# On h-Fibrations

Mehdi Tajik1<EMAIL_ADDRESS>Behrooz Mashayekhy1<EMAIL_ADDRESS>Ali
Pakdaman2<EMAIL_ADDRESS>1Department of Pure Mathematics, Center of
Excellence in Analysis on Algebraic Structures,
Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran.
2Department of Mathematics, Faculty of Science, University of Golestan,
P.O.Box 155, Gorgan, Iran.

###### Abstract

In this paper, we study h-fibrations, a weak homotopical version of fibrations
which have weak covering homotopy property. We present some homotopical
analogue of the notions related to fibrations and characterize h-fibrations
using them. Then we construct some new categories by h-fibrations and deduce
some results in these categories such as the existence of products and
coproducts.

###### keywords:

fiber homotopy, weak covering homotopy property, h-fibration, homotopically
path lifting.

###### MSC:

[2010]55R65, 55R05, 55R35, 57M12.

††journal:

##  1 Introduction

###  1.1 Motivation

A map $p:E\to B$ is said to be a fibration (Hurewicz fibration) if it has
covering homotopy property with respect to every space, that is, for every
space $X$, every map $\widetilde{f}:X\to E$ and every homotopy $F:X\times I\to
B$ with $p\circ\widetilde{f}=F\circ J_{0}$, there exists a homtopy
$\widetilde{F}:X\times I\to E$ such that $p\circ\widetilde{F}=F$ and
$\widetilde{F}\circ J_{0}=\widetilde{f}$, where $J_{0}:X\to X\times I$ is
$J_{0}(x)=(x,0)$.

Covering homotopy property is not invariant under fiber homotopy equivalence
and hence any fiber homotopic map to a fibration is not necessarily a
fibration. E. Fadell [2] introduced a new type of fibrations which do not have
this defect. Also, Dold [1] considered a weak version of covering homotopy
property introduced by Fuchs [3] which enjoys useful property of covering
homotopy property such as exact homotopy sequence and spectral sequence and
also, it is invariant under fiber homotopy equivalence.

A fiber homotopy is a kind of homotopy which preserves points in their fibers
during the homotopy ([4, 8]) and weak covering homotopy property is obtained
by replacing $\widetilde{F}\circ J_{0}=\widetilde{f}$ by $\widetilde{F}\circ
J_{0}\simeq_{p}\widetilde{f}$ in the definition of covering homotopy property,
(see [1, 3, 5, 6]). Dold proved that under a weak local contractibility
condition for a space $B$, a map $p:E\to B$ has weak covering homotopy
property if and only if it is locally fiber homotopically trivial [1, Theorem
6.4]. A map $p:E\to B$ is called h-fibration or Dold fibration if it has weak
covering homotopy property with respect to every space. A good
characterization of fibrations and h-fibrations can be found in [1, 6, 8].

A map $p:E\longrightarrow B$ is said to have unique path lifting property
(upl) if for given paths $\alpha$ and $\alpha^{\prime}$ in $E$ such that
$p\circ\alpha=p\circ\alpha^{\prime}$ and $\alpha(0)=\alpha^{\prime}(0)$, we
have $\alpha=\alpha^{\prime}$ (see [8]). Unique path lifting property has an
important role for fibrations, because make them very close to covering
projections and also implies lifting theorem [8, Theorem 2.4.5]. In [7], the
authors presented a homotopical version of unique path lifting property and
studied it’s properties for fibrations.

Here, after rehabilitating the definition of fiber homotopic maps with respect
to arbitrary maps (instead of fibrations), we study h-fibrations with weakly
unique path homotopically lifting property and give a sufficient condition
which makes an h-fibration to be a fibration. We prove that an h-fibration has
the homotopically path lifting property, which is a homotopical version of
path lifting property. Also, we show that an h-fibration has homotopically
lifting function. In Section 3, by proving that the composition of
h-fibrations is an h-fibration, we introduce some new categories by
h-fibrations $\mathrm{hFib}$, $\mathrm{hFibu}$ and $\mathrm{hFibwu}$. Then we
compare them by the categories constructed by fibrations $\mathrm{Fib}$,
$\mathrm{Fibu}$ and $\mathrm{Fibwu}$ (see [7, 8]). Moreover, we show that
these new categories have products and coproducts by introducing them.

###  1.2 Preliminaries

Throughout this paper, all spaces are path connected, unless otherwise stated.
A _map_ $f:X\longrightarrow Y$ means a continuous function. A map
$\alpha:I\longrightarrow X$ is called a path from $x_{0}=\alpha(0)$ to
$x_{1}=\alpha(1)$ and it’s inverse $\alpha^{-1}$ is defined by
$\alpha^{-1}(t)=\alpha(1-t)$. For two paths $\alpha,\beta:I\longrightarrow X$
with $\alpha(1)=\beta(0)$, $\alpha*\beta$ denotes the usual concatenation of
the two paths. Also, all homotopies between paths are assumed to be relative
to end points.

For given maps $p:E\to B$ and $f:X\to B$, a map $\widetilde{f}:X\to E$ is
called a lift of $f$ if $p\circ\widetilde{f}=f$. When $F:X\times
I\longrightarrow Y$ is a map, we say that $F$ is a homotopy from $F_{0}$ to
$F_{1}$ and write $F:F_{0}\simeq F_{1}$, where $F_{i}:X\longrightarrow Y$ is
$F_{i}(x)=F(x,i)$, for $i=0,1$. The constant map from $X$ to $Y$ which sends
all the points to $y\in Y$ is denoted by $C_{y}$.

For a toplological space $Y$, $Y^{I}$ is the space of paths in $Y$ and for a
given map $f:X\to Y$, $P_{f}$ is the mapping path space, that is,
$P_{f}={\\{(x,\alpha)\in X\times Y^{I}|f(x)=\alpha(0)}\\}$. Also, $p:P_{f}\to
X$ by $p(x,\alpha)=x$ is a fibration which is called the mapping path
fibration (see [8]).

##  2 h-Fibrations

For the definition of fiber homotopic maps with respect to a fibration and the
definition of fiber homotopy equivalent fibrations, see [8]. We give here
similar definitions for an arbitrary map and a few basic results that we need
in sequel.

######  Definition 2.1.

Let $p:E\to B$ be a map. Two maps $f_{0},f_{1}:X\to E$ are said to be fiber
homotopic with respect to $p$, denoted by $f_{0}\simeq_{p}f_{1}$ if there is a
homotopy $F:f_{0}\simeq f_{1}$ such that $p\circ F(x,t)=p\circ f_{0}(x)=p\circ
f_{1}(x)$ for every $x\in X$ and any $t\in I$.

We recall that for given maps $p_{1}:E_{1}\to B$ and $p_{2}:E_{2}\to B$, a map
$f:E_{1}\to E_{2}$ is called fiber-preserving if $p_{1}=p_{2}\circ f$.

######  Definition 2.2.

Two maps $p_{1}:E_{1}\to B$ and $p_{2}:E_{2}\to B$ are said to be fiber
homotopy equivalent, if there exist fiber preserving maps $f:E_{1}\to E_{2}$
and $g:E_{2}\to E_{1}$ such that $g\circ f\simeq_{p_{1}}1_{E_{1}}$ and $f\circ
g\simeq_{p_{2}}1_{E_{2}}$. Each of the maps $f$ and $g$ is called a fiber
homotopy equivalence.

We have the following proposition for the fiber homotopy property.

######  Proposition 2.3.

Let $p:E\to B$ be a map.
(i) If $p^{\prime}:E^{\prime}\to E$ and $f_{0},f_{1}:X\to E^{\prime}$ are maps
such that $f_{0}\simeq_{p^{\prime}}f_{1}$, then $f_{0}\simeq_{p\circ
p^{\prime}}f_{1}$.
(ii) The fiber homotopy with respect to $p$ is an equivalence relation on the
set of maps from $X$ to $E$.
(iii) If $f,g:X\to E$ and $h:Z\to X$ are maps such that $f\simeq_{p}g$, then
$f\circ h\simeq_{p}g\circ h$.
(iv) If $f,g:X\to E^{\prime}$ and $p^{\prime}:E^{\prime}\to E$ are maps such
that $f\simeq_{p\circ p^{\prime}}g$, then $p^{\prime}\circ
f\simeq_{p}p^{\prime}\circ g$.

###### Proof.

(i) Let $H:f_{0}\simeq_{p^{\prime}}f_{1}$. Since $p^{\prime}\circ
H(x,t)=p^{\prime}\circ f_{0}(x)=p^{\prime}\circ f_{1}(x),$ we have

$(p\circ p^{\prime})\circ H(x,t)=(p\circ p^{\prime})\circ f_{0}(x)=(p\circ
p^{\prime})\circ f_{1}(x),$

which implies that $H:f_{0}\simeq_{p\circ p^{\prime}}f_{1}$.
(ii) It is similar to the proof of ordinary homotopy relation.
(iii) Let $H:X\times I\to E$ be a fiber homotopy from $f$ to $g$ with respect
to $p$. Define $H^{\prime}:Z\times I\to E$ by $H^{\prime}(z,t)=H(h(z),t)$.
Then $H^{\prime}$ is a homotopy from $f\circ h$ to $g\circ h$ and

$p\circ H^{\prime}(z,t)=p\circ H(h(z),t)=p\circ f(h(z))=p\circ g(h(z)).$

(iv) Let $H:X\times I\to E^{\prime}$ be the fiber homotopy $H:f\simeq_{p\circ
p^{\prime}}g$ and define $H^{\prime}:X\times I\to E$ by
$H^{\prime}(x,t)=p^{\prime}\circ H(x,t)$. Then $p\circ H^{\prime}(x,t)=p\circ
p^{\prime}\circ H(x,t)=p\circ p^{\prime}\circ f(x)=p\circ p^{\prime}\circ
g(x)$ and so $H^{\prime}:p^{\prime}\circ f\simeq_{p}p^{\prime}\circ g$. ∎

A map $p:E\to B$ has weak covering homotopy property, abbreviated by wchp, if
for every space X and any maps $\widetilde{f}:X\to E$, $F:X\times I\to B$ with
$p\circ\widetilde{f}=F\circ J_{0}$, there exists a homotopy
$\widetilde{F}:X\times I\to E$ such taht $p\circ\widetilde{F}=F$ and
$\widetilde{F}\circ J_{0}\simeq_{p}\widetilde{f}$. In Fact, Dold’s definition
was a bit different [1]. A map $p:E\to B$ is called an h-fibration if it has
wchp [6]. By [1, Proposition 5.2], if a map is fiber homotopy equivalent to a
fibration, it has wchp and so is an h-fibration. Also, in [5] it is mentioned
that every h-fibration is fiber homotopy equivalent to a fibration (see [6,
Proposition 1.15]). Since one can not find a detailed proof for this fact, we
are going to give a proof for it. First, for $f:X\rightarrow Y$, let $P_{f}$
be the mapping path space, $p:P_{f}\to X$ be the mapping path fibration and
$h:X\to P_{f}$ be the map $h(x)=(x,C_{f(x)})$. Then $h$ is a section of $p$,
moreover, $h$ and $p$ are homotopy inverse ([8, Theorem 2.8.9]).

######  Theorem 2.4.

A map is an h-fibration if and only if it is fiber homotopy equivalent to a
fibration.

