<s>
In	O
category	O
theory	O
,	O
a	O
category	O
is	O
Cartesian	B-Application
closed	I-Application
if	O
,	O
roughly	O
speaking	O
,	O
any	O
morphism	O
defined	O
on	O
a	O
product	O
of	O
two	O
objects	O
can	O
be	O
naturally	O
identified	O
with	O
a	O
morphism	O
defined	O
on	O
one	O
of	O
the	O
factors	O
.	O
</s>
<s>
They	O
are	O
generalized	O
by	O
closed	O
monoidal	O
categories	O
,	O
whose	O
internal	O
language	O
,	O
linear	O
type	O
systems	O
,	O
are	O
suitable	O
for	O
both	O
quantum	B-Architecture
and	O
classical	O
computation	O
.	O
</s>
<s>
The	O
category	O
C	O
is	O
called	O
Cartesian	B-Application
closed	I-Application
if	O
and	O
only	O
if	O
it	O
satisfies	O
the	O
following	O
three	O
properties	O
:	O
</s>
<s>
The	O
third	O
condition	O
is	O
equivalent	O
to	O
the	O
requirement	O
that	O
the	O
functor	B-Language
–	O
×Y	O
(	O
i.e.	O
</s>
<s>
the	O
functor	B-Language
from	O
C	O
to	O
C	O
that	O
maps	O
objects	O
X	O
to	O
X×Y	O
and	O
morphisms	O
φ	O
to	O
φ×idY	O
)	O
has	O
a	O
right	O
adjoint	O
,	O
usually	O
denoted	O
–	O
Y	O
,	O
for	O
all	O
objects	O
Y	O
in	O
C	O
.	O
</s>
<s>
Take	O
care	O
to	O
note	O
that	O
a	O
Cartesian	B-Application
closed	I-Application
category	I-Application
need	O
not	O
have	O
finite	O
limits	O
;	O
only	O
finite	O
products	O
are	O
guaranteed	O
.	O
</s>
<s>
If	O
a	O
category	O
has	O
the	O
property	O
that	O
all	O
its	O
slice	O
categories	O
are	O
Cartesian	B-Application
closed	I-Application
,	O
then	O
it	O
is	O
called	O
locally	O
cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
Note	O
that	O
if	O
C	O
is	O
locally	O
Cartesian	B-Application
closed	I-Application
,	O
it	O
need	O
not	O
actually	O
be	O
Cartesian	B-Application
closed	I-Application
;	O
that	O
happens	O
if	O
and	O
only	O
if	O
C	O
has	O
a	O
terminal	O
object	O
.	O
</s>
<s>
If	O
ΓY(p )	O
exists	O
for	O
every	O
morphism	O
p	O
with	O
codomain	O
Y	O
,	O
then	O
it	O
can	O
be	O
assembled	O
into	O
a	O
functor	B-Language
ΓY	O
:	O
C/Y	O
→	O
C	O
on	O
the	O
slice	O
category	O
,	O
which	O
is	O
right	O
adjoint	O
to	O
a	O
variant	O
of	O
the	O
product	O
functor	B-Language
:	O
</s>
<s>
Examples	O
of	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
include	O
:	O
</s>
<s>
The	O
category	O
Set	O
of	O
all	O
sets	O
,	O
with	O
functions	O
as	O
morphisms	O
,	O
is	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
The	O
adjointness	O
is	O
expressed	O
by	O
the	O
following	O
fact	O
:	O
the	O
function	O
f	O
:	O
X×Y	O
→	O
Z	O
is	O
naturally	O
identified	O
with	O
the	O
curried	B-Application
function	I-Application
g	O
:	O
X	O
→	O
ZY	O
defined	O
by	O
g(x )	O
(	O
y	O
)	O
=	O
f(x,y )	O
for	O
all	O
x	O
in	O
X	O
and	O
y	O
in	O
Y	O
.	O
</s>
<s>
The	O
category	O
of	O
finite	O
sets	O
,	O
with	O
functions	O
as	O
morphisms	O
,	O
is	O
Cartesian	B-Application
closed	I-Application
for	O
the	O
same	O
reason	O
.	O
</s>
<s>
If	O
G	O
is	O
a	O
group	O
,	O
then	O
the	O
category	O
of	O
all	O
G-sets	O
is	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
The	O
category	O
of	O
finite	O
G-sets	O
is	O
also	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
The	O
category	O
Cat	O
of	O
all	O
small	O
categories	O
(	O
with	O
functors	B-Language
as	O
morphisms	O
)	O
is	O
Cartesian	B-Application
closed	I-Application
;	O
the	O
exponential	O
CD	O
is	O
given	O
by	O
the	O
functor	B-Language
category	O
consisting	O
of	O
all	O
functors	B-Language
from	O
D	O
to	O
C	O
,	O
with	O
natural	O
transformations	O
as	O
morphisms	O
.	O
</s>
<s>
If	O
C	O
is	O
a	O
small	O
category	O
,	O
then	O
the	O
functor	B-Language
category	O
SetC	O
consisting	O
of	O
all	O
covariant	O
functors	B-Language
from	O
C	O
into	O
the	O
