<s>
that	O
is	O
equivariant	O
with	O
respect	O
to	O
the	O
conjugation	B-Algorithm
action	O
of	O
on	O
itself	O
:	O
</s>
<s>
The	O
first	O
mention	O
of	O
the	O
second	O
identity	O
for	O
a	O
crossed	B-Algorithm
module	I-Algorithm
seems	O
to	O
be	O
in	O
footnote	O
25	O
on	O
p.422	O
of	O
J	O
.	O
H	O
.	O
C	O
.	O
Whitehead	O
's	O
1941	O
paper	O
cited	O
below	O
,	O
while	O
the	O
term	O
'	O
crossed	B-Algorithm
module	I-Algorithm
 '	O
is	O
introduced	O
in	O
his	O
1946	O
paper	O
cited	O
below	O
.	O
</s>
<s>
These	O
ideas	O
were	O
well	O
worked	O
up	O
in	O
his	O
1949	O
paper	O
'	O
Combinatorial	O
homotopy	O
II	O
 '	O
,	O
which	O
also	O
introduced	O
the	O
important	O
idea	O
of	O
a	O
free	O
crossed	B-Algorithm
module	I-Algorithm
.	O
</s>
<s>
Whitehead	O
's	O
ideas	O
on	O
crossed	B-Algorithm
modules	I-Algorithm
and	O
their	O
applications	O
are	O
developed	O
and	O
explained	O
in	O
the	O
book	O
by	O
Brown	O
,	O
Higgins	O
,	O
Sivera	O
listed	O
below	O
.	O
</s>
<s>
Some	O
generalisations	O
of	O
the	O
idea	O
of	O
crossed	B-Algorithm
module	I-Algorithm
are	O
explained	O
in	O
the	O
paper	O
of	O
Janelidze	O
.	O
</s>
<s>
is	O
a	O
crossed	B-Algorithm
module	I-Algorithm
with	O
the	O
conjugation	B-Algorithm
action	O
of	O
on	O
.	O
</s>
<s>
For	O
any	O
group	O
H	O
,	O
the	O
homomorphism	O
from	O
H	O
to	O
Aut(H )	O
sending	O
any	O
element	O
of	O
H	O
to	O
the	O
corresponding	O
inner	B-Algorithm
automorphism	I-Algorithm
is	O
a	O
crossed	B-Algorithm
module	I-Algorithm
.	O
</s>
<s>
together	O
with	O
the	O
action	O
of	O
on	O
defines	O
a	O
crossed	B-Algorithm
module	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
central	O
extensions	O
can	O
be	O
seen	O
as	O
special	O
crossed	B-Algorithm
modules	I-Algorithm
.	O
</s>
<s>
Conversely	O
,	O
a	O
crossed	B-Algorithm
module	I-Algorithm
with	O
surjective	O
boundary	O
defines	O
a	O
central	O
extension	O
.	O
</s>
<s>
from	O
the	O
second	O
relative	O
homotopy	O
group	O
to	O
the	O
fundamental	O
group	O
,	O
may	O
be	O
given	O
the	O
structure	O
of	O
crossed	B-Algorithm
module	I-Algorithm
.	O
</s>
<s>
may	O
be	O
given	O
the	O
structure	O
of	O
crossed	B-Algorithm
module	I-Algorithm
.	O
</s>
<s>
These	O
examples	O
suggest	O
that	O
crossed	B-Algorithm
modules	I-Algorithm
may	O
be	O
thought	O
of	O
as	O
"	O
2-dimensional	O
groups	O
"	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
a	O
crossed	B-Algorithm
module	I-Algorithm
is	O
essentially	O
the	O
same	O
as	O
a	O
categorical	O
group	O
or	O
2-group	O
:	O
that	O
is	O
,	O
a	O
group	O
object	O
in	O
the	O
category	O
of	O
categories	O
,	O
or	O
equivalently	O
a	O
category	O
object	O
in	O
the	O
category	O
of	O
groups	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
concept	O
of	O
crossed	B-Algorithm
module	I-Algorithm
is	O
one	O
version	O
of	O
the	O
result	O
of	O
blending	O
the	O
concepts	O
of	O
"	O
group	O
"	O
and	O
"	O
category	O
"	O
.	O
</s>
<s>
This	O
allows	O
one	O
to	O
prove	O
that	O
(	O
pointed	O
,	O
weak	O
)	O
homotopy	O
2-types	O
are	O
completely	O
described	O
by	O
crossed	B-Algorithm
modules	I-Algorithm
.	O
</s>