###### Proof.

Let $f:X\to Y$ be fiber homotopy equivalent to a fibration
$f^{\prime}:X^{\prime}\to Y$. There exist two maps $s:X\to X^{\prime}$ and
$s^{\prime}:X^{\prime}\to X$ such that $s\circ
s^{\prime}\simeq_{f^{\prime}}1_{X^{\prime}}$, $s^{\prime}\circ
s\simeq_{f}1_{X}$, $f\circ s^{\prime}=f^{\prime}$ and $f^{\prime}\circ s=f$ .
If $\widetilde{f}:Z\to X$ and $F:Z\times I\to Y$ are maps such that
$f\circ\widetilde{f}=F\circ J_{0}$, then $f^{\prime}\circ
s\circ\widetilde{f}=F\circ J_{0}$. Therefore by assumption, there is a
homotopy $\widetilde{F}:Z\times I\to X^{\prime}$ such that
$f^{\prime}\circ\widetilde{F}=F$ and $\widetilde{F}\circ
J_{0}=s\circ\widetilde{f}$. Let $\widetilde{G}=s^{\prime}\circ\widetilde{F}$.
Then

$f\circ\widetilde{G}=f\circ
s^{\prime}\circ\widetilde{F}=f^{\prime}\circ\widetilde{F}=F,$

and by Proposition 2.3

$\widetilde{G}\circ J_{0}=s^{\prime}\circ\widetilde{F}\circ
J_{0}=s^{\prime}\circ
s\circ\widetilde{f}\simeq_{f}1_{X}\circ\widetilde{f}=\widetilde{f}.$

Hence $f$ is an h-fibration.
Conversely, let $f:X\to Y$ be an h-fibration. Define a map $p_{1}:P_{f}\to Y$
by $p_{1}(x,\alpha)=\alpha(1)$ which is a fibration (see [8, Theorem 2.8.9]).
We show that $f$ and $p_{1}$ are fiber homotopy equivalent. Let
$\gamma:P_{f}\times I\to Y$ be defined by $\gamma(x,\alpha,t)=\alpha(t)$ .
Note that

$f\circ p(x,\alpha)=f(x)=\alpha(0)=\gamma(x,\alpha,0)=\gamma\circ
J_{0}(x,\alpha),$

where $J_{0}:P_{f}\to P_{f}\times I$ is the map
$J_{0}(x,\alpha)=(x,\alpha,0)$. Since $f$ is an h-fibration, there exist
homotopies $\widetilde{\gamma}:P_{f}\times I\to X$ and $T:P_{f}\times I\to X$
such that $f\circ\widetilde{\gamma}=\gamma$ and $T:\widetilde{\gamma}\circ
J_{0}\simeq_{f}p$. Let $g:P_{f}\to X$ be defined by
$g(x,\alpha)=\widetilde{\gamma}(x,\alpha,1)$. It is sufficient to show that
$g\circ h\simeq_{f}1_{X}$ and $h\circ g\simeq_{p_{1}}1_{P_{f}}$. We have that
$p_{1}\circ h=f$ and also $f\circ g=p_{1}$, because $f\circ
g(x,\alpha)=f\circ\widetilde{\gamma}(x,\alpha,1)=\gamma(x,\alpha,1)=\alpha(1)=p_{1}(x,\alpha)$.
Let $F:X\times I\to X$ be the map $F(x,t)=\widetilde{\gamma}(x,C_{f(x)},t)$.
Since $F:F_{0}\simeq F_{1}$ and

$f\circ
F(x,t)=f\circ\widetilde{\gamma}(x,C_{f(x)},t)=\gamma(x,C_{f(x)},t)=C_{f(x)}(t)=f(x),$
$f\circ F_{0}(x)=f\circ
F(x,0)=f\circ\widetilde{\gamma}(x,C_{f(x)},0)=\gamma(x,C_{f(x)},0)=C_{f(x)}(0)=f(x),$
$f\circ F_{1}(x)=f\circ
F(x,1)=f\circ\widetilde{\gamma}(x,C_{f(x)},1)=\gamma(x,C_{f(x)},1)=C_{f(x)}(1)=f(x),$
$F_{1}(x)=F(x,1)=\widetilde{\gamma}(x,C_{f(x)},1)=g(x,C_{f(x)})=g\circ h(x),$

we have $F:F_{0}\simeq_{f}g\circ h$.
Define $T^{\prime}:X\times I\to X$ by $T^{\prime}(x,t)=T(x,C_{f(x)},t)$. Then
$T^{\prime}:F_{0}\simeq_{f}1_{X}$ since

$T^{\prime}(x,0)=T(x,C_{f(x)},0)=\widetilde{\gamma}\circ
J_{0}(x,C_{f(x)})=F(x,0)=F_{0}(x),$
$T^{\prime}(x,1)=T(x,C_{f(x)},1)=p(x,C_{f(x)})=x=1_{X}(x),$ $f\circ
T^{\prime}(x,t)=f\circ T(x,C_{f(x)},t)=f\circ\widetilde{\gamma}\circ
J_{0}(x,C_{f(x)})=f\circ F_{0}(x),$ $f\circ T^{\prime}(x,t)=f\circ
T(x,C_{f(x)},t)=f\circ p(x,C_{f(x)})=f(x)=f\circ 1_{X}(x).$

Hence, transitivity of fiber homotopy implies that $g\circ h\simeq_{f}1_{X}$.
For the second fiber homotopy, define $H:P_{f}\times I\to P_{f}$ by
$H(x,\alpha,s)=(\widetilde{\gamma}(x,\alpha,s),\alpha_{s})$, for in which
$\alpha_{s}$ is the path $\alpha_{s}(t)=\alpha(s+t-st)$, for every $s,t\in I$.
Clearly, $H:H_{0}\simeq H_{1}$. Moreover,

$p_{1}\circ
H(x,\alpha,s)=p_{1}(\widetilde{\gamma}(x,\alpha,s),\alpha_{s})=\alpha_{s}(1)=\alpha(1),$
$p_{1}\circ H_{0}(x,\alpha)=p_{1}\circ
H(x,\alpha,0)=p_{1}(\widetilde{\gamma}(x,\alpha,0),\alpha_{0})=\alpha_{0}(1)=\alpha(1),$
$p_{1}\circ H_{1}(x,\alpha)=p_{1}\circ
H(x,\alpha,1)=p_{1}(\widetilde{\gamma}(x,\alpha,1),\alpha_{1})=p_{1}(\widetilde{\gamma}(x,\alpha,1),C_{\alpha(1)})=C_{\alpha(1)}(1)=\alpha(1),$

and hence $H:H_{0}\simeq_{p_{1}}H_{1}.$ On the other hand,

$h\circ
g(x,\alpha)=h(\widetilde{\gamma}(x,\alpha,1))=(\widetilde{\gamma}(x,\alpha,1),C_{f\circ\widetilde{\gamma}(x,\alpha,1)})=$
$(\widetilde{\gamma}(x,\alpha,1),C_{\gamma(x,\alpha,1)})=(\widetilde{\gamma}(x,\alpha,1),C_{\alpha(1)})=H_{1}(x,\alpha).$

Thus $h\circ g=H_{1}$ and so $H:H_{0}\simeq_{p_{1}}h\circ g$. Now, define
$T^{\prime\prime}:P_{f}\times I\to P_{f}$ by
$T^{\prime\prime}(x,\alpha,s)=(T(x,\alpha,s),\alpha)$. Note that
$T^{\prime\prime}$ is well-define because

$f\circ T(x,\alpha,s)=f\circ p(x,\alpha)=f(x)=\alpha(0).$

Moreover, $T^{\prime\prime}:H_{0}\simeq_{p_{1}}1_{P_{f}}$ since

$T^{\prime\prime}(x,\alpha,0)=(T(x,\alpha,0),\alpha)=(\widetilde{\gamma}(x,\alpha,0),\alpha)=H_{0}(x,\alpha),$
$T^{\prime\prime}(x,\alpha,1)=(T(x,\alpha,1),\alpha)=(p(x,\alpha),\alpha)=(x,\alpha)=1_{P_{f}}(x,\alpha),$
$p_{1}\circ
T^{\prime\prime}(x,\alpha,s)=p_{1}(T(x,\alpha,s),\alpha)=\alpha(1),$
$p_{1}\circ H_{0}(x,\alpha)=p_{1}\circ
H(x,\alpha,0)=p_{1}(\widetilde{\gamma}(x,\alpha,0),\alpha)=\alpha(1),$
$p_{1}\circ 1_{P_{f}}(x,\alpha)=p_{1}(x,\alpha)=\alpha(1).$

Therefore, using $H$ and $T^{\prime\prime}$, we have $h\circ
g\simeq_{p_{1}}1_{P_{f}}$. ∎

For fibrations with path connected base space, any two fibers have the same
homotopy type. Since, every h-fibration is fiber homotopy equivalent to a
fibration, and a fiber homotopy equivalence can be thought as a family of
homotopy equivalences between corresponding fibers ([4, Page 406]), hence we
have another proof for the following proposition.

######  Proposition 2.5.

([6, Proposition 1.12]). The fibers of an h-fibration have the same homotopy
type.

######  Corollary 2.6.

If an h-fibration has the path connected base space with a path connected
fiber, then its total space is also path connected.

###### Proof.

Let $p:E\to B$ be an h-fibration with a path connected fiber. Then by
Proposition 2.5, every fiber of $p$ is path connected. Let
$p^{\prime}:E^{\prime}\to B$ be a fibration which is fiber homotopy equivalent
to $p$. Then the fibers of $p^{\prime}$ are path connected and so by [8,
Exercise 2.8.E.2], $E^{\prime}$ is path connected. By definition, there exist
fiber preserving maps $f:E\to E^{\prime}$ and $g:E^{\prime}\to E$. If $x,y\in
E$, then there exists a path $\alpha$ in $E^{\prime}$ from $f(x)$ to $f(y)$.
Since $g\circ f\simeq_{p}1_{E}$, we have $x,g(f(x))\in p^{-1}(x)$, also
$y,g(f(y))\in p^{-1}(y)$. Let $\beta$ be a path in $p^{-1}(x)$ from $x$ to
$g(f(x))$ and $\gamma$ be a path in $p^{-1}(y)$ from $g(f(y))$ to $y$.
Therefore $\beta\ast(g\circ\alpha)\ast\gamma$ is a path in $E$ from $x$ to
$y$. ∎

By definitions, every fibration is an h-fibration. But an h-fibration is not
necessarily a fibration (for an example see [1]). In order to find a
sufficient condition which makes an h-fibration a fibration, first consider
the following lemma.

######  Lemma 2.7.

If a map $p:E\to B$ has upl and $f_{0},f_{1}:X\to E$ are fiber homotopic with
respect to $p$, then $f_{0}=f_{1}$.

###### Proof.

Let $H:f_{0}\simeq_{p}f_{1}$. Then for every $x\in X$ and every $t\in I$,
$p\circ H(x,t)=p\circ f_{0}(x)=p\circ f_{1}(x)$. For a fix $x\in X$, $H(x,-)$
is a path in the fiber $p^{-1}(p\circ f_{0}(x))$ and so $p\circ
H(x,-)=C_{p\circ f_{0}(x)}$. Since $p\circ C_{f_{0}(x)}=C_{p\circ f_{0}(x)}$,
$H(x,0)=f_{0}(x)=C_{f_{0}(x)}(0)$ and $p$ has upl, we have
$H(x,-)=C_{f_{0}(x)}$. Hence $f_{0}(x)=H(x,0)=H(x,1)=f_{1}(x)$, as desired. ∎

######  Theorem 2.8.

Every h-fibration with upl is a fibration.

