$ be a subcategory of Sh∗ consists of all pointed topological spaces having bi-expansions. In follow, we conclude some results in the subcategory $\widetilde{Sh}_{*}$. It is well-known that the pair $(\Sigma,\Omega)$ is an adjoint pair of functors on hTop∗. In the following theorem we prove similar result on subcategory $\widetilde{Sh}_{*}$.

Theorem 3.8**.**

For every $(Y,y)\in\widetilde{Sh}_{*}$ and $(X,x)\in Sh_{*}$, there is a natural bijection

[TABLE]

Proof.

Let $\mathbf{p}:(X,x)\rightarrow(\mathbf{X},\mathbf{x})$ be an HPol∗-expansion of $(X,x)$ and $\mathbf{q}:(Y,y)\rightarrow(\mathbf{Y},\mathbf{y})$ be a bi-expansion of $(Y,y)$. We define

[TABLE]

by $\tau_{XY}(F)=F^{\sharp}$ and

[TABLE]

by $\theta_{XY}(G)=G^{\flat}$. By Lemmas 3.6 and 3.7, the maps $\tau_{XY}$ and $\theta_{XY}$ are well-defined. It is easy to see that $\theta_{XY}\circ\tau_{XY}=id$, $\tau_{XY}\circ\theta_{XY}=id$ and $\tau_{XY}$ is natural in each variable. Hence the result holds. ∎

Using natural bijection (7), one can see that the functor $Sh((I,\dot{I}),-)$ preserves inverse limits such as products, pullbacks, kernels, nested intersections and completions, provided inverse limit exists in the subcategory $\widetilde{Sh}_{*}$. Also, the functor $\Sigma$ preserves direct limits of connected spaces in this subcategory. Hence if $(X\times Y,(x,y))$ is a product of pointed spaces $(X,x)$ and $(Y,y)$ in the subcategory $\widetilde{Sh}_{*}$, then

[TABLE]

and so

[TABLE]

Lemma 3.9**.**

The mappings $\tau_{XY}$ and $\theta_{XY}$ are continuous.

Proof.

First, we show that $\tau_{XY}$ is continuous. Let $V^{F}_{\mu}$ be a basis element of $Sh((X,x),(Sh((I,\dot{I}),(Y,y)),e_{y}))$ containing $F$. We will show that $\tau_{XY}(V^{F^{\flat}}_{\mu})\subseteq V^{F}_{\mu}$. Let $G\in V^{F{{}^{\flat}}}_{\mu}$. By definition, $q_{\mu}\circ F^{\flat}=q_{\mu}\circ G$ as homotopy classes to $Y_{\mu}$, or equivalently $f_{\mu}^{\flat}\circ\Sigma p_{f(\mu)}\simeq g_{\mu}\circ\Sigma p_{g(\mu)}\ \ (rel\{\Sigma x\})$. It is sufficient to show that $(q_{\mu})_{*}\circ F=(q_{\mu})_{*}\circ G^{\sharp}$ as homotopy classes to $Sh(I,Y_{\mu})$ or equivalently $f_{\mu}\circ p_{f(\mu)}\simeq g_{\mu}^{\sharp}\circ p_{g(\mu)}\ \ (rel\{x\})$. For every $x\in X$,

[TABLE]

and for every $t\in I$,

[TABLE]

Also

[TABLE]

where $\mathcal{S}(f^{\prime}_{\mu p_{f(\mu)}(x)})=f_{\mu}(p_{f(\mu)}(x))$. Since $f_{\mu}^{\flat}\circ\Sigma p_{f(\mu)}\simeq g_{\mu}\circ\Sigma p_{g(\mu)}\ \ (rel\{\Sigma x\})$, by the above equalities, $l_{p_{g(\mu)}(x)\mu}\simeq f^{\prime}_{\mu p_{f(\mu)}(x)}\ \ (rel\{\dot{I}\})$. Thus

[TABLE]

So $\tau_{XY}(G)=G^{\sharp}\in V^{F}_{\mu}$, and therefore $\tau_{XY}$ is continuous. Similarly, $\theta_{XY}$ is continuous. ∎

In particular, we can conclude that for any pointed topological space $(X,x)$, $Sh((I,\dot{I}),(Sh((I,\dot{I}),(X,x)),e_{x}))\cong Sh((I^{2},\dot{I^{2}}),(X,x))$. We know that for any pointed space $(X,x)$ and for all $1\leq k\leq n-1$, $\pi_{n}(X,x)\cong\pi_{n-k}(\Omega(X,x),e_{x})$. As a result of Theorem 3.8, we have the following corollary:

Corollary 3.10**.**

Let $(X,x)$ be a pointed topological space in $\widetilde{Sh}_{*}$. Then for all $1\leq k\leq n-1$

[TABLE]

Proof.

By the definition of the shape homotopy group and using Theorem 3.8 and Lemma 3.9, we have

[TABLE]

∎

In follow, we exhibit an example in which the above corollary and therefore Theorem 3.8 do not hold in the category Sh∗.

Remark 3.11**.**

The pair $(\Sigma,Sh((I,\dot{I}),-))$ is not an adjoint pair of functors on the category Sh∗. By contrary, if the pair $(\Sigma,Sh((I,\dot{I}),-))$ is an adjoint pair on Sh∗, with the same argument we obtain $\check{\pi}_{n}(X,x)\cong\check{\pi}_{n-k}(Sh((S^{k},*),(X,x)),e_{x})$, for all $1\leq k\leq n-1$ and for all pointed topological space $(X,x)$. But this isomorphism does not hold in general. Put $X=S^{2}$ and $n=2$, we have $\check{\pi}_{2}(S^{2})=\pi_{2}(S^{2})=\mathbb{Z}$ while $\check{\pi}_{1}(Sh(S^{1},S^{2}))$ is trivial. Note that, $S^{2}$ is a polyhedron and so $Sh(S^{1},S^{2})$ is discrete by [13, Theorem 4.4]. Hence $\check{\pi}_{1}(Sh(S^{1},S^{2}))$ is trivial.

Nasri et al. in [14] showed that for any pointed topological space $(X,x)$, $\pi_{n}^{qtop}(X,x)\cong\pi_{n-k}^{qtop}(\Omega^{k}(X,x),e_{x})$, for all $1\leq k\leq n-1$. In the following corollary we prove this result for $\check{\pi}_{n}^{top}$. The following result is an immediate consequence of Corollary 3.10 and Lemma 3.9.

Corollary 3.12**.**

Let $(X,x)$ be a pointed topological space in $\widetilde{Sh}_{*}$. Then for all $1\leq k\leq n-1$

[TABLE]

Acknowledgements

This research was supported by a grant from Ferdowsi University of Mashhad-Graduate Studies (No. 2/43171).