A Baues fibration category of $\mathfrak{P}$ -spaces

Definition 1

A pair $(X\text{,}\ast )$ is called a $\mathfrak{P}$ -space where X is a separable, completely metrizable space and $\ast$ is a closed operation on X. $$\ast :X\times X\to X\text{,}\text{such that}(x_1\text{,}{x}_2)\mapsto x_1\ast x_2\in X\text{,}$$ $\forall (x_1\text{,}{x}_2)\in X\times X$ . A $\mathfrak{P}$ -map $f:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ between two $\mathfrak{P}$ -spaces is a continuous map $f:X\to Y$ that preserves the operations, i.e., $f(x_1{\ast }_X{x}_2)=f(x_1){\ast }_{Y}f(x_2)$ for all $x_1\text{,}{x}_2\in X$ .

Remark 1

1. Clearly, the composition of two $\mathfrak{P}$ -maps is a $\mathfrak{P}$ -map. The identity map ${\mathit{id}}_X:X\to X$ is a $\mathfrak{P}$ -map $(X\text{,}\ast )\to (X\text{,}\ast )$ for any closed operation on X.

2. Since the spaces under study are metrizable, (and hence they are Hausdorff and paracompact), the diagonal set $\mathrm{\Gamma }=\{(x_1\text{,}{x}_2)\in X\times X|x_1=x_2\}$ is closed in $X\times X$ . Therefore the pair $(X\text{,}\mathrm{\Gamma })$ is a $\mathfrak{P}$ -space and the set of $\mathfrak{P}$ -maps $(X\text{,}\mathrm{\Gamma })\to (Y\text{,}\ast )$ is the same as the set of continuous maps $X\to Y$ . Moreover, X is said to have a $G_{\delta }$ -diagonal since $\mathrm{\Gamma }$ is $G_{\delta }$ -set ((Gruenhage, 2014)).

3. The two natural projections given by ${\pi }_1(x_1\text{,}{x}_2)=x_1$ and ${\pi }_2(x_1\text{,}{x}_2)=x_2$ for all $x_1\text{,}{x}_2\in X$ define closed operations that make X into the $\mathfrak{P}$ -space $(X\text{,}{\pi }_1)$ and $(X\text{,}{\pi }_2)$ .

Example 1

The space $C(X\text{,}Y)$ of continuous maps $X\to Y$ with the compact-open topology is separable, completely metrizable Kechris, 2012, Theorem 4.19. Then $\left(C(X\text{,}Y)\text{,}\circ \right)$ is a $\mathfrak{P}$ -space, where $\circ$ is the composition of maps.

Definition 2

Let $(X\text{,}{\ast }_X)$ be a $\mathfrak{P}$ -space and let A be a subset of X. Then $(A\text{,}{\ast }_A)$ is a $\mathfrak{P}$ -subspace of $(X\text{,}{\ast }_X)$ provided A is a $G_{\delta }$ (in relative topology), and $a_1{\ast }_A{a}_2=a_1{\ast }_X{a}_2$ for all $a_1\text{,}{a}_2\in A$ .

Propostion 1

The category $\mathfrak{P}$ is closed under pullbacks.

Proof

Let $g:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ and $f:(Z\text{,}{\ast }_Z)\to (Y\text{,}{\ast }_Y)$ be two $\mathfrak{P}$ -maps. Consider the fiber product space $P=Z{\times }_{Y}X=\{(z\text{,}x)\in Z\times X|f(z)=g(x)\}$ with the operation $\ast$ defined by $(z_1\text{,}{x}_1)\ast (z_2\text{,}{x}_2)=(z_1{\ast }_Z{z}_2\text{,}{x}_1{\ast }_X{x}_2)$ Then $(P\text{,}\ast )$ is a $\mathfrak{P}$ space. Indeed, the product of Polish spaces is Polish (Kechris, 2012), and the induced operation $\ast$ on P is closed as a subset of $P\times P$ . Consider the diagram in $\mathfrak{P}$ .

