8.7.2 The van Kampen theorem with a set of basepoints

8.7.2 The van Kampen theorem with a set of basepoints

The improvement of van Kampen we present now is closely analogous to a

similar

improvement in classical

algebraic topology

, where

A

is equipped with a

set

S

of

base points

.
In fact, it turns out to be unnecessary for our proof to assume that the “set of basepoints” is a

set

— it might just as well be an arbitrary type; the utility of assuming

S

is a set arises later, when applying the

theorem

to obtain computations.
What is important is that

S

contains at least one point in each

connected component

of

A

.
We state this in

type theory

by saying that we have a type

S

and a

function

k

:

S

→

A

which is

surjective

, i.e.

(

-

1

)

-

connected

.
If

S

≡

A

and

k

is the

identity function

, then we will recover the naive

van Kampen theorem

.
Another example to keep in mind is when

A

is pointed and (0-)connected, with

k

:

𝟏

→

A

the point: by

\autoref

thm:minusoneconn-surjective,

\autoref

thm:connected-pointed this map is surjective just when

A

is 0-connected.

Let

A

,

B

,

C

,

f

,

g

,

P

,

i

,

j

,

h

be as in the previous

section

.
We now define, given our surjective map

k

:

S

→

A

, an auxiliary type which improves the connectedness of

k

.
Let

T

be the

higher inductive type

generated by

•

A function

ℓ

:

S

→

T

, and

•

For each

s

,

s

′

:

S

, a function

m

:

(

k

s

=

A

k

s

′

)

→

(

ℓ

s

=

T

ℓ

s

′

)

.

There is an obvious induced function

k

¯

:

T

→

A

such that

k

¯

⁢

ℓ

=

k

, and any

p

:

k

⁢

s

=

k

⁢

s

′

is equal to the composite

k

⁢

s

=

k

¯

⁢

ℓ

⁢

s

⁢

=

k

¯

⁢

m

⁢

p

⁢

k

¯

⁢

ℓ

⁢

s

′

=

k

⁢

s

′

.

Lemma 8.7.1

.

k

¯

is 0-connected.

Proof.

We must show that for all

a

:

A

, the 0-truncation of the type

∑

(

t

:

T

)

(

k

¯

t

=

a

)

is

contractible

.
Since

contractibility

is a mere proposition and

k

is

(

-

1

)

-connected, we may assume that

a

=

k

⁢

s

for some

s

:

S

.
Now we can take the center of

contraction

to be

|

(

ℓ

⁢

s

,

q

)

|

0

where

q

is the

equality

k

¯

⁢

ℓ

⁢

s

=

k

⁢

s

.

It remains to show that for any

ϕ

:

∥

∑

(

t

:

T

)

(

k

¯

t

=

k

s

)

∥

0

we have

ϕ

=

|

(

ℓ

s

,

q

)

|

0

.
Since the latter is a mere proposition, and in particular a set, we may assume that

ϕ

=

|

(

t

,

p

)

|

0

for

t

:

T

and

p

:

k

¯

⁢

t

=

k

⁢

s

.

Now we can do

induction

on

t

:

T

.
If

t

≡

ℓ

⁢

s

′

, then

k

⁢

s

′

=

k

¯

⁢

ℓ

⁢

s

′

⁢

=

𝑝

⁢

k

⁢

s

yields via

m

an equality

ℓ

⁢

s

=

ℓ

⁢

s

′

.
Hence by definition of

k

¯

and of equality in homotopy fibers, we obtain an equality

(

k

⁢

s

′

,

p

)

=

(

k

⁢

s

,

q

)

, and thus

|

(

k

s

′

,

p

)

|

0

=

|

(

k

s

,

q

)

|

0

.
Next we must show that as

t

varies along

m

these equalities agree.
But they are equalities in a set (namely

∥

∑

(

t

:

T

)

(

k

¯

t

=

k

s

)

∥

0

), and hence this is automatic.
∎

Remark 8.7.2

.