category	O
of	O
sets	O
,	O
with	O
natural	O
transformations	O
as	O
morphisms	O
,	O
is	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
If	O
F	O
and	O
G	O
are	O
two	O
functors	B-Language
from	O
C	O
to	O
Set	O
,	O
then	O
the	O
exponential	O
FG	O
is	O
the	O
functor	B-Language
whose	O
value	O
on	O
the	O
object	O
X	O
of	O
C	O
is	O
given	O
by	O
the	O
set	O
of	O
all	O
natural	O
transformations	O
from	O
to	O
F	O
.	O
</s>
<s>
The	O
category	O
of	O
all	O
directed	O
graphs	O
is	O
Cartesian	B-Application
closed	I-Application
;	O
this	O
is	O
a	O
functor	B-Language
category	O
as	O
explained	O
under	O
functor	B-Language
category	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
category	O
of	O
simplicial	O
sets	O
(	O
which	O
are	O
functors	B-Language
X	O
:	O
op	O
Set	O
)	O
is	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
Even	O
more	O
generally	O
,	O
every	O
elementary	O
topos	O
is	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
In	O
algebraic	O
topology	O
,	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
are	O
particularly	O
easy	O
to	O
work	O
with	O
.	O
</s>
<s>
Neither	O
the	O
category	O
of	O
topological	O
spaces	O
with	O
continuous	O
maps	O
nor	O
the	O
category	O
of	O
smooth	B-Architecture
manifolds	I-Architecture
with	O
smooth	O
maps	O
is	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
Substitute	O
categories	O
have	O
therefore	O
been	O
considered	O
:	O
the	O
category	O
of	O
compactly	O
generated	O
Hausdorff	O
spaces	O
is	O
Cartesian	B-Application
closed	I-Application
,	O
as	O
is	O
the	O
category	O
of	O
Frölicher	O
spaces	O
.	O
</s>
<s>
In	O
order	O
theory	O
,	O
complete	O
partial	O
orders	O
(	O
cpos	O
)	O
have	O
a	O
natural	O
topology	O
,	O
the	O
Scott	O
topology	O
,	O
whose	O
continuous	O
maps	O
do	O
form	O
a	O
Cartesian	B-Application
closed	I-Application
category	I-Application
(	O
that	O
is	O
,	O
the	O
objects	O
are	O
the	O
cpos	O
,	O
and	O
the	O
morphisms	O
are	O
the	O
Scott	O
continuous	O
maps	O
)	O
.	O
</s>
<s>
Both	O
currying	B-Application
and	O
apply	O
are	O
continuous	O
functions	O
in	O
the	O
Scott	O
topology	O
,	O
and	O
currying	B-Application
,	O
together	O
with	O
apply	O
,	O
provide	O
the	O
adjoint	O
.	O
</s>
<s>
A	O
Heyting	O
algebra	O
is	O
a	O
Cartesian	B-Application
closed	I-Application
(	O
bounded	O
)	O
lattice	O
.	O
</s>
<s>
This	O
poset	O
is	O
a	O
Cartesian	B-Application
closed	I-Application
category	I-Application
:	O
the	O
"	O
product	O
"	O
of	O
U	O
and	O
V	O
is	O
the	O
intersection	O
of	O
U	O
and	O
V	O
and	O
the	O
exponential	O
UV	O
is	O
the	O
interior	O
of	O
.	O
</s>
<s>
A	O
category	O
with	O
a	O
zero	O
object	O
is	O
Cartesian	B-Application
closed	I-Application
if	O
and	O
only	O
if	O
it	O
is	O
equivalent	O
to	O
a	O
category	O
with	O
only	O
one	O
object	O
and	O
one	O
identity	O
morphism	O
.	O
</s>
<s>
In	O
particular	O
,	O
any	O
non-trivial	O
category	O
with	O
a	O
zero	O
object	O
,	O
such	O
as	O
an	O
abelian	O
category	O
,	O
is	O
not	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
So	O
the	O
category	O
of	O
modules	O
over	O
a	O
ring	O
is	O
not	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
However	O
,	O
the	O
functor	B-Language
tensor	O
product	O
with	O
a	O
fixed	O
module	O
does	O
have	O
a	O
right	O
adjoint	O
.	O
</s>
<s>
Examples	O
of	O
locally	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
include	O
:	O
</s>
<s>
Every	O
elementary	O
topos	O
is	O
locally	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
The	O
category	O
LH	O
whose	O
objects	O
are	O
topological	O
spaces	O
and	O
whose	O
morphisms	O
are	O
local	O
homeomorphisms	O
is	O
locally	O
Cartesian	B-Application