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where $\tilde{f}$ and $\tilde{g}$ are projections, and hence, $\mathfrak{P}$ -maps. To show that $(P\text{,}\ast )$ has the universal property of the pullback, let $(Q\text{,}{\ast }_Q)$ be a $\mathfrak{P}$ -space and let $\alpha :(Q\text{,}{\ast }_Q)\to (X\text{,}{\ast }_X)$ and $\beta :(Q\text{,}{\ast }_Q)\to (Z\text{,}{\ast }_Z)$ be two $\mathfrak{P}$ -maps such that $g\circ \alpha =f\circ \beta$ . Assume that there exists a unique continuous map $\mu :(Q\text{,}{\ast }_Q)\to (P\text{,}\ast )$ such that $\alpha =\tilde{f}\circ \mu$ and $\beta =\tilde{g}\circ \mu$ . To show that $\mu$ is a $\mathfrak{P}$ -map; let $q\in Q$ . Since the spaces are Polish, and both $\alpha$ and $\beta$ are $\mathfrak{P}$ -maps, then there exist neighborhoods U and V of q such that $\alpha$ is compatible with the operations on U and $\beta$ is compatible with the operations on V. Since $U\cap V$ is a neighborhood of q, and since $\alpha$ and $\beta$ are compatible with the operations on $U\cap V$ , then $\mu$ is compatible with the operations on $U\cap V$ . Therefore $(P\text{,}\ast )$ is the pullback of g and f in $\mathfrak{P}$ .  □

Definition 3

Two $\mathfrak{P}$ -maps $f\text{,}g:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ are said to be $\mathfrak{P}$ -homotopic, (notation: $f{\simeq }_{\mathfrak{p}}g$ ), if there exists a $\mathfrak{P}$ -map $H:(X\text{,}{\ast }_X)\to (Y^I\text{,}{\ast }_{Y^I})$ called a $\mathfrak{P}$ -homotopy joining f and g such that for all $x\in X$ ; $H(x)(0)=f(x)$ and $H(x)(1)=g(x)$ .

Definition 4

A $\mathfrak{P}$ -map $f:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ is a $\mathfrak{P}$ -homotopy equivalence if there exists a $\mathfrak{P}$ -map $g:(Y\text{,}{\ast }_Y)\to (X\text{,}{\ast }_X)$ such that $g\circ f:(X\text{,}{\ast }_X)\to (X\text{,}{\ast }_X)$ and $f\circ g:(Y\text{,}{\ast }_Y)\to (Y\text{,}{\ast }_Y)$ are homotopic to ${\mathit{id}}_X$ and ${\mathit{id}}_Y$ respectively.

Theorem 1

1. The relation ${\simeq }_{\mathfrak{P}}$ is an equivalence relation.

2. Let $f_1\text{,}{f}_2:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ and $g_1\text{,}{g}_2:(Y\text{,}{\ast }_Y)\to (Z\text{,}{\ast }_Z)$ be $\mathfrak{P}$ -maps. If $f_1{\simeq }_{\mathfrak{P}}{f}_2$ and $g_1{\simeq }_{\mathfrak{P}}{g}_2$ , then $g_1{f}_1{\simeq }_{\mathfrak{P}}{g}_2{f}_2$ .

3. Every invertible $\mathfrak{P}$ -map in $\mathfrak{P}$ is a $\mathfrak{P}$ -homotopy equivalence. Moreover, if two of the $\mathfrak{P}$ -maps $f\text{,}h$ , and $f\circ h$ are $\mathfrak{P}$ -homotopy equivalence, then so is the third.

Proof

(1) symmetry and reflexivity are straightforward. To check transitivity: Let $f{\simeq }_{\mathfrak{P}}g$ and $g{\simeq }_{\mathfrak{P}}h$ by $\mathfrak{P}$ -homotopies H and F respectively, and let K be given by $$K(x)(t)=\left\{\begin{array}{c}H(x)(2t)\text{,}0⩽t⩽1/2\hfill \\ F(x)(2t-1)\text{,}1/2⩽t⩽1\text{.}\hfill \end{array}\right.$$ Then K is continuous and also a $\mathfrak{P}$ -homotopy between f and h.