###### Proof.

Let $p:E\to B$ be an h-fibration. Also let $X$ be a topological space,
$\widetilde{f}:X\to E$ and $F:X\times I\to B$ be maps such that
$p\circ\widetilde{f}=F\circ J_{0}$. Then, there exists a homotopy
$\widetilde{F}:X\times I\to E$ such that $p\circ\widetilde{F}=F$ and
$\widetilde{F}\circ J_{0}\simeq_{p}\widetilde{f}$. By Lemma 2.7,
$\widetilde{F}\circ J_{0}=\widetilde{f}$ which implies that $p$ is a
fibration. ∎

Unique path lifting property is important in the study of fibrations because
make them a covering map, when the base space is locally nice, i.e, locally
path connected and semi-locally simply connected [8]. The authors introduced a
homotopical version of upl in [7] and studied its role in fibrations. Since
here we are working with a weak homotopical version of fibrations, we are
going to study h-fibrations with the homotopical version of upl.

A map $p:E\to B$ is said to have _weakly unique path homotopically lifting
property_ abbreviated by wuphl, if by given two paths $\alpha$ and $\beta$ in
$E$ with $\alpha(0)=\beta(0),\ \alpha(1)=\beta(1)$ and $p\circ\alpha\simeq
p\circ\beta,\ \mathrm{rel}\ \dot{I}$, then it follows that
$\alpha\simeq\beta,\ \mathrm{rel}\ \dot{I}$ (see [7]).
The unique path lifting property for fibrations is equivalent to the fact that
every path in any fiber is constant [8, Theorem 2.2.5]. Also, the weakly
unique path homotopically lifting property for fibrations is equivalent to the
fact that every loop in any fiber is nullhomotopic [7, Theorem 3.4]. In the
following, we show that these facts hold for h-fibrations.

######  Proposition 2.9.

An h-fibration $p:E\to B$ has upl if and only if every path in any fiber is
constant.

###### Proof.

If $p$ has upl, then it is easy to see that every path in any fiber is
constant. For the converse, let $\alpha,\beta:I\to E$ be two lifts of a path
in $B$ started from the same point $\alpha(0)=\beta(0)$. Let $t\in I$ and
consider the path $\gamma_{t}$ in $E$ from $\alpha(t)$ to $\beta(t)$ by

$\displaystyle\gamma_{t}(t^{\prime})=\begin{cases}\alpha((1-2t^{\prime})t),&t^{\prime}\in[0,\frac{1}{2}]\\\
\beta((2t^{\prime}-1)t),&t^{\prime}\in[\frac{1}{2},1].\end{cases}$

By assumption $p\circ\alpha=p\circ\beta$, then there exists a homotopy
$F:p\circ\gamma_{t}\simeq C_{p\circ\alpha(t)}$, rel $\dot{I}$. Since $p$ has
wchp, there exist homotopies $\widetilde{F},H:I\times I\to E$ with
$p\circ\widetilde{F}=F$ and $H:\widetilde{F}\circ J_{0}\simeq_{p}\gamma_{t}$.
Thus

$p\circ\widetilde{F}(0,t)=F(0,t)=C_{p\circ\alpha(t)}(0)=p\circ\alpha(t),$
$p\circ\widetilde{F}(t,1)=F(t,1)=C_{p\circ\alpha(t)}(t)=p\circ\alpha(t),$

and

$p\circ\widetilde{F}(1,t)=F(1,t)=C_{p\circ\alpha(t)}(1)=p\circ\alpha(t).$

Therefore,
$\widetilde{F}(0,-)\ast\widetilde{F}(-,1)\ast(\widetilde{F}(1,-))^{-1}$ is a
path in the fiber $p^{-1}(p\circ\alpha(t))$. So by assumption it is constant,
which implies that $\widetilde{F}(0,0)=\widetilde{F}(1,0)$. Now, note that
$H(0,-)$ is a path from $\widetilde{F}(0,0)$ to $\gamma_{t}(0)=\alpha(t)$ in
the fiber $p^{-1}(p\circ\gamma_{t}(0))$, and also $H(1,-)$ is a path from
$\widetilde{F}(1,0)$ to $\gamma_{t}(1)=\beta(t)$ in the fiber
$p^{-1}(p\circ\gamma_{t}(1))$. Then, since
$\widetilde{F}(0,0)=\widetilde{F}(1,0)$ and

$p^{-1}(p\circ\gamma_{t}(0))=p^{-1}(p\circ\alpha(t))=p^{-1}(p\circ\beta(t))=p^{-1}(p\circ\gamma_{t}(1)),$

there exists a path in this fiber from $\alpha(t)$ to $\beta(t)$, which by
assumption it must be constant. Then, $\alpha(t)=\beta(t)$ and since $t$ is
arbitrary we will have $\alpha=\beta$. ∎

######  Proposition 2.10.

An h-fibration $p:E\to B$ has wuphl if and only if every loop in any fiber is
nullhomotopic.

###### Proof.

Necessity is trivial. For the sufficiency, let
$\widetilde{\alpha},\widetilde{\beta}:I\longrightarrow E$ be two paths with
$\widetilde{\alpha}(0)=\widetilde{\beta}(0)$,
$\widetilde{\alpha}(1)=\widetilde{\beta}(1)$ and
$p\circ\widetilde{\alpha}\simeq p\circ\widetilde{\beta}$, rel $\dot{I}$. Let
$\gamma:=\widetilde{\alpha}*\widetilde{\beta}^{-1}$ which is a loop at
$\widetilde{\alpha}(0)$. Put $\tilde{x}_{0}=\widetilde{\alpha}(0)$ and
$x_{0}=p(\tilde{x}_{0})$, then we have

$p\circ\gamma=p\circ(\widetilde{\alpha}*\widetilde{\beta}^{-1})=(p\circ\widetilde{\alpha})*(p\circ\widetilde{\beta}^{-1})=(p\circ\widetilde{\alpha})*(p\circ\widetilde{\beta})^{-1}\simeq
C_{x_{0}},\ \mathrm{rel}\ \dot{I}.$

Let $F:p\circ\gamma\simeq C_{x_{0}}$, rel $\dot{I}$. Since $p$ is an
h-fibration, there exist homotopies $\widetilde{F},H:I\times I\longrightarrow
E$ such that $p\circ\widetilde{F}=F$ and $H:\gamma\simeq_{p}\widetilde{F}\circ
J_{0}$. Let $f:=\widetilde{F}(0,-),\ f^{\prime}:=H(0,-),\
g:=\widetilde{F}(-,1),\ h:=\widetilde{F}(1,-)$ and $h^{\prime}:=H(1,-)$ which
are paths in $E$ with $f^{\prime}(1)=f(0),\ f(1)=g(0)$, $g(1)=h(1)=h^{-1}(0)$
and $h^{-1}(1)=h^{\prime-1}(0)$, so we can define $\eta:=f*g*h^{-1}$ and
$\delta:=f^{\prime}\ast\eta\ast h^{\prime-1}$. Note that $\delta$ is a closed
path because

$\delta(0)=f^{\prime}(0)=H(0,0)=\gamma(0)=\widetilde{\alpha}(0)=\widetilde{\beta}(0)=$
$\widetilde{\beta}^{-1}(1)=\gamma(1)=H(1,0)=h^{\prime}(0)=h^{\prime-1}(1)=\delta(1).$

Also, since

$\displaystyle p\circ f^{\prime}(t)=p\circ H(0,t)=p\circ\widetilde{F}\circ
J_{0}(0)=p\circ\widetilde{F}(0,0)=F(0,0)=x_{0}=C_{x_{0}}(t),\quad$
$\displaystyle p\circ\eta=(p\circ f)*(p\circ g)*(p\circ
h)^{-1}=F(0,-)*F(-,1)*(F(1,-))^{-1}=\quad$ $\displaystyle
C_{x_{0}}*C_{x_{0}}*(C_{x_{0}})^{-1}=C_{x_{0}},\quad$ $\displaystyle p\circ
h^{\prime}(t)=p\circ H(1,t)=p\circ\widetilde{F}\circ
J_{0}(1)=p\circ\widetilde{F}(1,0)=F(1,0)=x_{0}=C_{x_{0}}(t),\quad$

we have $p\circ\delta=(p\circ f^{\prime})\ast(p\circ\eta)\ast(p\circ
h^{\prime-1})=C_{x_{0}}$. Hence $\delta$ belongs to the fiber $p^{-1}(x_{0})$
and so by assumption, $\delta$ is null. On the other hand, by definitions of
$\gamma,\delta$ and $H,\widetilde{F}$ $\gamma\ast\delta\simeq
C_{\tilde{x}_{0}}$, rel $\dot{I}$, which implies $\gamma\simeq
C_{\tilde{x}_{0}}$, rel $\dot{I}$ and so
$\widetilde{\alpha}\simeq\widetilde{\beta}$, rel $\dot{I}$. ∎

Obviously, if every loop in fibers of an h-fibration $p$ is constant, then $p$
has wuphl, but the converse is not necessarily true. For example, the
h-fibration $pr_{1}:X\times Y\to X$, when $Y$ is any non-singleton simply
connected space, has wuphl and also has nonconstant paths in its fibers. Since
the fibers of two fiber homotopy equivalence fibrations (h-fibrations) have
the same homotopy type, by Proposition 2.10 we have the following result.

######  Corollary 2.11.

(i) If two h-fibrations are fiber homotoy equivalent and one of them has
wuphl, then so has the other one.
(ii) If an h-fibration is fiber homotoy equivalent to a fibration and one of
them has wuphl, then so has the other one.

A map $p:E\to B$ has path lifting property if for a given path $\alpha:I\to B$
with $\alpha(0)\in p(E)$ and every $e\in p^{-1}(\alpha(0))$ there exists a
path $\widetilde{\alpha}$ in $E$ started at $e$, such that
$p\circ\widetilde{\alpha}=\alpha$. We know that every fibration has the path
lifting property and the following example shows that an h-fibration does not
necessarily have the path lifting property.

######  Example 2.12.

Let $E=([-1,0]\times[-1,0])\cup([0,1]\times[0,1])$, $B=[-1,1]$ and $p$ be the
projection on the first component. Then $p$ is an h-fibration because for
given maps $F:X\times I\to B$ and $\widetilde{f}:X\to E$ with
$p\circ\widetilde{f}=F\circ J_{0}$ it suffices to define
$\widetilde{F}:X\times I\to E$ by $\widetilde{F}(x,t)=(F(x,t),0)$. But, there
is no lift for the path $\alpha(t)=t$, started from $(0,-\dfrac{1}{2})$.

In the following we give a homotopical analogue of path lifting property and
show that h-fibrations enjoy this property.

######  Definition 2.13.

A map $p:E\to B$ has homotopically path lifting property if for a given $b\in
B$, $e\in p^{-1}(b)$ and a path $\alpha$ in $B$ beginning at $b$, there exists
a path $\widetilde{\alpha}$ in $E$ such that $\widetilde{\alpha}(0)=e$ and
$p\circ\widetilde{\alpha}\simeq\alpha$, rel $\dot{I}$.

######  Theorem 2.14.

An h-fibration has homotopically path lifting property.