T

can be regarded as the (

homotopy

)

coequalizer

of the “

kernel pair

” of

k

.
If

S

and

A

were sets, then the

(

-

1

)

-connectivity of

k

would imply that

A

is the

0

-truncation of this coequalizer (see

\autoref

cha:set-math).
For general types, higher topos theory suggests that

(

-

1

)

-connectivity of

k

will imply instead that

A

is the

colimit

(a.k.a. “

geometric realization

”) of the “simplicial kernel” of

k

.
The type

T

is the colimit of the “1-skeleton” of this simplicial kernel, so it makes sense that it improves the connectivity of

k

by

1

.
More generally, we might expect the colimit of the

n

-

skeleton

to improve connectivity by

n

.

Now we define

𝖼𝗈𝖽𝖾

:

P

→

P

→

𝒰

by double induction as follows

•

𝖼𝗈𝖽𝖾

⁢

(

i

⁢

b

,

i

⁢

b

′

)

is now a set-quotient of the type of

sequences

(

b

,

p

0

,

x

1

,

q

1

,

y

1

,

p

1

,

x

2

,

q

2

,

y

2

,

p

2

,

…

,

y

n

,

p

n

,

b

′

)

where

–

n

:

ℕ

,

–

x

k

:

S

and

y

k

:

S

for

0

<

k

≤

n

,

–

p

0

:

Π

1

⁢

B

⁢

(

b

,

f

⁢

k

⁢

x

1

)

and

p

n

:

Π

1

⁢

B

⁢

(

f

⁢

k

⁢

y

n

,

b

′

)

for

n

>

0

, and

p

0

:

Π

1

⁢

B

⁢

(

b

,

b

′

)

for

n

=

0

,

–

p

k

:

Π

1

⁢

B

⁢

(

f

⁢

k

⁢

y

k

,

f

⁢

k

⁢

x

k

+

1

)

for

1

≤

k

<

n

,

–

q

k

:

Π

1

⁢

C

⁢

(

g

⁢

k

⁢

x

k

,

g

⁢

k

⁢

y

k

)

for

1

≤

k

≤

n

.

The

quotient

is generated by the following equalities (see

\autoref

rmk:naive):

(

…

,

q

k

,

y

k

,

𝗋𝖾𝖿𝗅

f

⁢

y

k

,

y

k

,

q

k

+

1

,

…

)

=

(

…

,

q

k

⁢

\centerdot

⁢

q

k

+

1

,

…

)

(

…

,

p

k

,

x

k

,

𝗋𝖾𝖿𝗅

g

⁢

x

k

,

x

k

,

p

k

+

1

,

…

)

=

(

…

,

p

k

⁢

\centerdot

⁢

p

k

+

1

,

…

)

(

…

,

p

k

-

1

⁢

\centerdot

⁢

f

⁢

w

,

x

k

′

,

q

k

,

…

)

=

(

…

,

p

k

-

1

,

x

k

,

g

⁢

w

⁢

\centerdot

⁢

q

k

,

…

)

(for $w:\Pi_{1}A(kx_{k},kx_{k}^{\prime})$)

(

…

,

q

k

⁢

\centerdot

⁢

g

⁢

w

,

y

k

′

,

p

k

,

…

)

=

(

…

,

q

k

,

y

k

,

f

⁢

w

⁢

\centerdot

⁢

p

k

,

…

)

.

(for $w:\Pi_{1}A(ky_{k},ky_{k}^{\prime})$)

We will need below the definition of the case of

𝖽𝖾𝖼𝗈𝖽𝖾

on such a sequence, which as before concatenates all the paths

p

k

and

q

k

together with

instances

of

h

to give an element of

Π

1

⁢

P

⁢

(

i

⁢

f

⁢

b

,

i

⁢

f

⁢

b

′

)

, cf. (

LABEL:eq:decode

).
As before, the other three point cases are nearly identical.

•

For

a

:

A

and

b

:

B

, we require an

equivalence

𝖼𝗈𝖽𝖾

⁢

(

i

⁢

b

,

i

⁢

f

⁢

a

)

≃

𝖼𝗈𝖽𝖾

⁢

(

i

⁢

b

,

j

⁢

g

⁢

a

)

.