(2) Let $H:(X\text{,}{\ast }_X)\to (Y^I\text{,}{\ast }_{Y^I})$ be a $\mathfrak{P}$ -homotopy joining $f_1$ and $f_2$ . Similarly, let $F:(Y\text{,}B)\to (Z^I\text{,}{\ast }_{Z^I})$ be a $\mathfrak{P}$ -homotopy joining $g_1$ and $g_2$ . Define $$K(x)(t)=\left\{\begin{array}{c}{g}_{1}H(x)(2t)\text{,}0⩽t⩽1/2\hfill \\ F(f_2(x))(2t-1)\text{,}1/2⩽t⩽1\text{.}\hfill \end{array}\right.$$ Then $K(x)(t)$ is a $\mathfrak{P}$ -homotopy $(X\text{,}A)\to (Z^I\text{,}{\ast }_{Z^I})$ joining $g_1{f}_1$ and $g_2{f}_2$ .

(3) Follows from the corresponding fact for isomorphisms and from the case of Top category.  □

Theorem 2

Given two $\mathfrak{P}$ -maps $f\text{,}g:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ . If f and g are $\mathfrak{P}$ -homotopic, then $f\text{,}g:X\to Y$ are homotopic in Top.

Proof

Since f and g are $\mathfrak{P}$ -homotopic through a $\mathfrak{P}$ -homotopy $F:(X\text{,}{\ast }_X)\to (Y^I\text{,}{\ast }_{Y^I})$ , then the function $H:X\times I\to Y$ defined by $H(x\text{,}t)=F(x)(t)$ for every $(x\text{,}t)\in X\times I$ , is a homotopy between the maps $f\text{,}g:X\to Y$ . This is implied by the properties of the compact-open topology on the path space $Y^I$ and the Hausdorffness of the Polish spaces.  □

Example 2

Given a $\mathfrak{P}$ -map $f:(\mathbb{T}\text{,}{\pi }_1)\to (\mathbb{I}\text{,}·)$ where $\mathbb{T}$ is the unit circle with the first projection operation, and $\mathbb{I}$ is the unit interval with the usual multiplication. Both $\mathbb{I}$ and $\mathbb{T}$ are connected Polish spaces. Then, for every $y\in \mathbb{T}$ $f({\pi }_1(y\text{,}y))=f(y)·f(y)=f(y)$ . Because f is by definition continuous and the spaces are connected, this implies that the value of $f(y)$ is either 0 or 1 and hence, f must be constant. Consequently, there are only two $\mathfrak{P}$ -maps between $(\mathbb{T}\text{,}{\pi }_1)$ and $(\mathbb{I}\text{,}·)$ . Let $f_0$ and $f_1$ be the constant $\mathfrak{P}$ -maps that send $\mathbb{T}$ into 0 and 1 respectively. In a similar manner, we can show that there are two $\mathfrak{P}$ -maps between $(\mathbb{T}\text{,}{\pi }_1)$ and $({\mathbb{I}}^I\text{,}{\ast }_{{\mathbb{I}}^I})$ . The space $({\mathbb{I}}^I\text{,}{\ast }_{{\mathbb{I}}^I})$ is Hausdorff with distinct points, whereas $\mathbb{T}$ is connected. Thus, the $\mathfrak{P}$ -maps $f_0$ and $f_1$ are homotopic but not $\mathfrak{P}$ -homotopic.

Theorem 3

The $\mathfrak{P}$ -maps $f\text{,}g:(X\text{,}{\pi }_i)\to (Y\text{,}{\pi }_i)\text{,}(i=1\text{,}2)$ , are $\mathfrak{P}$ -homotopic if and only if the maps $f\text{,}g:X\to Y$ are homotopic.

Proof

We will prove the case when the projection is ${\pi }_1$ . The case of ${\pi }_2$ will follow similarly.