closed	I-Application
,	O
since	O
LH/X	O
is	O
equivalent	O
to	O
the	O
category	O
of	O
sheaves	O
.	O
</s>
<s>
However	O
,	O
LH	O
does	O
not	O
have	O
a	O
terminal	O
object	O
,	O
and	O
thus	O
is	O
not	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
If	O
C	O
has	O
pullbacks	O
and	O
for	O
every	O
arrow	O
p	O
:	O
X	O
→	O
Y	O
,	O
the	O
functor	B-Language
p*	O
:	O
C/Y	O
→	O
C/X	O
given	O
by	O
taking	O
pullbacks	O
has	O
a	O
right	O
adjoint	O
,	O
then	O
C	O
is	O
locally	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
If	O
C	O
is	O
locally	O
Cartesian	B-Application
closed	I-Application
,	O
then	O
all	O
of	O
its	O
slice	O
categories	O
C/X	O
are	O
also	O
locally	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
Non-examples	O
of	O
locally	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
include	O
:	O
</s>
<s>
Cat	O
is	O
not	O
locally	O
Cartesian	B-Application
closed	I-Application
.	O
</s>
<s>
In	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
,	O
a	O
"	O
function	O
of	O
two	O
variables	O
"	O
(	O
a	O
morphism	O
f	O
:	O
X×Y	O
→	O
Z	O
)	O
can	O
always	O
be	O
represented	O
as	O
a	O
"	O
function	O
of	O
one	O
variable	O
"	O
(	O
the	O
morphism	O
λf	O
:	O
X	O
→	O
ZY	O
)	O
.	O
</s>
<s>
In	O
computer	B-General_Concept
science	I-General_Concept
applications	O
,	O
this	O
is	O
known	O
as	O
currying	B-Application
;	O
it	O
has	O
led	O
to	O
the	O
realization	O
that	O
simply-typed	O
lambda	O
calculus	O
can	O
be	O
interpreted	O
in	O
any	O
Cartesian	B-Application
closed	I-Application
category	I-Application
.	O
</s>
<s>
The	O
Curry	B-Application
–	O
Howard	O
–	O
Lambek	O
correspondence	O
provides	O
a	O
deep	O
isomorphism	O
between	O
intuitionistic	O
logic	O
,	O
simply-typed	O
lambda	O
calculus	O
and	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
.	O
</s>
<s>
Certain	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
,	O
the	O
topoi	O
,	O
have	O
been	O
proposed	O
as	O
a	O
general	O
setting	O
for	O
mathematics	O
,	O
instead	O
of	O
traditional	O
set	O
theory	O
.	O
</s>
<s>
The	O
renowned	O
computer	O
scientist	O
John	O
Backus	O
has	O
advocated	O
a	O
variable-free	O
notation	O
,	O
or	O
Function-level	B-Application
programming	I-Application
,	O
which	O
in	O
retrospect	O
bears	O
some	O
similarity	O
to	O
the	O
internal	O
language	O
of	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
.	O
</s>
<s>
CAML	B-Language
is	O
more	O
consciously	O
modelled	O
on	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
.	O
</s>
<s>
Let	O
C	O
be	O
a	O
locally	O
Cartesian	B-Application
closed	I-Application
category	I-Application
.	O
</s>
<s>
Taking	O
pullbacks	O
along	O
p	O
gives	O
a	O
functor	B-Language
p*	O
:	O
C/Y	O
→	O
C/X	O
which	O
has	O
both	O
a	O
left	O
and	O
a	O
right	O
adjoint	O
.	O
</s>
<s>
The	O
reason	O
for	O
these	O
names	O
is	O
because	O
,	O
when	O
interpreting	O
P	O
as	O
a	O
dependent	O
type	O
,	O
the	O
functors	B-Language
and	O
correspond	O
to	O
the	O
type	O
formations	O
and	O
respectively	O
.	O
</s>
<s>
In	O
every	O
Cartesian	B-Application
closed	I-Application
category	I-Application
(	O
using	O
exponential	O
notation	O
)	O
,	O
(	O
XY	O
)	O
Z	O
and	O
(	O
XZ	O
)	O
Y	O
are	O
isomorphic	O
for	O
all	O
objects	O
X	O
,	O
Y	O
and	O
Z	O
.	O
</s>
<s>
One	O
may	O
ask	O
what	O
other	O
such	O
equations	O
are	O
valid	O
in	O
all	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
.	O
</s>
<s>
Bicartesian	O
closed	O
categories	O
extend	O
Cartesian	B-Application
closed	I-Application
categories	I-Application
with	O
binary	O
coproducts	O
and	O
an	O
initial	O
object	O
,	O
with	O
products	O
distributing	O
over	O
coproducts	O
.	O
</s>