###### Proof.

Let $p:E\to B$ be an h-fibration, $\alpha$ be a path in $B$ and $e\in
p^{-1}(\alpha(0))$. Also, let $F:{\\{\ast}\\}\times I\to B$ be the homotopy
$F(\ast,t)=\alpha(t)$ and $\widetilde{f}:{\\{\ast}\\}\to E$ be the map
$\widetilde{f}(\ast)=e$. Then $p\circ\widetilde{f}=F\circ J_{0}$ and since $p$
is an h-fibration, there is a homotopy $\widetilde{F}:{\\{\ast}\\}\times I\to
E$ and a fiber homotopy $H:{\\{\ast}\\}\times I\to E$ such that
$p\circ\widetilde{F}=F$ and $H:\widetilde{F}\circ
J_{0}\simeq_{p}\widetilde{f}$. Let $\widetilde{\alpha}$ be the path in $E$
defined by $\widetilde{\alpha}(t)=\widetilde{F}(\ast,t)$. Then
$H(\ast,0)=\widetilde{F}\circ J_{0}(\ast)=\widetilde{\alpha}(0)$,
$H(\ast,1)=\widetilde{f}(\ast)=e$ and

$\ p\circ H(\ast,t)=p\circ\widetilde{F}\circ
J_{0}(\ast)=p\circ\widetilde{f}(\ast)=\alpha(0).$

Let $\widetilde{\gamma}:=H(\ast,-)$ which is a path in the fiber
$p^{-1}(\alpha(0))$ from $\widetilde{\alpha}(0)$ to $e$. Then
$\widetilde{\beta}=\widetilde{\gamma}^{-1}\ast\widetilde{\alpha}$ is a
homotopical lift of $\alpha$ started from $e$, because

$p\circ\widetilde{\beta}=p\circ(\widetilde{\gamma}^{-1}\ast\widetilde{\alpha})=(p\circ\widetilde{\gamma}^{-1})\ast(p\circ\widetilde{\alpha})\simeq
C_{\alpha(0)}\ast\alpha\simeq\alpha,\ rel\ \dot{I}.$

∎

We know that restriction of a fibration on each of whose path components is a
fibration and for maps with locally path connected total space, we have the
converse (see, [8, Lemma 2.3.1 and Theorem 2.3.2]). These results are
satisfied for h-fibrations with a simulated proof which is left to the
readers.

######  Proposition 2.15.

Let $p:E\to B$ be a map. If $E$ is locally path connected, then $p$ is an
h-fibration if and only if for each path component $A$ of $E$, $p(A)$ is a
path component of $B$ and $p|_{A}:A\to p(A)$ is an h-fibration.

Let $p:E\to B$ be a map and define a subspace $\overline{B}\subseteq E\times
B^{I}$ as follows:

$\overline{B}={\\{(e,\omega)\in E\times B^{I}|\omega(0)=p(e)}\\}.$

Recall that, a lifting function for $p$ is a map $\lambda:\overline{B}\to
E^{I}$ which assigns to each point $e\in E$ and any path $\omega$ in $B$
starting at $p(e)$ a path $\lambda(e,\omega)$ in $E$ starting at $e$ that is a
lift of $\omega$. Existence of a lifting function for a map $p:E\to B$ is
equivalent to $p$ is a fibration (see [8, Theorem 2.7.8]). For h-fibrations we
introduce a homotopical version of lifting function and show that every
h-fibration has one of them.

######  Definition 2.16.

A _homotopically lifting function for $p$_ is a map $\lambda:\overline{B}\to
E^{I}$ which assigns to each point $e\in E$ and any path $\omega$ in $B$
starting at $p(e)$ a path $\lambda(e,\omega)$ in $E$ starting at $e$ that is a
homotopical lift of $\omega$.

######  Theorem 2.17.

Every h-fibration has a homotopically lifting function.

###### Proof.

Let $p:E\to B$ be an h-fibration. Define two maps
$\widetilde{f}:\overline{B}\to E$ and $F:\overline{B}\times I\to B$ by
$\widetilde{f}(e,\omega)=e$ and $F((e,\omega),t)=\omega(t)$, respectively.
Since $F\circ
J_{0}(e,\omega)=F((e,\omega),0)=\omega(0)=p(e)=p\circ\widetilde{f}(e,\omega)$
and $p$ is an h-fibration, there exist homotopies
$\widetilde{F},H:\overline{B}\times I\to E$ such that $p\circ\widetilde{F}=F$
and $H:\widetilde{F}\circ J_{0}\simeq_{p}\widetilde{f}$. Define
$\lambda:\overline{B}\to E^{I}$ by
$\lambda(e,\omega)(t)=\widetilde{F}((e,\omega),t)$ which is continuous. Let
$\tilde{x}:=\lambda(e,\omega)(0)$. Then $\tilde{x}\in p^{-1}(p(e))$ because
$p(\tilde{x})=p\circ\lambda(e,\omega)(0)=p\circ\widetilde{F}\circ
J_{0}(e,\omega)=p\circ\widetilde{f}(e,\omega)=p(e)$. Similar to the proof of
Proposition 2.14, there is a path $\gamma$ in the fiber $p^{-1}(p(e))$ from
$e$ to $\tilde{x}$. Define $\lambda^{\prime}:\overline{B}\to E^{I}$ by
$\lambda^{\prime}(e,\omega)=\gamma\ast\lambda(e,\omega)$. Then
$\lambda^{\prime}(e,\omega)(0)=\gamma(0)=e$ and

$p\circ\lambda^{\prime}(e,\omega)=p\circ(\gamma\ast\lambda(e,\omega))=(p\circ\gamma)\ast(p\circ\lambda(e,\omega))=C_{p(e)}\ast
p\circ\lambda(e,\omega)$ $=C_{p(e)}\ast
p\circ\widetilde{F}((e,\omega),-)=C_{p(e)}\ast
F((e,\omega),-)=C_{p(e)}\ast\omega\simeq\omega,\ \ \\\ rel\ \dot{I}.$

Therefore $\lambda^{\prime}$ is a homotopically lifting function for $p$. ∎

######  Remark 2.18.

The converse of Theorem 2.17 is not true. As an example, let $E=I\times
I-{\\{(0,\frac{1}{2})}\\}$, $B=I$ and $p$ be the projection on the first
component. Since the fibers of $p$ do not have the same homotopy, $p$ is not
an h-fibration. However, $p$ has a homotopically lifting function. For, let
$e\in E$ and $\omega$ be a path in $B$ starting at $p(e)$. Also define two
paths $\alpha,\beta$ in $E$ by $\alpha(t)=(1-t)e+tA$ and
$\beta(t)=(1-t)A+t(\omega(1),0)$, where $A=(1,\frac{1}{2})$. Define
$\lambda:\overline{B}\to E^{I}$ such that
$\lambda(e,\omega)(t)=(\alpha\ast\beta)(t)$. Then $\lambda(e,\omega)$ is a
path starting at $\lambda(e,\omega)(0)=\alpha(0)=e$. Moreover, since

$p\circ\lambda(e,\omega)(0)=p\circ\alpha(0)=p(e)=\omega(0),$
$p\circ\lambda(e,\omega)(1)=p\circ\beta(1)=p(\omega(1),0)=\omega(1)$

and $B$ is simply connected, $p\circ\lambda(e,\omega)(t)\simeq\omega(t)\ rel\
\dot{I}$, as desired.

##  3 Category of h-Fibrations

In this section, $\mathrm{Fib}$ and $\mathrm{Fib(B)}$ are the category of
fibrations and fibrations over $B$, and have the categories $\mathrm{Fibu}$
and $\mathrm{Fibu(B)}$ (with the extra assumption unique path lifting) as
subcategory, respectively (see [8]). When we deal with fibrations with wuphl
instead of upl, we have the categories $\mathrm{Fibwu}$ and
$\mathrm{Fibwu(B)}$ [7], for which

$\mathrm{Fibu}\leq\mathrm{Fibwu},\ \ \ \mathrm{Fibu(B)}\leq\mathrm{Fibwu(B)}.$

To construct new categories by h-fibrations, we need to the following
essential proposition.

######  Proposition 3.1.

Composition of two h-fibrations is an h-fibration.

###### Proof.

Let $p^{\prime}:E^{\prime}\to E$ and $p:E\to B$ be two h-fibrations,
$\widetilde{f}:X\to E^{\prime}$ and $F:X\times I\to B$ be two maps such that
$(p\circ p^{\prime})\circ\widetilde{f}=F\circ J_{0}$. Since $p$ is an
h-fibration, there exist homotopies $\widetilde{F},H:X\times I\to E$ such that
$p\circ\widetilde{F}=F$ and
$H:p^{\prime}\circ\widetilde{f}\simeq_{p}\widetilde{F}\circ J_{0}$. Since
$p^{\prime}\circ\widetilde{f}=H\circ J_{0}$ and $p^{\prime}$ is an
h-fibration, there exist homotopies $\widetilde{H},D:X\times I\to E^{\prime}$
such that $p^{\prime}\circ\widetilde{H}=H$ and
$D:\widetilde{f}\simeq_{p^{\prime}}\widetilde{H}\circ J_{0}$. Let $J_{1}:X\to
X\times I$ be the map $J_{1}(x)=(x,1)$. Since
$p^{\prime}\circ(\widetilde{H}\circ
J_{1}(x))=p^{\prime}\circ\widetilde{H}(x,1)=H(x,1)=\widetilde{F}\circ
J_{0}(x)$ and $p^{\prime}$ is an h-fibration, there exist homotopies
$\widetilde{K},T:X\times I\to E^{\prime}$ such that
$p^{\prime}\circ\widetilde{K}=\widetilde{F}$ and $T:\widetilde{H}\circ
J_{1}\simeq_{p^{\prime}}\widetilde{K}\circ J_{0}$. Since, $(p\circ
p^{\prime})\circ\widetilde{K}=p\circ\widetilde{F}=F$, $\widetilde{K}$ is the
desired homotopy. Also, by Proposition 2.3, $D:\widetilde{f}\simeq_{p\circ
p^{\prime}}\widetilde{H}_{0}$ and $T:\widetilde{H}_{1}\simeq_{p\circ
p^{\prime}}\widetilde{K}_{0}$. Moreover,
$(p\circ p^{\prime})\circ\widetilde{H}(x,t)=p\circ H(x,t)=p\circ
H(x,0)=(p\circ p^{\prime})\circ\widetilde{H}(x,0)=(p\circ
p^{\prime})\circ\widetilde{H}_{0}(x)$,
$(p\circ p^{\prime})\circ\widetilde{H}(x,t)=p\circ H(x,t)=p\circ
H(x,1)=(p\circ p^{\prime})\circ\widetilde{H}(x,1)=(p\circ
p^{\prime})\circ\widetilde{H}_{1}(x)$, which imply that
$\widetilde{H}:\widetilde{H}_{0}\simeq_{p\circ p^{\prime}}\widetilde{H}_{1}$.
Since fiber homotopy is an equivalence relation, $\widetilde{f}\simeq_{p\circ
p^{\prime}}\widetilde{K}\circ J_{0}$ and so the result holds. ∎

It is straightforward that composition of two maps with wuphl is a map with
wuphl [7, Proposition 4.1] and so we have the following proposition.

######  Proposition 3.2.

Composition of h-fibrations with wuphl is an h-fibration with wuphl.

