Category with Families
Definition.
A
category with families
(cwf) consists of type formers
Con
Subst
Ty
Tm
:
Set
:
Con
→
Con
→
Set
:
Con
→
Set
:
(
Γ
:
Con
)
→
Ty
Γ
→
Set
such that (
Con
,
Subst
) forms a category,
Ty
is a presheaf on (
Con
,
Subst
), and
Tm
is a
displayed presheaf
over
Ty
−
[
−
]
−
[
−
]
:
{
Δ
Γ
:
Con
}
→
Subst
Δ
Γ
→
Ty
Γ
→
Ty
Δ
:
{
Δ
Γ
:
Con
,
A
:
Ty
Γ
}
→
(
γ
:
Subst
Δ
Γ
)
→
Tm
Γ
A
→
Tm
Δ
A
[
γ
]
with the following
comprehension structure
.
(
−
▹
−
)
(
pq
,
(
−
,
−
))
:
(
Γ
:
Con
)
→
Ty
Γ
→
Con
:
{
Δ
Γ
:
Con
,
A
:
Ty
Γ
}
→
Subst
Δ
(
Γ
▹
A
)
≅
(
γ
:
Subst
Δ
Γ
)
×
(
Tm
Δ
A
[
γ
])