(8.7.3)

Since

𝖼𝗈𝖽𝖾

is set-valued, by

\autoref

thm:kbar we may assume that

a

=

k

¯

⁢

t

for some

t

:

T

.
Next, we can do induction on

t

.
If

t

≡

ℓ

⁢

s

for

s

:

S

, then we define (

8.7.3

) as in

\autoref

sec:naive-vankampen:

(

…

,

y

n

,

p

n

,

f

⁢

k

⁢

s

)

↦

(

…

,

y

n

,

p

n

,

s

,

𝗋𝖾𝖿𝗅

g

⁢

k

⁢

s

,

g

⁢

k

⁢

s

)

,

(

…

,

x

n

,

p

n

,

s

,

𝗋𝖾𝖿𝗅

f

⁢

k

⁢

s

,

f

⁢

k

⁢

s

)

\mapsfrom

⁢

(

…

,

x

n

,

p

n

,

g

⁢

k

⁢

s

)

.

These respect the

equivalence relations

, and define

quasi-inverses

just as before.
Now suppose

t

varies along

m

s

,

s

′

⁢

(

w

)

for some

w

:

k

⁢

s

=

k

⁢

s

′

; we must show that (

8.7.3

) respects transporting along

k

¯

⁢

m

⁢

w

.
By definition of

k

¯

, this essentially boils down to transporting along

w

itself.
By the

characterization

of transport in path types, what we need to show is that

w

*

⁢

(

…

,

y

n

,

p

n

,

f

⁢

k

⁢

s

)

=

(

…

,

y

n

,

p

n

⁢

\centerdot

⁢

f

⁢

w

,

f

⁢

k

⁢

s

′

)

is mapped by (

8.7.3

) to

w

*

⁢

(

…

,

y

n

,

p

n

,

s

,

𝗋𝖾𝖿𝗅

g

⁢

k

⁢

s

,

g

⁢

k

⁢

s

)

=

(

…

,

y

n

,

p

n

,

s

,

𝗋𝖾𝖿𝗅

g

⁢

k

⁢

s

⁢

\centerdot

⁢

g

⁢

w

,

g

⁢

k

⁢

s

′

)

But this follows directly from the new

generators

we have imposed on the set-quotient

relation

defining

𝖼𝗈𝖽𝖾

.

•

The other three requisite equivalences are defined similarly.

•

Finally, since the

commutativity

(

LABEL:eq:bfa-bga-comm

) is a mere proposition, by

(

-

1

)

-connectedness of

k

we may assume that

a

=

k

⁢

s

and

a

′

=

k

⁢

s

′

, in which case it follows exactly as before.

Theorem 8.7.4

(van Kampen with a set of basepoints).

For all

u

,

v

:

P

there is an equivalence

Π

1

⁢

P

⁢

(

u

,

v

)

≃

𝖼𝗈𝖽𝖾

⁢

(

u

,

v

)

.

with

code

defined as in this section.

Proof.

Basically just like before.
To show that

𝖽𝖾𝖼𝗈𝖽𝖾

respects the new generators of the quotient relation, we use the naturality of

h

.
And to show that

𝖽𝖾𝖼𝗈𝖽𝖾

respects the equivalences such as (

8.7.3

), we need to induct on

k

¯

and on

T

in order to decompose those equivalences into their definitions, but then it becomes again simply functoriality of

f

and

g

.
The rest is easy.
In particular, no additional

argument

is required for

𝖾𝗇𝖼𝗈𝖽𝖾

∘

𝖽𝖾𝖼𝗈𝖽𝖾

, since the goal is to prove an equality in a set, and so the case of

h

is

trivial

.
∎

\autoref

thm:van-Kampen allows us to calculate the

fundamental group

of a space

A

,
even when

A

is not a set, provided

S

is a set, for in that case,
each

𝖼𝗈𝖽𝖾

⁢

(

u

,

v

)

is, by definition, a set-quotient of a

set

by a
relation. In that respect, it is an improvement over

\autoref

thm:naive-van-kampen.

Example 8.7.5

.

Suppose

S

:

≡

1

, so that

A

has a basepoint

a

:

≡

k

(

⋆

)

and is connected.
Then code for loops in the

pushout

can be identified with

alternating

sequences of loops in

π

1

⁢

(

B

,

f

⁢

(

a

)

)

and

π

1

⁢

(

C

,

g

⁢

(

a

)

)

, modulo an equivalence relation which allows us to slide elements of

π

1

⁢

(

A

,

a

)

between them (after applying

f

and

g

respectively).
Thus,

π

1

⁢

(

P

)

can be identified with the

amalgamated free product

π

1

⁢

(

B

)

*

π

1

⁢

(

A

)

π

1

⁢

(

C

)

(the pushout in the category of groups), as constructed in

\autoref

sec:free-algebras.
This (in the case when

B

and

C

are open subspaces of

P

and

A

their

intersection

) is probably the most classical version of the van Kampen theorem.