Now, we can define category of h-fibrations, $\mathrm{hFib}$ and its
subcategory, category of h-fibrations with wuphl, $\mathrm{hFibwu}$ in which
the objects are topological spaces and morphisms are h-fibrations and
h-fibrations with wuphl, respectively. Moreover, for a given space $B$, we can
consider other categories, $\mathrm{hFib(B)}$ and $\mathrm{hFibwu(B)}$, whose
objects are h-fibrations and h-fibrations with wuphl over $B$ and morphisms
are the commutative triangles.

By Theorem 2.8, since every fibration is an h-fibration, we have the following
diagram of inclusion relations between categories.

FibuFibwuFibhFibuhFibwuhFib .

It is notable that we have a similar diagram for the categories constructed
over the base space $B$. Also, note that in the above diagram, the inclusions
are proper. The first row is proper [7, Example 3.3] and the second row is
proper, since a fibration is an h-fibration. Moreover, Example 2.12 shows that
the second and the third column are proper.

Now, we study the existence of products and coproducts for these categories.

######  Proposition 3.3.

Product of two h-fibrations is an h-fibration.

###### Proof.

Let $p:E\to B$ and $p^{\prime}:E^{\prime}\to B^{\prime}$ be two h-fibrations,
$\widetilde{f}:X\to E\times E^{\prime}$ and $F:X\times I\to B\times
B^{\prime}$ be maps such that $(p\times p^{\prime})\circ\widetilde{f}=F\circ
J_{0}$. Since $p\circ pr_{1}\circ\widetilde{f}=(pr_{1}\circ F)\circ J_{0}$,
$p^{\prime}\circ pr_{2}\circ\widetilde{f}=(pr_{2}\circ F)\circ J_{0}$ and $p$
and $p^{\prime}$ are h-fibration, there exist $\widetilde{F}_{1}:X\times I\to
E$ and $\widetilde{F}_{2}:X\times I\to E^{\prime}$ such that
$p\circ\widetilde{F}_{1}=pr_{1}\circ F$,
$p^{\prime}\circ\widetilde{F}_{2}=pr_{2}\circ F$, $\widetilde{F}_{1}\circ
J_{0}\simeq_{p}pr_{1}\circ\widetilde{f}$ and $\widetilde{F}_{2}\circ
J_{0}\simeq_{p^{\prime}}pr_{2}\circ\widetilde{f}$. Define
$\widetilde{F}:X\times I\to E\times E^{\prime}$ by
$\widetilde{F}(x,t)=(\widetilde{F}_{1}(x,t),\widetilde{F}_{2}(x,t))$. Then

$(p\times
p^{\prime})\circ\widetilde{F}=(p\circ\widetilde{F}_{1},p^{\prime}\circ\widetilde{F}_{2})=(pr_{1}\circ
F,pr_{2}\circ F)=F,$

and

$\widetilde{F}\circ J_{0}=(\widetilde{F}_{1}\circ J_{0},\widetilde{F}_{2}\circ
J_{0})\simeq_{p\times
p^{\prime}}(pr_{1}\circ\widetilde{f},pr_{2}\circ\widetilde{f})=\widetilde{f}.$

∎

It is easy to see that product of two maps with wuphl is a map with wuphl.
Hence we have the following result.

######  Proposition 3.4.

The categories $\mathrm{hFib}$ and $\mathrm{hFibwu}$ have the product.

To present products for $\mathrm{hFib(B)}$ and $\mathrm{hFibwu(B)}$, consider
the Whitney sum of h-fibrations (with wuphl). If ${\\{p_{j}:E_{j}\to B|j\in
J}\\}$ is an indexed collection of h-fibrations (with wuphl ) over the space
$B$, define

$\oplus_{B,J}E_{j}={\\{(e_{j})_{j}\in\sqcap_{j}{E_{j}}|e_{j}\in E_{j},\
\mathrm{and}\ p_{j}(e_{j})=p_{i}(e_{i}),\ \mathrm{for}\ i,j\in J}\\}$

and also define

$\oplus_{B,J}p_{j}:\oplus_{B,J}E_{j}\to B$ $(e_{j})_{j}\rightarrowtail
p_{j}(e_{j}).$

######  Proposition 3.5.

Let ${\\{p_{j}:E_{j}\to B|j\in J}\\}$ be an indexed collection of h-fibrations
(with wuphl) on the space $B$. Then $\oplus_{B,J}p_{j}$ is an h-fibration
(with wuphl).

###### Proof.

Let $E:=\oplus_{B,J}E_{j}$ and $p:=\oplus_{B,J}p_{j}$. Also, let
$\widetilde{f}:X\to E$ and $F:X\times I\to B$ be two maps such that
$p\circ\widetilde{f}=F\circ J_{0}$. Then
$\widetilde{f}=(\widetilde{f}_{j})_{j},\ j\in J$, where
$\widetilde{f}_{j}:X\to E_{j}$ is the projection of $\widetilde{f}$ over the
j-th component. By definition of $p$, $p_{j}\circ\widetilde{f}_{j}=F\circ
J_{0}$ and since $p_{j}$ is an h-fibration, there exist homotopies
$\widetilde{F}_{j}:X\times I\to E_{j}$ and $H_{j}:X\times I\to E_{j}$ such
that $p_{j}\circ\widetilde{F}_{j}=F$ and $H_{j}:\widetilde{F}_{j}\circ
J_{0}\simeq_{p_{j}}\widetilde{f}_{j}$. Since
$p_{j}\circ\widetilde{F}_{j}=F=p_{i}\circ\widetilde{F}_{i}$, we can define
$\widetilde{F}:X\times I\to E$ by
$\widetilde{F}(x,t)=(\widetilde{F}_{j}(x,t))_{j}$. Hence,

$p\circ\widetilde{F}=p\circ(\widetilde{F}_{j})_{j}=p_{j}\circ\widetilde{F}_{j}=F.$

Also, since $p_{j}\circ
H_{j}(x,t)=p_{j}\circ\widetilde{f}_{j}(x)=p_{i}\circ\widetilde{f}_{i}(x)=p_{i}\circ
H_{i}(x,t)$, $Im(H)\subseteq E$ and so we can define $H:X\times I\to E$ by
$H(x,t)=({H}_{j}(x,t))_{j}$. Now $H:\widetilde{F}\circ
J_{0}\simeq_{p}\widetilde{f}$ since

$\displaystyle H(x,0)=(H_{j}(x,0))_{j}=(\widetilde{F}_{j}\circ
J_{0}(x))_{j}=(\widetilde{F}_{j})_{j}\circ J_{0}(x)=\widetilde{F}\circ
J_{0}(x),\quad$ $\displaystyle
H(x,1)=(H_{j}(x,1))_{j}=(\widetilde{f}_{j}(x))_{j}=\widetilde{f}(x),\quad$
$\displaystyle p\circ H(x,t)=p\circ(H_{j}(x,t))_{j}=p_{j}\circ
H_{j}(x,t)=p_{j}\circ(\widetilde{F}_{j}\circ J_{0})(x)=\quad$
$\displaystyle(p_{j}\circ\widetilde{F}_{j})\circ
J_{0}(x)=(p\circ\widetilde{F})\circ J_{0}(x)=p\circ(\widetilde{F}\circ
J_{0})(x),\quad$ $\displaystyle p\circ
H(x,t)=p\circ(H_{j}(x,t))_{j}=p_{j}\circ
H_{j}(x,t)=p_{j}\circ\widetilde{f}_{j}(x)=p\circ\widetilde{f}(x).$

Therefore $p$ is an h-fibration. Moreover, if every $p_{j}$ has wuphl, since
the fibers of $p$ are the product of the fibers of $p_{j}$’s, then by
Proposition 2.10, $p$ has wuphl. ∎

The following result is a consequence of the above proposition.

######  Theorem 3.6.

The categories $\mathrm{hFib(B)}$ and $\mathrm{hFibwu(B)}$ have products.

Suppose ${\\{p_{j}:E_{j}\to B_{j}|j\in J}\\}$ is an indexed collection of
morphisms in $\mathrm{hFib}$ (or $\mathrm{hFibwu}$), $E:=\sqcup_{j}E_{j}$ and
$B:=\sqcup_{j}B_{j}$ are disjoint union of $E_{j}$’s and $B_{j}$’s,
respectively. Define $q:E\longrightarrow B$ by $q|_{E_{j}}=p_{j}$. Then $q$ is
an h-fibration (with wuphl). Because let $\widetilde{f}:X\to E$ and $F:X\times
I\to B$ be the maps such that $q\circ\widetilde{f}=F\circ J_{0}$. If $x_{0}\in
X$, then there exists one and only one $j\in J$ such that
$\widetilde{f}(x_{0})\in E_{j}$ and $F\circ J_{0}(x_{0})\in B_{j}$. Since
$E_{j}$’s and $B_{j}$’s are disjoint, continuity of $\widetilde{f}$ and
$F\circ J_{0}$ yields that for every $x\in X$ and every $t\in I$,
$\widetilde{f}(x)\in E_{j}$, $F\circ J_{0}(x)\in B_{j}$ and $F(x,t)\in B_{j}$
which imply that $p_{j}\circ\widetilde{f}(x)=F\circ J_{0}(x)$. By assumption,
there exist homotopies $\widetilde{F}_{j}:X\times I\to E_{j}$ and
$H_{j}:X\times I\to E_{j}$ such that $p_{j}\circ\widetilde{F}_{j}=F$ and
$H_{j}:\widetilde{F}_{j}\circ J_{0}\simeq_{p_{j}}\widetilde{f}$. Define
$\widetilde{F},H:X\times I\to E$ by
$\widetilde{F}(x,t)=\widetilde{F}_{j}(x,t)$ and $H(x,t)=H_{j}(x,t)$, if
$\widetilde{f}(x)\in E_{j}$. Therefore
$q\circ\widetilde{F}(x,t)=q\circ\widetilde{F}_{j}(x,t)=p_{j}\circ\widetilde{F}_{j}(x,t)=F(x,t)$.
Also, $H:\widetilde{F}\circ J_{0}\simeq_{q}\widetilde{f}$ because if
$\widetilde{f}(x)\in E_{j}$, then

$\displaystyle H(x,0)=H_{j}(x,0)=\widetilde{F}_{j}\circ J_{0}(x),\quad$
$\displaystyle H(x,1)=H_{j}(x,1)=\widetilde{f}(x),\quad$ $\displaystyle q\circ
H(x,t)=q\circ H_{j}(x,t)=p_{j}\circ
H_{j}(x,t)=p_{j}\circ(\widetilde{F}_{j}\circ J_{0})(x)=\quad$
$\displaystyle(p_{j}\circ\widetilde{F}_{j})\circ
J_{0}(x)=(q\circ\widetilde{F})\circ J_{0}(x)=q\circ(\widetilde{F}\circ
J_{0})(x),\quad$ $\displaystyle q\circ H(x,t)=q\circ H_{j}(x,t)=p_{j}\circ
H_{j}(x,t)=p_{j}\circ\widetilde{f}(x)=q\circ\widetilde{f}(x).$

Hence $q$ is an h-fibration. Moreover, if every $p_{j}$ has wuphl, since a
fiber of $q$ is a fiber of one of ${p_{j}}^{,}s$, Proposition 2.10 follows
that $q$ has wuphl.
Similarly, if ${\\{p_{j}:E_{j}\to B|j\in J}\\}$ is an indexed collection of
objects in $\mathrm{hFib(B)}$ (or $\mathrm{hFibwu(B)}$),
$q:\sqcup_{j}E_{j}\longrightarrow B$ defined by $q|_{E_{j}}=p_{j}$ is also an
h-fibration (with wuphl), because it is sufficient that for every $j\in J$,
let $B_{j}:=B$. Therefore, we have the following result.