For one assertion, assume that $F:X\times I\to Y$ is a homotopy joining f and g. To show the existence of a $\mathfrak{P}$ -homotopy between $(X\text{,}{\pi }_1)$ and $(Y^I\text{,}{\pi }_1)$ ; recall that by the definition of ${\pi }_1$ we have: ${({\pi }_{1}Y)}^I={\pi }_1(Y^I)=Y^I$ . Hence, define a map $H:(X\text{,}{\pi }_1)\to (Y^I\text{,}{\pi }_1)$ by $H(x)(t)=F(x\text{,}t)$ for every $x\in X$ and every $t\in I$ . On the other hand, the assertion that $\mathfrak{P}$ -homotopy implies homotopy is given by Theorem 2.

In the proofs of the last two theorems we applied the fact that for any topological spaces A and B, the map $f:A\times I\to B$ implies the existence of a map $g:A\to B^I$ defined by $g(a)=f(a\text{,}t)$ for all $a\in A$ and $t\in I$ ([Strom, 2011]). Since I is compact and regular, then the statement is reversible.  □

Remark 2

As a consequence of Theorem 3, it is clear that the functor ${\mathcal{G}}_i$ reflects the homotopy while the functor $\mathcal{F}$ does not in general.

Definition 5

A $\mathfrak{P}$ -map $f:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ is a $\mathfrak{P}$ -fibration with respect to every $\mathfrak{P}$ -space $(Z\text{,}{\ast }_Z)$ if for a $\mathfrak{P}$ -map $g:(Z\text{,}{\ast }_Z)\to (X\text{,}{\ast }_X)$ and a $\mathfrak{P}$ -homotopy $H:(Z\text{,}{\ast }_Z)\to (Y^I\text{,}{\ast }_{Y^I})$ with $H(z)(0)=f\circ g$ , there exists a $\mathfrak{P}$ -map $F:(Z\text{,}{\ast }_Z)\to (X^I\text{,}{\ast }_{X^I})$ such that $F(z)(0)=g$ and $f(F(z)(t))=H(z)(t)$ for every $z\in Z$ and every $t\in I$ .

Theorem 4

The $\mathfrak{P}$ -map $f:(X\text{,}{\pi }_1)\to (Y\text{,}{\pi }_1)$ is a $\mathfrak{P}$ -fibration if and only if $f:X\to Y$ is a Hurewicz fibration.

Proof

Since the definition of fibration and $\mathfrak{P}$ -fibration depend on homotopy and $\mathfrak{P}$ -homotopy respectively, then the result follows easily by applying Theorem 3.  □

Lemma 1

Let $f:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ be a $\mathfrak{P}$ -fibration. For any $\mathfrak{P}$ -subspace $(Z\text{,}{\ast }_Y)$ of $(Y\text{,}\ast )$ . The $\mathfrak{P}$ -map $\bar{f}:(f^{-1}(Z)\text{,}{\ast }_X)\to (Z\text{,}{\ast }_Y)$ is a $\mathfrak{P}$ -fibration.

Proof

Since the spaces are paracompact, the result follows from the general case of fibration (Hu, 1959, Proposition 8.1 p. 73) and the property of Polish subspaces, see e.g., Berberian, 1988, Proposition A.1.13).  □

Definition 6

A $\mathfrak{P}$ -map $\mathfrak{F}:(\lambda \text{,}{\ast }_{\lambda })\to (X^I\text{,}{\ast }_{X^I})$ is called a $\mathfrak{P}$ -lifting function if:

1. $\mathfrak{F}(x\text{,}∊)(0)=x\text{,}\forall (x\text{,}∊)\in \lambda$ ,

2. $f\circ \mathfrak{F}(x\text{,}∊)=∊\text{,}\forall (x\text{,}∊)\in \lambda$ .

Remark 3

The $\mathfrak{P}$ -extension property of $\mathfrak{P}$ -cofibrations is dual to the $\mathfrak{P}$ -lifting property that is used to define $\mathfrak{P}$ -fibrations. However, as the focus of this work is $\mathfrak{P}$ -fibrations and not $\mathfrak{P}$ -cofibrations; $\mathfrak{P}$ -extension property was not found to be of critical importance to the contribution of the paper. This will serve as a topic for further studies in future.

