A universe in a cwf
Definition.
A universe internal to a cwf (
Con
,
Subst
,
Ty
,
Tm
) consists of
U
El
:
{
Γ
:
Con
}
→
Ty
Γ
:
{
Γ
:
Con
}
→
Tm
Γ
U
→
Ty
Γ
Definition.
A universe internal to a cwf is said to
have products
if we have
prod
(
pair
,
proj
)
:
Tm
Γ
U
→
Tm
Γ
U
→
Tm
Γ
U
:
Tm
Γ
(
El
A
)
×
Tm
Γ
(
El
B
)
≅
Tm
Γ
(
El
(
prod
A
B
))
Definition.
A universe internal to a cwf is sait to
have dependent products
if we have
dprod
(
dpair
,
dproj
)
:
(
A
:
Tm
Γ
U
)
→
(
B
:
Tm
(
Γ
▹
El
A
)
U
)
→
Tm
Γ
U
:
(
a
:
Tm
Γ
(
El
A
))
×
Tm
Γ
(
El
B
[(
id
,
a
)])
≅
Tm
Γ
(
dprod
A
B
)