######  Theorem 3.7.

The categories $\mathrm{hFib}$, $\mathrm{hFibwu}$, $\mathrm{hFib(B)}$ and
$\mathrm{hFibwu(B)}$ have coproducts.

References

## References

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  * [3] K. Fuchs, Homotopiefaserungen. Dissertation, Universitat Saarbrucken.
  * [4] A. Hatcher, Algebraic topology, Cambridge University Press, 2001.
  * [5] P. Pave$\check{s}$i$\acute{c}$ , A note on trivial fibrations, Glasnik Matematicki, 46(2011), 513 - 519.
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w(\mathcal{T}_{1,i})S(\mathcal{T}_{1,i})\right).$

By the induction hypothesis, each $w(\mathcal{T}_{1,i})S(\mathcal{T}_{1,i})$
has a good tail. As proved in the previous section,
$w(\dot{\mathcal{H}})S(\dot{\mathcal{H}})$ also has a good tail. Combining
this with Lemma 7.2 and Lemma (5.5) as in the proof of Lemma 5.11, we conclude
that $w(\mathcal{H}^{m,l})S(\mathcal{H}^{m,l})$ has a good tail.

###  7.3. Proof of Lemma 6.11 for $M_{l}(\dot{\mathcal{H}})$

Following the proof of Lemma 5.3, we derive a recusive bound for
$M_{l}(\dot{\mathcal{H}})$.

######  Lemma 7.3.

Under the hypothesis of Lemma 6.11, we have

$\displaystyle\frac{1}{2}w(\dot{\mathcal{H}}^{m,l})M_{l}(\dot{\mathcal{H}}^{m,l})$
$\displaystyle\leq$ $\displaystyle\lambda(\Sigma_{2}+\Sigma_{m})$

where

$\displaystyle\Sigma_{2}$ $\displaystyle=$ $\displaystyle\left[1+\lambda
w(\dot{\mathcal{H}}^{m-1,l})S(\dot{\mathcal{H}}^{m-1,l})\right]\Bigg{[}\sum_{i=1}^{D_{2}}w_{2,i}M_{l-1}(\mathcal{T}_{2,i})\prod_{j\neq
i,1\leq j\leq D_{2}}(1+\lambda
w(\mathcal{T}_{2,j})S(\mathcal{T}_{2,j}))\Bigg{]}$
$\displaystyle+w(\dot{\mathcal{H}}^{m-1,l})M_{l}(\dot{\mathcal{H}}^{m-1,l})\prod_{j=1}^{D_{2}}(1+\lambda
w(\mathcal{T}_{2,j})S(\mathcal{T}_{2,j}))$

and similarly for $\Sigma_{m}$.

###### Proof of Lemma 7.4.

We shall reuse the notations in the proof of Lemma 7.1. We denote the
corresponding expected total number of infections at depth-$l$ leaves of the
processes $(X_{2,t}^{\sharp})$ and $(X_{2,t}^{\otimes})$ by $M_{2}^{\sharp}$
and $M_{2}^{\otimes}$.

Similar to the derivation of (25), we observe that after the recovery of the
first infection from $v_{2}$ in $(X^{\sharp}_{2,t})$, the number of excursions
of $(X_{2,t}^{\otimes})$ included in a single excursion of
$(X^{\sharp}_{2,t})$ is the same as a geometric random variable with success
probability $(1+\lambda(D_{2}+2)\theta_{2}^{\otimes})^{-1}$. Therefore, the
expected number of infections at depth-$l$ leaves after time $t_{1}$ is

(58)
$\lambda(D_{2}+2)\theta_{2}^{\otimes}M_{2}^{\otimes}(\dot{\mathcal{H}}^{m,l}).$

By the same argument as in (56), we bound from above the expected number of
infections at depth-$l$ leaves of $(X_{t})$ during $[0,t_{1}]$ by

(59) $\frac{\lambda
w_{m}(\dot{\mathcal{H}}^{m,l})}{1+\lambda(D_{2}+2)}M_{l,m}(\dot{\mathcal{H}}^{m,l})+\theta_{2}^{\otimes}M_{2}^{\otimes}.$

Now we attempt to control $M^{\otimes}(\dot{\mathcal{H}}^{m,l})$ in terms of
$\\{M(\mathcal{T}_{2,i})\\}$ and $M(\dot{\mathcal{H}}^{m-1,l})$ where
$\dot{\mathcal{H}}^{m-1,l}\sim\textsf{\footnotesize{{GWC}}}^{2}(\xi,m-1)^{l}$
contains the cycle $v_{2},v_{3},\dots,v_{m}$ and the trees $\mathcal{T}_{i}$
attached to $v_{3},\dots,v_{m}$. Let $\mathcal{L}$ be the collection of
depth-$l$ leaves of $\dot{\mathcal{H}}^{m,l}$. We observe that

$\begin{split}\lim_{t_{0}\rightarrow\infty}\frac{1}{t_{0}}\sum_{v\in\mathcal{L}}\mathbb{E}_{\textsc{cp}}&\left[\,\left|\left\\{s\in[0,t_{0}]:\;X_{2,s}^{\otimes}(v)=1^{*}\textnormal{
and }X_{2,s-}^{\otimes}(v)=0\right\\}\right|\,\right]\\\
&=\frac{M_{2}^{\otimes}(\dot{\mathcal{H}}^{m,l})}{(\lambda(D_{2}+2)\theta_{2}^{\otimes})^{-1}+S_{2}^{\otimes}(\dot{\mathcal{H}}^{m,l})}.\end{split}$

On the other hand, by splitting $\mathcal{L}$ into parts corresponding to
$\mathcal{T}_{2,i}$ and $\dot{\mathcal{H}}^{m-1,l}$, the above is also equal
to

$\sum_{i=1}^{D_{2}}\frac{M_{l-1}(\mathcal{T}_{2,i})}{(\lambda
w(\mathcal{T}_{2,i}))^{-1}+S(\mathcal{T}_{2,i})}+\frac{M_{l}(\dot{\mathcal{H}}^{m-1,l})}{(\lambda
w(\dot{\mathcal{H}}^{m-1,l}))^{-1}+S(\dot{\mathcal{H}}^{m-1,l})}.$

Combining these equations and (57), we get

$\displaystyle(D_{2}+2)\theta_{2}^{\otimes}M_{2}^{\otimes}(\dot{\mathcal{H}}^{m,l})$
$\displaystyle=$ $\displaystyle\left[1+\lambda
w(\dot{\mathcal{H}}^{m-1,l})S(\dot{\mathcal{H}}^{m-1,l})\right]\Bigg{[}\sum_{i=1}^{D_{2}}w(\mathcal{T}_{2,i})M_{l-1}(\mathcal{T}_{2,i})\prod_{j\neq
i,1\leq j\leq D_{2}}(1+\lambda
w(\mathcal{T}_{2,j})S(\mathcal{T}_{2,j}))\Bigg{]}$
$\displaystyle+w(\dot{\mathcal{H}}^{m-1,l})M_{l}(\dot{\mathcal{H}}^{m-1,l})\prod_{j=1}^{D_{2}}(1+\lambda
w(\mathcal{T}_{2,j})S(\mathcal{T}_{2,j}))=\Sigma_{2}$

By this together with (58), (59), and $M^{\sharp}(\dot{\mathcal{H}}^{m,l})\geq
M_{l}(\dot{\mathcal{H}}^{m,l})$, we conclude that

$\displaystyle M_{l,2}(\dot{\mathcal{H}}^{m,l})$ $\displaystyle\leq$
$\displaystyle\frac{\lambda
w_{m}(\dot{\mathcal{H}}^{m,l})}{1+\lambda(D_{2}+2)}M_{l,m}(\dot{\mathcal{H}}^{m,l})+\frac{1+\lambda(D_{2}+2)}{D_{2}+2}\Sigma_{2}.$

Thus,

$\displaystyle w_{2}(\dot{\mathcal{H}}^{m,l})M_{l,2}(\dot{\mathcal{H}}^{m,l})$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}w_{m}(\dot{\mathcal{H}}^{m,l})M_{l,m}(\dot{\mathcal{H}}^{m,l})+\lambda\Sigma_{2}.$

Adding this with the analog for $M_{l,m}(\dot{\mathcal{H}}^{m,l})$ and
rearranging, we obtain

$\displaystyle w(\dot{\mathcal{H}}^{m,l})M_{l}(\dot{\mathcal{H}}^{m,l})$
$\displaystyle\leq$ $\displaystyle 2\lambda\Sigma_{2}+2\lambda\Sigma_{m}$

as claimed. ∎

To conclude that $(1.5)^{l-1}w(\mathcal{H}^{m,l})M_{l}(\mathcal{H}^{m,l})$ has
a good tail, we proceed as in the proof of Lemma 5.12. We set

$W_{i}=\max\\{(1.5)^{l-1}w(\mathcal{T}_{2,i})M_{l-1}(\mathcal{T}_{2,i}),w(\mathcal{T}_{2,i})S(\mathcal{T}_{2,i})\\},\quad
i=1,\dots,D_{2}$

and

$W_{0}=\max\left\\{(1.5)^{l-1}w(\dot{\mathcal{H}}^{m-1,l})M_{l}(\dot{\mathcal{H}}^{m-1,l}),w(\dot{\mathcal{H}}^{m-1,l})S(\dot{\mathcal{H}}^{m-1,l})\right\\},$

where we used the subscript $0$ in place of $\dot{\mathcal{H}}^{m-1,l}$ for
notational convenience. Then, the quantity $\Sigma_{2}$ in Lemma 7.3 satisfies

$2(1.5)^{l-1}\lambda\Sigma_{2}\leq 3\sum_{i=0}^{D_{2}}\lambda
W_{i}\prod_{0\leq j\leq D_{2},j\neq i}(1+\lambda
W_{j})\leq\frac{3}{2}\left(\prod_{i=0}^{D_{2}}(1+2\lambda W_{i})-1\right).$

We have by the induction hypothesis that $W_{i}$ has a good tail for all
$0\leq i\leq D_{2}$. By Lemma 5.5, $\lambda\Sigma_{2}$ has a strong tail.
Similarly, $\lambda\Sigma_{m}$ also has a strong tail. And so does,
$(1.5)^{l-1}w(\dot{\mathcal{H}}^{m,l})M_{l}(\dot{\mathcal{H}}^{m,l})$.

###  7.4. Proof of Lemma 6.11 for $M_{l}(\mathcal{H}^{m,l})$

For the tail of $M_{l}(\mathcal{H}^{m,l})$, we have the following recursive
inequality which can be derived analogously as Lemma 5.3.

######  Lemma 7.4.

Under the hypothesis of Lemma 6.11, we have

$\displaystyle w(\mathcal{H}^{m,l})M_{l}(\mathcal{H}^{m,l})\leq
w(\dot{\mathcal{H}}^{m,l})M_{l}(\dot{\mathcal{H}}^{m,l})\prod_{j=1}^{D_{1}}\left(1+\lambda
w(\mathcal{T}_{1,j})S(\mathcal{T}_{1,j})\right)$
$\displaystyle\quad\quad\quad+\left[1+\lambda
w(\dot{\mathcal{H}}^{m,l})S(\dot{\mathcal{H}}^{m,l})\right]\left[\sum_{i=1}^{D_{1}}w(\mathcal{T}_{1,i})M_{l-1}(\mathcal{T}_{1,i})\prod_{j\neq
i,1\leq j\leq D_{1}}\left(1+\lambda
w(\mathcal{T}_{1,j})S(\mathcal{T}_{1,j})\right)\right].$

From here, the conclusion that
$(1.5)^{l}w(\mathcal{H}^{m,l})M_{l}(\mathcal{H}^{m,l})$ has a good tail
follows from the same argument as above.