Example 8.7.6

.

As a special case of

\autoref

eg:clvk, suppose additionally that

C

:

≡

1

, so that

P

is the cofiber

B

/

A

.
Then every loop in

C

is equal to

reflexivity

, so the relations on path codes allow us to collapse all sequences to a single loop in

B

.
The additional relations require that multiplying on the left, right, or in the middle by an

element

in the image of

π

1

⁢

(

A

)

is the

identity

.
We can thus identify

π

1

⁢

(

B

/

A

)

with the quotient of the group

π

1

⁢

(

B

)

by the

normal subgroup generated by

the image of

π

1

⁢

(

A

)

.

Example 8.7.7

.

As a further special case of

\autoref

eg:cofiber, let

B

:

≡

S

1

∨

S

1

, let

A

:

≡

S

1

, and let

f

:

A

→

B

pick out the composite loop

p

⁢

\centerdot

⁢

q

⁢

\centerdot

⁢

p

-

1

⁢

\centerdot

⁢

q

-

1

, where

p

and

q

are the generating loops in the two copies of

S

1

comprising

B

.
Then

P

is a

presentation

of the torus

T

2

.
Indeed, it is not hard to identify

P

with the presentation of

T

2

as described in

\autoref

sec:hubs-spokes, using the cone on a particular loop.
Thus,

π

1

⁢

(

T

2

)

is the quotient of the

free group

on two generators (i.e.,

π

1

⁢

(

B

)

) by the relation

p

⁢

\centerdot

⁢

q

⁢

\centerdot

⁢

p

-

1

⁢

\centerdot

⁢

q

-

1

=

1

.
This clearly yields the free

abelian

group on two generators, which is

Z

×

Z

.

Example 8.7.8

.

More generally, any CW complex can be obtained by repeatedly “coning off” spheres, as described in

\autoref

sec:hubs-spokes.
That is, we start with a set

X

0

of points (“0-cells”), which is the “0-skeleton” of the CW complex.
We take the pushout

\xymatrix

S_1

×

𝕊

^0

\ar

[r]^-f_1

\ar

[d] &
X_0

\ar

[d]

1

\ar

[r] &
X_1

for some set

S

1

of 1-cells and some family

f

1

of “

attaching maps

”, obtaining the “1-skeleton”

X

1

.
Then we take the pushout

\xymatrix

S_2

×

𝕊

^1

\ar

[r]^f_2

\ar

[d] &
X_1

\ar

[d]

1

\ar

[r] &
X_2

for some set

S

2

of 2-cells and some family

f

2

of attaching maps, obtaining the 2-skeleton

X

2

, and so on.
The fundamental group of each pushout can be calculated from the van Kampen theorem: we obtain the group presented by generators derived from the 1-skeleton, and relations derived from

S

2

and

f

2

.
The pushouts after this stage do not alter the fundamental group, since

π

1

⁢

(

S

n

)

is trivial for

n

>

1

(see

\autoref

sec:pik-le-n).

Example 8.7.9

.

In particular, suppose given any presentation of a (set-)group

G

=

⟨

X

∣

R

⟩

, with

X

a set of generators and

R

a set of words in these generators.
Let

B

:

≡

⋁

X

S

1

and

A

:

≡

⋁

R

S

1

, with

f

:

A

→

B

sending each copy of

S

1

to the corresponding word in the generating loops of

B

.
It follows that

π

1

⁢

(

P

)

≅

G

; thus we have constructed a connected type whose fundamental group is

G

.
Since any group has a presentation, any group is the fundamental group of some type.
If we 1-truncate such a type, we obtain a type whose only nontrivial

homotopy group

is

G

; this is called an

Eilenberg–Mac Lane space

K

⁢

(

G

,

1

)

.

Title

8.7.2 The van Kampen theorem with a set of basepoints

\metatable

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