Lemma 2

Every $\mathfrak{P}$ -fibration is regular.

Proof

Let $f:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ be a $\mathfrak{P}$ -fibration. Using the fact that $Y\times Y$ is a normal space and that $\mathrm{\Gamma }$ is a $G_{\delta }$ -set (Remark 1), then $(Y\text{,}{\ast }_Y)$ admits a $\varphi$ -function (Tulley, 1965, Theorem 3.1, p.134). Therefore, f is a regular $\mathfrak{P}$ -fibration.  □

Theorem 5

1. Every isomorphism of $\mathfrak{P}$ -maps is a $\mathfrak{P}$ -fibration.

2. The composition of two $\mathfrak{P}$ -fibrations is a $\mathfrak{P}$ -fibration.

3. If $(g:X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ is a $\mathfrak{P}$ -fibration and $f:(Z\text{,}{\ast }_Z)\to (Y\text{,}{\ast }_Y)$ is any $\mathfrak{P}$ -map, then the $\mathfrak{P}$ -map $\tilde{g}:(P\text{,}\ast )\to (Z\text{,}{\ast }_Z)$ is a $\mathfrak{P}$ -fibration. Moreover, if g (resp., f) is a $\mathfrak{P}$ -homotopy equivalence, so is $\tilde{g}$ (resp., $\tilde{f}$ ).

Proof

(1) is obvious.

(2) Given $\mathfrak{P}$ -fibrations $u:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ and $v:(Y\text{,}{\ast }_Y)\to (Z\text{,}{\ast }_Z)$ . Let $v\circ u:(X\text{,}{\ast }_X)\to (Z\text{,}{\ast }_Z)$ be a $\mathfrak{P}$ -map and let $(S\text{,}{\ast }_S)$ be any $\mathfrak{P}$ -space. Assume that we are given a $\mathfrak{P}$ -map $h:(S\text{,}{\ast }_S)\to (X\text{,}{\ast }_X)$ and that $H:(S\text{,}{\ast }_S)\to (Z^I\text{,}{\ast }_{Z^I})$ is a $\mathfrak{P}$ -homotopy with $H_0=g\circ u\circ h$ . Since v is a $\mathfrak{P}$ -fibration, this implies the existence of a $\mathfrak{P}$ -homotopy $F:(S\text{,}{\ast }_S)\to (Y^I\text{,}{\ast }_{Y^I})$ such that $F_0=\widehat{h}:(S\text{,}{\ast }_S)\to (Y\text{,}{\ast }_Y)$ and $v(F(s)(t))=H(x)(t)\mathit{foralls}\in S\text{,}t\in I$ Since u is a $\mathfrak{P}$ -fibration, then there is a $\mathfrak{P}$ -homotopy $G:(S\text{,}{\ast }_S)\to (X^I\text{,}{\ast }_{X^I})$ such that $G_0=h$ and $u(G(x)(t))=F(x)(t)\text{for all}s\in S\text{,}t\in I$ . Hence, we have the composition $$\begin{array}{cc}v[u(G(x)(t))]\hfill & =\mathit{vu}[G(x)(t)]\hfill \\ \hfill & =H(x)(t)\hfill \end{array}$$ for all $s\in S\text{,}t\in I$ . This implies that $v\circ u:(X\text{,}{\ast }_X)\to (Z\text{,}{\ast }_Z)$ is a $\mathfrak{P}$ -fibration.

(3) The second assertion follows easily from the Top category case, see e.g. [Baues, 1989]. The proof of the first assertion follows by analyzing the diagram

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where $P=Z{\times }_{g}X$ is as defined in Proposition 1.