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# Deep Curriculum Learning for PolSAR Image Classification

Hamidreza Mousavi, Maryam Imani, Hassan Ghassemian
Image processing and Information Analysis Lab
Faculty of Electrical and Computer Engineering, Tehran, Iran
{h.mousavi, maryam.imani<EMAIL_ADDRESS>
###### Abstract

Following the great success of curriculum learning in the area of machine
learning, a novel deep curriculum learning method proposed in this paper,
entitled DCL, particularly for the classification of fully polarimetric
synthetic aperture radar (PolSAR) data. This method utilizes the entropy-alpha
target decomposition method to estimate the degree of complexity of each
PolSAR image patch before applying it to the convolutional neural network
(CNN). Also, an accumulative mini-batch pacing function is used to introduce
more difficult patches to CNN. Experiments on the widely used data set of
AIRSAR Flevoland reveal that the proposed curriculum learning method can not
only increase classification accuracy but also lead to faster training
convergence.

###### Index Terms:

Polarimetric SAR image classification, Curriculum learning, Deep learning,
Convloutional neural network.

††publicationid: pubid: 978-1-6654-1216-2/22/$31.00 © 2022 IEEE

##  I Introduction

Deep neural networks have been extensively employed in the area of PolSAR
image classification. Specially convolutional neural networks (CNNs) have
proved to be a useful automatic feature extraction tool for PolSAR image
classification in recent years[1]. The main focus in this area of research is
on developing practical CNN architectures in terms of classification accuracy
and model complexity. For instance, Zhang et al.[2] extended a real-valued CNN
(RV-CNN) in a complex domain to exploit the potential of phase information in
PolSAR data and their proposed complex-valued CNN (CV-CNN) could remarkably
increase the classification accuracy over RV-CNN. The first architecture
utilizing 3D convolution and 3D pooling layers for PolSAR image classification
was introduced by Zhang et al.[3] and achieved a classification accuracy
better than conventional 2D CNNs. Lightweight 3D convolutions introduced by
Dong et al.[4] to reduce the redundancy of 3D convolutions and take advantage
of their excellent ability in spatial-polarimetry feature extraction.
Considerable attention has been focused on the CNN architecture evolution over
the last few years for increasing the feature extraction ability. In contrast,
to reduce the number of model parameters, prevent from the overfitting, and
even to eliminate the fully connected layers, less attention has been devoted
to improving the training process. Since neural network architectures are
inspired by the human brain, it seems reasonable that studying the learning
process should also be inspired by the biological processes of the human
brain. Starting from easy tasks before progressing to high and complicated
levels generally provides better learning results for Humans. Using a similar
approach for training a machine learning model, we can obtain great benefits
including faster convergence speed of the training process and better
accuracy. This strategy has been formalized by Bengio et al.[5] and called
curriculum learning in the context of machine learning. Curriculum learning
approaches have been successfully applied in various fields of machine
learning[6]. Each curriculum learning strategy needs a proper complexity
measurement criterion of training samples and a pacing function for
progressively introducing more complex samples to the machine learning
model[5, 6]. To our knowledge, there isn’t a criterion to measure the
difficulty of PolSAR patches before applying them to a deep learning model.
This motivates us to formalize this idea.
PolSAR data analysis is rich in accurate pixel information extraction thanks
to the publication of a large number of proposed methods in the target
decomposition theory. The Cloude-Pottier target decomposition method proposed
parameters such as entropy (H), mean alpha angle ($\bar{\alpha}$), and
anisotropy (AA), which provide the physical interpretation of the scattering
mechanisms of objects. The most attractive and practical application of the
H/$\bar{\alpha}$ method is an unsupervised classification of PolSAR data by
subdividing the H/$\bar{\alpha}$ plane into nine zones correspond to various
physical scattering characteristics. In this study, we use the
H/$\bar{\alpha}$ plane to introduce a novel patch complexity criterion (PaCC)
for PolSAR patches. The basic idea is that each patch is considered more
difficult for neural network to learn if it consists of pixels with a more
complex scattering process. Therefore, two criteria are needed to first
measure the complexity of each pixel in the patch and then determine the
complexity of the patch with the obtained values for pixels. Two parameters of
entropy H and mean alpha angle $\bar{\alpha}$, which yield direct information
about the scattering mechanism are used to measure the scattering complexity
of pixels and a criterion named pixel complexity criterion (PiCC) is defined.
Finally, by calculating the average of the obtained PiCC values, the PaCC
parameter is determined. The rest of this paper is structured as follows.
Section II presents the methodology. Experimental results are shown in Section
III, which illustrate the effectiveness of the proposed method. Finally, we
draw some conclusions in Section IV.

##  II METHODOLOGY

###  II-A Deep learning

Deep learning (DL) attempts to develop suitable models by mimicking the
structure and functioning of the human brain in processing data. From the
viewpoint of automation, the evolution of DL has transformed the handcrafted
design of features in machine learning to facilitate automatic feature
extraction from high-dimensional raw data. This remarkable ability of high-
dimensional data processing and automatic feature extraction has made
breakthroughs in the pattern recognition field. Most recently, The huge
success of DL in the computer vision area has motivated the remote-sensing
community to this technology and DL algorithms bring huge advancements in many
remote sensing image interpretation tasks including Image fusion[11], scene
classification [18], object detection[12, 16], surface terrian
classification[17], and other tasks. Convolutional neural networks (CNN) is a
specific type of DL model which is particularly useful for recognizing visual
patterns in images. It is inspired by the structure of the human and animal
visual system and designed to automatically extract low, mid and high-level
features from multi-dimensional data. Because of this characteristic, it is
applicable for processing multi-channel remote sensing images.

CNN has proven to have an exemplary performance to fill the gaps in
conventional machine learning algorithms.
Inspired by the lightweight operations for CNNs in the PolSAR image
classification field, in this paper, pseudo-3D(P3D) CNN introduced by [4] is
used in the experiments to investigate the performance of the proposed
curriculum learning algorithm. Lightweight P3D CNN was introduced to not only
perform feature learning in both the spatial and polarimetric dimensions but
also reduce network parameters and high computational costs of 3D CNNs[4, 15].

###  II-B Curriculum Learning

Curriculum Learning (CL) can be considered as an optimization technique, which
can improve optimization performance by discovering better relative minima of
a non-convex cost function[5, 7]. It is inspired based on imitating the
learning behaviour of humans when trying to learn concepts. They can master a
task when try to learn easy parts of a task and gradually increase complexity
level. Using a similar approach for training a machine learning model, we can
obtain great benefits including faster convergence speed of the training
process and better accuracy.
Since that samples are considered in random order during training of machine
learning model, the hard or noisy samples, especially in the initial stage of
the training process may result in over-fitting the train set and leading to
poor generalization. This learning strategy has been empirically proved to be
robust to noisy samples and be effective in avoiding getting stuck in poor
local minima[14]. PolSAR images also are heavily affected by speckle noise.
This noise diminishes the visual quality of PolSAR images for interpretation
and limits image processing algorithms during information extraction from
PolSAR data. Hence, CL can effectively help machine learning algorithms to
suppress the effect of noisy PolSAR samples during the training process and
achieve more reliable results. However, the implementation of CL in machine
learning is subject to finding a proper computational complexity measurement
criterion of training samples and a pacing function for progressively
introducing more complex samples to the machine learning model. This is the
principal motivation behind the design and implementation of our CL framework
to improve the classification performance of SAR polarimetry data.
The complexity of the scattering process complicates the direct analysis of
coherency matrix T. Hence, the ultimate purpose of the incoherent
decompositions tools is to simplify the interpretation of the scattering
process by expanding the second-order matrices into a weighted addition of
basis matrices describing scattering responses of simpler objects. Two
important average parameters (the entropy H and mean angle $\bar{\alpha}$)
were introduced by Cloude and Pottier[7] using the eigenvalue/eigenvector
analysis of T. Entropy H is defined as a function of the eigenvalues:

$\displaystyle H$ $\displaystyle=-\sum^{3}_{i=1}p_{i}\log_{3}(p_{i})$ ,
$\displaystyle p_{i}$
$\displaystyle=\dfrac{\lambda_{i}}{\sum^{3}_{j=1}\lambda_{j}}$ (1)

Entropy H can go above 0 and below 1. H = 0 corresponds to only one nonzero
eigenvalue in T. It physically indicates that there is only a single
scattering mechanism caused by the surface scattering. When H equals to 1, T
has three equal nonzero eigenvalues, which statically means three scattering
mechanisms occur simultaneously. From a physical standpoint, this shows that
the behaviour of scatter is completely random and this mechanism occurs over
the forest and vegetated surfaces. As H increases from 0 to 1, the randomness
of the scattering mechanism increases, which means the ability to distinguish
between scattering mechanisms decreases[8, 9]. In short, the the complexity of
every pixel in the PolSAR image is proportional to entropy H.
Mean alpha angle $\bar{\alpha}$ is obtained from parameterizing of the
eigenvectors of T versus some of the angle parameters which $\alpha$ is one of
them and given by

$\bar{\alpha}=\sum^{3}_{i=1}p_{i}{\alpha_{i}}$ (2)

The mean alpha angle $\bar{\alpha}$ indicates the type of dominant scattering
mechanism and lies within the range of $0^{\circ}$ to $90^{\circ}$.
$\bar{\alpha}$ = $0^{\circ}$ corresponds to the surface scattering caused by
water body or bare soil, $\bar{\alpha}$ = $45^{\circ}$ to the volumetric
scattering arisen from woodlands, and $\bar{\alpha}$ = $90^{\circ}$ to double-
bounce scattering as a result of interaction between the incident wave and
street/building classes in urban areas. By continuously increasing
$\bar{\alpha}$ from 0 to 90, the scattering mechanism changes from isotropic
simple scattering($\bar{\alpha}$ = 0) to random volume scattering
($\bar{\alpha}$ = 60) and finally $\bar{\alpha}$ = 90, is related to double
bounce scattering[7, 8]. For the measurement of complexity based on the alpha
angle, if we consider complexity range from 0 to 1, by increasing
$\bar{\alpha}$ from 0 to 60, complexity increases to 1 and by changing alpha
from 60 to 90, the complexity decreases from the maximum value to the median
one, which equals to 0.5. Based on the above analysis, the PaCC parameter is
proposed to measure the complexity of each PolSAR image patch by averaging the
complexity of whole pixels exist in that patch. The complexity of each pixel
is calculated by using Euclidean distance in the two-dimensional (2D)
H/$\bar{\alpha}$ plane. The details of the patch-ranking algorithm are shown
in Algorithm 1 and the details of the method are expressed as follows.
First, we introduce a metric, called pixel complexity criterion (PiCC) to
measure the complexity of every pixel in a patch using Euclidean distance as
follows:

$\displaystyle PiCC(\textbf{v}^{(i)})=$
$\displaystyle\sqrt{H_{i}^{2}+(1-|\dfrac{\bar{\alpha}_{i}-60}{60}|)^{2}}\quad
i=1,2,...,m_{1}m_{2}$ (3)

