<s>
A	O
"	O
hooked	O
arrow	O
"	O
(	O
)	O
is	O
sometimes	O
used	O
in	O
place	O
of	O
the	O
function	O
arrow	O
above	O
to	O
denote	O
an	O
inclusion	B-Algorithm
map	I-Algorithm
;	O
thus	O
:	O
</s>
<s>
This	O
and	O
other	O
analogous	O
injective	O
functions	O
from	O
substructures	O
are	O
sometimes	O
called	O
natural	B-Algorithm
injections	I-Algorithm
.	O
</s>
<s>
Inclusion	B-Algorithm
maps	I-Algorithm
tend	O
to	O
be	O
homomorphisms	O
of	O
algebraic	O
structures	O
;	O
thus	O
,	O
such	O
inclusion	B-Algorithm
maps	I-Algorithm
are	O
embeddings	O
.	O
</s>
<s>
More	O
precisely	O
,	O
given	O
a	O
substructure	O
closed	O
under	O
some	O
operations	O
,	O
the	O
inclusion	B-Algorithm
map	I-Algorithm
will	O
be	O
an	O
embedding	O
for	O
tautological	O
reasons	O
.	O
</s>
<s>
Inclusion	B-Algorithm
maps	I-Algorithm
are	O
seen	O
in	O
algebraic	O
topology	O
where	O
if	O
is	O
a	O
strong	O
deformation	O
retract	O
of	O
the	O
inclusion	B-Algorithm
map	I-Algorithm
yields	O
an	O
isomorphism	O
between	O
all	O
homotopy	O
groups	O
(	O
that	O
is	O
,	O
it	O
is	O
a	O
homotopy	O
equivalence	O
)	O
.	O
</s>
<s>
Inclusion	B-Algorithm
maps	I-Algorithm
in	O
geometry	O
come	O
in	O
different	O
kinds	O
:	O
for	O
example	O
embeddings	O
of	O
submanifolds	O
.	O
</s>