Given a $\mathfrak{P}$ -homotopy $H:(M\text{,}{\ast }_M)\to (Z^I\text{,}{\ast }_{Z^I})$ and a $\mathfrak{P}$ -lifting $F_0:(M\text{,}{\ast }_M)\to (P^I\text{,}{\ast }_{P^I})$ of $H_0$ , i.e., $j_0\circ F_0=H_0$ . Then we can find a $\mathfrak{P}$ -homotopy $F:(M\text{,}{\ast }_M)\to (P^I\text{,}{\ast }_{P^I})$ such that the first coordinate is given by H, and the second coordinate is given by the lift of $\beta \circ H$ . Since g is a $\mathfrak{P}$ -fibration, this implies that any $\mathfrak{P}$ -homotopy $(M\text{,}{\ast }_M)\to (Y^I\text{,}{\ast }_{Y^I})$ gets lifted to a $\mathfrak{P}$ -homotopy $(M\text{,}{\ast }_M)\to (X^I\text{,}{\ast }_{X^I})$ .

Propostion 2

Let $g:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ be an arbitrary $\mathfrak{P}$ -map. Then g can be factorized into a $\mathfrak{P}$ -homotopy equivalence $f:(X\text{,}{\ast }_X)\to (Z\text{,}{\ast }_Z)$ followed by a $\mathfrak{P}$ -fibration $h:(Z\text{,}{\ast }_Z)\to (Y\text{,}{\ast }_Y)$ .

Proof

Let $(Z\text{,}{\ast }_Z)$ be the $\mathfrak{P}$ -space $(\lambda \text{,}{\ast }_{\lambda })$ where $\lambda =X{\times }_Y{Y}^I$ . To show that f is a $\mathfrak{P}$ -homotopy equivalence; for any point $y\in Y$ , there is a constant path given by $\delta :I\to Y$ such that ${\delta }_y(t)=y$ . Define f by $f(x)=(x\text{,}{\delta }_{g(x)})\mathrm{for}\mathrm{each}x\in X$ Now, let $\pi :(\lambda \text{,}{\ast }_{\lambda })\to (X\text{,}{\ast }_X)$ be the projection. Then $\pi \circ f={\mathit{id}}_X$ . Let $H:(\lambda \text{,}{\ast }_{\lambda })\to ({\lambda }^I\text{,}{\ast }_{{\lambda }^I})$ be a $\mathfrak{P}$ -homotopy given by $$H(x\text{,}∊)(t)=\left\{\begin{array}{cc}\hfill (x\text{,}{\delta }_{∊(0)})\text{,}\hfill & \hfill 0⩽t⩽\frac{1}{2}\hfill \\ \hfill (x\text{,}{∊}_{2t-1})\text{,}\hfill & \hfill \frac{1}{2}⩽t⩽1\hfill \end{array}\right.$$ where ${∊}_t:I\to Y$ is given by ${∊}_t(s)=∊(\mathit{st})\text{,}s\in I$ . This implies that $f\circ \pi {\simeq }_{\mathfrak{P}}{\mathit{id}}_{\lambda }$ . Thus, f is a $\mathfrak{P}$ -homotopy equivalence.

To show that h is a $\mathfrak{P}$ -fibration; consider the following diagram.

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where the $\mathfrak{P}$ -maps $\alpha$ and H are given such that $h\circ \alpha =j_{\circ }H$ . We need to construct a lift F of H. Let $\alpha (m)=({\alpha }_1(m)\text{,}{\alpha }_2(m))\in \lambda$ and let $$F(m)(t)=({\alpha }_1(m)\text{,}\mathfrak{f}(m\text{,}t))$$ where $$\mathfrak{f}(m\text{,}t)(s)=\left\{\begin{array}{cc}\hfill {\alpha }_2(m)(s+\mathit{st})\text{,}\hfill & \hfill 0⩽s⩽\frac{1}{(1+t)}\hfill \\ \hfill H(m\text{,}s+\mathit{ts}-1)\text{,}\hfill & \hfill \frac{1}{(1+t)}⩽s⩽1\text{.}\hfill \end{array}\right.$$

Then $F(m)(t)$ is a $\mathfrak{P}$ -map because its components are all $\mathfrak{P}$ -maps. It follows that h is a $\mathfrak{P}$ -fibration.