Where $\textbf{v}^{(i)}$ is i-th pixel in a PolSAR patch, which defined as a
vector as shown in Equation (4) :

$\displaystyle\textbf{v}=\\{t_{11},t_{22},t_{33},\Re{\\{t_{12}\\}},\Im{\\{t_{12}\\}},\Re{\\{t_{13}\\}},\Im{\\{t_{13}\\}},$
(4) $\displaystyle\Re{\\{t_{23}\\}},\Im{\\{t_{23}\\}}\\}$

In equation (3), $H_{i}$ and $\bar{\alpha}_{i}$ are the entropy and mean alpha
angle of pixel $\textbf{v}^{(i)}$, which exist in a PolSAR image patch with
$m_{1}\times m_{2}\times 9$ size. Finally, by averaging on all the calculated
PiCCs for the patch, the patch complexity criterion (PaCC) for arbitrary
PolSAR image patch $\emph{{X}}\in\mathbb{R}^{m_{1}\times m_{2}\times 9}$ is
defined by the following formula:

$PaCC(\emph{{X}})=\dfrac{1}{m_{1}m_{2}}\sum_{i=1}^{m_{1}m_{2}}PiCC(\textbf{v}^{(i)})\\\
=\dfrac{1}{m_{1}m_{2}}\sum_{i=1}^{m_{1}m_{2}}\sqrt{H_{i}^{2}+(1-|\dfrac{\bar{\alpha}_{i}-60}{60}|)^{2}}$
(1)

Any patch with a higher PaCC value is considered a more difficult patch for
training. Therefore, after calculating the PaCC value for all the patches in
the training set $\mathbb{L}$, it is necessary to arrange the PaCC values in
ascending order. Finally, the training samples are reordered to form the
sorted training set $\mathbb{L}^{\dagger}$. The argsort operator in the eighth
line of Algorithm 1 returns indices of the training samples after sorting PaCC
values of them in an ascending order.

Another key part of any curriculum learning method is task scheduling, which
specifies how to gradually increase the complexity of tasks and update the
training process. Several scheduling strategies have been proposed, where
batching, sampling and weighing are the most extensively employed ones[6]. In
this study, we introduce a new batching strategy, called batch accumulation
scheduler function. The main idea is that in each iteration after random
sampling, we sort the training samples from easy to hard. Then, the sorted
training set is split into n same size batches. The fine-tuning process starts
on the easiest batch (k = 1), the second batch is concatenated with the
previous one and applied to the deep learning model for updating (k = 2). This
process continues until all mini-batches are concatenated together.
Eventually, the deep learning model is updated with all training samples
existing in the training set (k = n). In step k for gradient updating, the
sliced batch from the sorted training set $\mathbb{L}^{\dagger}$ is sliced as
follows:

$\mathbb{B}_{k}=\\{(\emph{{X}}_{i},y_{i})\\}_{i=j_{1}}^{j_{m}},m=k\dfrac{N}{n},k=1,2,...,n$
(6)

where $\mathbb{B}_{k}$ and m are sliced batch and batch size at k-th step,
respectively. This pacing function is also inspired by curricula taught in
education systems around the world that usually enables learning concepts in
progressively increasing levels of complexity while considering and reviewing
previously learned concepts.

Input: $\mathbb{L}=\\{(\emph{{X}}_{i},y_{i})\\}_{i=1}^{N}$: PolSAR image patch
training set.

Output: $\mathbb{L}^{\dagger}=\\{(\emph{{X}}_{j},y_{j})\\}_{i=j_{1}}^{j_{N}}$:
Sorted PolSAR image patch training set.

1 $\textbf{c}_{i}=0$, $i\in\\{1,2,3,...,N\\}$

2 for  _i $\leftarrow$ 1 to N_ do

3 $\textbf{d}_{k}=0$, $k\in\\{1,2,3,...,m_{1}m_{2}\\}$

4 for  _k $\leftarrow$ 1 to $m_{1}m_{2}$ _ do

5 Calculate $H_{k}$ and $\bar{\alpha}_{k}$ by formulas 1 and 2 using
$\textbf{v}^{(k)}$ as input

6
$\textbf{d}_{k}\leftarrow\sqrt{H_{k}^{2}+(1-|\dfrac{\bar{\alpha}_{k}-60}{60}|)^{2}}$

7
$\textbf{c}_{i}\leftarrow\dfrac{1}{m_{1}m_{2}}\sum_{k=1}^{m_{1}m_{2}}\textbf{d}_{k}$

8$\textbf{j}=\underset{i}{\mathrm{arg\,sort}}\textbf{ c}_{i}$

9return
_$\mathbb{L}^{\dagger}=\\{(\emph{{X}}_{i},y_{i})\\}_{i=j_{1}}^{j_{N}}$_

10

Algorithm 1 Patch-ranking by $H/\bar{\alpha}$ decomposition

###  II-C Deep Curriculum Learning (DCL)

Algorithm 2 presents the pseudo-code of the DCL method. First, the weights of
P3D CNN initialized by random values. In the iterative phase, $N_{t}$ samples
randomly chose from all labeled training set $\mathbb{U}$ and these samples
are added to the training set $\mathbb{L}$. Training samples in the training
set sorted from easy to hard based on Algorithm 1. Then training set divided
into n batches and we fine-tune P3D CNN by incrementally accumulating batches.

Input:

$\mathbb{U}$: All labeled patch set.

$N_{t}$: The number of patches to be selected in each stage of DCL , where t =
0,1,…, T, T is number of stages.

$n$: Training set splitting parameter

Output: Trained P3D CNN model.

Initalization:

Initilize weights of P3D CNN by random values.

$\mathbb{S}_{0}$ = {}.

$\mathbb{L}$ = {}.

1

2

3

4 for  _t $\leftarrow$ 0 to T_ do

5 Select $N_{t}$ patches randomly from $\mathbb{U}$ and build
$\mathbb{S}_{t}$.

6

7 Remove Selected patched from $\mathbb{U}$, i.e,
$\mathbb{U}\leftarrow\mathbb{U}\setminus\mathbb{S}_{t}$.

8

9 $\mathbb{L}\leftarrow\mathbb{L}\cup\mathbb{S}_{t}$.

10

11 Sort patches in $\mathbb{L}$ using Algorithm 1 to obtain sorted patch set
$\mathbb{L}^{\dagger}$.

12

13 for  _k $\leftarrow$ 1 to n_ do

14 Slice Batch $\mathbb{B}_{k}$ by Eq. (6).

15 P3D CNN traning ($c=1$) or fine-tuning($c>1$) using batch $\mathbb{B}_{k}$.

16

return _Trained P3D CNN model._

Algorithm 2 Deep Curriculum learning (DCL) for PolSAR Image Classification

##  III EXPERIMENTAL RESULTS

In this section, to analyze the performance of the algorithms mentioned in
this paper, we conduct experiments on the publicly available PolSAR data set,
which is collected by the AIRSAR sensor over the Flevoland area. The overall
accuracy (OA) and runtime are also used to evaluation of classification
performance.

###  III-A Experimental data and parameter setting

The AIRSAR data is a four-look full polarized image acquired by NASA/JPL using
L- band PolSAR sensor was mounted on a NASA DC-8 aircraft on 16 August 1989.
It reflects an agricultural area of Flevoland in the Netherlands and it is
widely used as a benchmark data for PolSAR image classification. The size of
this image is 1024 × 750 pixels and contains 15 categories, which are
building, water, forest, bare soil, lucerne, beet, potatoes, grass, rapeseed,
barley, three different types wheat, steam beans, peas. The RGB image formed
by the Pauli decomposition and the corresponding ground truth image,
containing a total of 157296 samples and legend of this data set are shown in
Fig. 1. Each colour denotes one category and white region stands for unknown
land covers. multi-look full polarized image

Figure 1: AIRSAR Floevoland data set adopted in the experiments.

We set the experimental settings in this work as follows: The lightweight P3D
CNN is implemented by Keras with the TensorFlow backend. During the CL cycle,
each data set is randomly divided into three sets. We randomly select 100
labeled samples in each iteration ($N_{t}=100$). To evaluate a model fit while
tuning network hyper-parameters, 10000 samples are selected as a validation
set and the remaining samples are used for further random sampling. We conduct
30 iterations of network training for the experiments. In each iteration, we
join 100 randomly selected training samples to the original training samples
to fine-tune the network using our proposed curriculum learning strategy.
Therefore, at the end of the training process, a total of 3000 training
samples are used. In the proposed curriculum, two hyper-parameters, including
splitting parameter n, and fine-tuning iterations need to be chosen. The
mentioned two hyper-parameters experimentally are set to 25 and 5,
respectively.
We compare our curriculum setting, with a no-curriculum setting in which
samples are not ordered by their complexity or applied to P3D CNN by
accumulative batching. In other words, during no-curriculum experiment, after
each random sampling and before applying to P3D CNN, training samples are
shuffled randomly and the training process done by a few epochs with a fixed-
batch size. These two hyper-parameters, which are related to no-curriculum
setting are set to 25 and 10, respectively. To assure the reliability of
experimental outcomes, we do ten independent tests over the data set and the
average results are reported.

###  III-B Classification Results

To completely study the influence of curriculum learning on the training
process, two groups of experiments are conducted, which are training P3D CNN
using curriculum and no-curriculum learning. We have monitored the values of
OA during training according to the number of training samples as shown in
Fig. 2. The main observations from this figure are as follows:
(a) The OA values with the curriculum, except in a few initial steps are
higher than without curriculum. This reflects that curriculum learning
provides better generalization to the unseen examples.
(b) With attention to the zoomed area in Fig. 2, the OA value curve for a
curriculum learning experiment is smoother than the no-curriculum experiment,
which shows that the output results of curriculum-based learning are more
reliable for PolSAR data classification.
The obtained OA and running time are given in Table I. From Table I, It is
shown that the proposed DCL has higher classification accuracy and lower
runtime, which illustrates the strength of the curriculum learning approach
and its positive impact on promoting the CNNs.

TABLE I: OA VALUES (%) AND RUN TIMES (SEC) WITH CURRICULUM AND NO-CURRICULUM STRATEGY ON FLEVOLAND AREA DATA SET. Method | OA (%) | run time (sec)
---|---|---
curriculum | 97.64 | 623
no-curriculum | 96.42 | 818

Finally, for comparative purposes, Fig. 3 (a) and (b) show the classification
maps by curriculum learning and no curriculum learning, respectively. It can
be observed that the result obtained by curriculum learning is remarkable,
which is better than that obtained by no curriculum learning.

Figure 2: The obtained overall accuracies (OAs, averaging from 10 independent
runs) using two different strategies for the AIRSAR Flevoland data set. In
each step, the original training samples are joined with 100 randomly selected
training samples. Therefore, at the end of the training process, a total of
3000 training samples are used.

##  IV CONCLUSION

In this paper, a novel deep curriculum learning method called DCL has been
designed for PolSAR image classification. Besides, a patch complexity
criterion, which is called PaCC, is introduced to estimate the learning
difficulty of the PolSAR patches before applying them to a CNN. The
experiments are conducted on benchmark AIRSAR Floevoland data set and the
experimental results show that the DCL is fast and can achieve better accuracy
than the methods which consider samples in a random order during training.

Figure 3: Classification results of DCL on AIRSAR Floevoland data set. (a)
Result of whole classification map using curriculum learning. (b) Result of
whole classification map using no curriculum learning.

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