8.5.1 Fibrations over pushouts

8.5.1 Fibrations over pushouts

We first start with a lemma explaining how to

construct

fibrations

over

pushouts

.

Lemma 8.5.1

.

Let

D

=

(

Y

←

𝑗

X

→

𝑘

Z

)

be a span and assume
that we have

•

Two fibrations

E

Y

:

Y

→

𝒰

and

E

Z

:

Z

→

𝒰

.

•

An

equivalence

e

X

between

E

Y

∘

j

:

X

→

𝒰

and

E

Z

∘

k

:

X

→

𝒰

, i.e.

e

X

:

∏

x

:

X

E

Y

⁢

(

j

⁢

(

x

)

)

≃

E

Z

⁢

(

k

⁢

(

x

)

)

.

Then we can construct a fibration

E

:

Y

⊔

X

Z

→

U

such that

•

For all

y

:

Y

,

E

⁢

(

𝗂𝗇𝗅

⁢

(

y

)

)

≡

E

Y

⁢

(

y

)

.

•

For all

z

:

Z

,

E

⁢

(

𝗂𝗇𝗋

⁢

(

z

)

)

≡

E

Z

⁢

(

z

)

.

•

For all

x

:

X

,

E

(

𝗀𝗅𝗎𝖾

(

x

)

)

=

𝗎𝖺

(

e

X

(

x

)

)

(note that both sides of
the equation are paths in

𝒰

from

E

Y

⁢

(

j

⁢

(

x

)

)

to

E

Z

⁢

(

k

⁢

(

x

)

)

).

Moreover, the

total space

of this fibration fits in the following pushout
square:

\xymatrix

⁢

∑

x

:

X

E

Y

⁢

(

j

⁢

(

x

)

)

⁢

\ar

⁢

[

r

]

∼

𝗂𝖽

×

e

X

⁢

\ar

⁢

[

d

]

j

×

𝗂𝖽

⁢

&

⁢

∑

x

:

X

E

Z

⁢

(

k

⁢

(

x

)

)

⁢

\ar

⁢

[

r

]

-

⁢

k

×

𝗂𝖽

⁢

&

⁢

∑

z

:

Z

E

Z

⁢

(

z

)

⁢

\ar

⁢

[

d

]

𝗂𝗇𝗋

⁢

∑

y

:

Y

E

Y

⁢

(

y

)

⁢

\ar

⁢

[

r

⁢

r

]

𝗂𝗇𝗅

⁢

&

⁢

&

⁢

∑

t

:

Y

⊔

X

Z

E

⁢

(

t

)

Proof.

We define

E

by the

recursion

principle of the pushout

Y

⊔

X

Z

. For
that, we need to specify the value of

E

on elements of the form

𝗂𝗇𝗅

⁢

(

y

)

,

𝗂𝗇𝗋

⁢

(

z

)

and the action of

E

on paths

𝗀𝗅𝗎𝖾

⁢

(

x

)

, so we can just choose the following
values:

E

⁢

(

𝗂𝗇𝗅

⁢

(

y

)

)

:

≡

E

Y

(

y

)

,

E

⁢

(

𝗂𝗇𝗋

⁢

(

z

)

)

:

≡

E

Z

(

z

)

,

E

(

𝗀𝗅𝗎𝖾

(

x

)

)

\coloneqq

⁢

𝗎𝖺

⁢

(

e

X

⁢

(

x

)

)

.

To see that the total space of this fibration is a pushout, we apply the
flattening lemma (

\autoref

thm:flattening) with the following values:

•

A

:

≡

Y

+

Z

,

B

:

≡

X

and

f

,

g

:

B

→

A

are defined by

f

(

x

)

:

≡

𝗂𝗇𝗅

(

j

(

x

)

)

,

g

(

x

)

:

≡

𝗂𝗇𝗋

(

k

(

x

)

)

,

•

the type family

C

:

A

→

𝒰

is defined by

C

(

𝗂𝗇𝗅

(

y

)

)

:

≡

E

Y

(

y

)

and

C

(

𝗂𝗇𝗋

(

z

)

)

:

≡

E

Z

(

z

)

,

•

the family of equivalences

D

:

∏

(

b

:

B

)

C

⁢

(

f

⁢

(

b

)

)

≃

C

⁢

(

g

⁢

(

b

)

)

is defined
to be

e

X

.

The base

higher inductive type

W

in the flattening lemma is

equivalent

to
the pushout

Y

⊔

X

Z

and the type family

P

:

Y

⊔

X

Z

→

𝒰

is
equivalent to the

E

defined above.

Thus the flattening lemma tells us that

∑

(

t

:

Y

⊔

X

Z

)

E

⁢

(

t

)

is equivalent
the higher inductive type

E

tot

′

with the following

generators

:

•

a

function

𝗓

:

∑

(

a

:

Y

+

Z

)

C

⁢

(

a

)

→

E

tot

′

,

•

for each

x

:

X

and

t

:

E

Y

⁢

(

j

⁢

(

x

)

)

, a path

𝗓

⁢

(

𝗂𝗇𝗅

⁢

(

j

⁢

(

x

)

)

,

t

)

=

𝗓

⁢

(

𝗂𝗇𝗋

⁢

(

k

⁢

(

x

)

)

,

e

C

⁢

(

t

)

)

.

Using the flattening lemma again or a direct computation, it is easy to see
that

∑

(

a

:

Y

+

Z

)

C

⁢

(

a

)

≃

∑

(

y

:

Y

)

E

Y

⁢

(

y

)

+

∑

(

z

:

Z

)

E

Z

⁢

(

z

)

, hence

E

tot

′

is equivalent to the higher inductive type

E

tot

with the following generators:

•

a function

𝗂𝗇𝗅

:

∑

(

y

:

Y

)

E

Y

⁢

(

y

)

→

E

tot

,

•

a function

𝗂𝗇𝗋

:

∑

(

z

:

Z

)

E

Z

⁢

(

z

)

→

E

tot

,

•

for each

(

x

,

t

)

:

∑

(

x

:

X

)

E

Y

⁢

(

j

⁢

(

x

)

)

a path

𝗀𝗅𝗎𝖾

⁢

(

x

,

t

)

:

𝗂𝗇𝗅

⁢

(

j

⁢

(

x

)

,

t

)

=

𝗂𝗇𝗋

⁢

(

k

⁢

(

x

)

,

e

X

⁢

(

t

)

)

.

Thus the total space of

E

is the pushout of the total spaces of

E

Y

and

E

Z

, as required.
∎

Title

8.5.1 Fibrations over pushouts

\metatable

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