Propostion 3

If the $\mathfrak{P}$ -map $f:(X\text{,}{\ast }_X)\to (Y\text{,}{\ast }_Y)$ is a trivial $\mathfrak{P}$ -fibration, then f has a section, i.e., a $\mathfrak{P}$ -map $s:(Y\text{,}{\ast }_Y)\to (X\text{,}{\ast }_X)$ such that $s\circ f{\simeq }_{\mathfrak{P}}{\mathit{id}}_X$ .

Proof

Let $g:(Y\text{,}{\ast }_Y)\to (X\text{,}{\ast }_X)$ be the inverse of f such that $f\circ g{\simeq }_{\mathfrak{P}}{\mathit{id}}_Y$ and $g\circ f{\simeq }_{\mathfrak{P}}{\mathit{id}}_X$ . Let H be the homotopy between $f\circ g$ and ${\mathit{id}}_Y$ . Since f is a $\mathfrak{P}$ -fibration, then there exists a $\mathfrak{P}$ -homotopy $G:(Y\text{,}{\ast }_Y)\to (X^I\text{,}{\ast }_{X^I})$ such that $G(y)(0)=g(y)$ and $f\circ G=H$ . Let $s:(Y\text{,}{\ast }_Y)\to (X\text{,}{\ast }_X)$ such that $s(y)=G(y)(1)$ . Then $s{\simeq }_{\mathfrak{P}}g$ and hence $s\circ f{\simeq }_{\mathfrak{P}}g\circ f{\simeq }_{\mathfrak{P}}{\mathit{id}}_X$ .  □

Definition 7

Definition 7 Baues, 1989

Let fib and $w.e$ . denote classes of morphisms in a category $\mathcal{C}$ , called fibrations and weak equivalences respectively. A fibration category is a category $\mathcal{C}$ with the structure $(\mathcal{C}\text{,}\mathit{fib}\text{,}w.e.)$ that satisfies the following axioms:

• (F1) Composition axiom. Isomorphisms are both fibrations and weak equivalences. For two maps in $\mathcal{C}$ $$X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$$ if any two of $f\text{,}g$ , and gf are weak equivalences, then so is the third. The composition of fibrations is a fibration.

• (F2) Pullback axiom. For a fibration $g:X\to Y$ and any map $f:Z\to Y$ , there exists a pullback diagram in $\mathcal{C}$

  

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  such that

  1. $\tilde{g}$ is a fibration,

  2. if f is a weak equivalence, so is $\tilde{f}$ ,

  3. if g is a weak equivalence, so is $\tilde{g}$ .

• (F3) Factorization axiom. Every map $g:X\to Y$ in $\mathcal{C}$ can be factorized into a weak equivalence f followed by a fibration h such that next diagram commutes.

  

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• (F4) Axioms on cofibrant models. For any object Y in $\mathcal{C}$ there exists a trivial fibration $X\to Y$ where X is cofibrant in $\mathcal{C}$ . An object X in $\mathcal{C}$ is a cofibrant model if every trivial fibration $f:E\to X$ admits a section.

Theorem 6

The category $\mathfrak{P}$ with the structure $$\begin{array}{cc}\mathit{fib}\hfill & =\mathfrak{P}-\mathit{fibrations}\mathit{with}\mathit{the}\mathit{unique}\mathit{path}\mathit{lifting}\mathit{in}\mathfrak{P}\text{,}\mathrm{and}\hfill \\ w.e.\hfill & =\mathfrak{P}-\mathit{homotopy}\mathit{equivalences}\mathit{in}\mathfrak{P}\text{,}\hfill \end{array}$$ is a fibration category in which all objects are $\mathfrak{P}$ -fibrant and $\mathfrak{P}$ -cofibrant.

Proof

Axiom $F1$ is proved in Theorem 1(3), Theorem 5(1), and Theorem 5(2). Combining Proposition 1 and Theorem 5(3) yields the proof of axiom $F2$ . $F3$ is Proposition 2. $F4$ follows from Proposition 3 together with the general case of Top.  □