6.6 Cell complexes

6.6 Cell complexes

In classical

topology

, a

cell complex

is a space obtained by successively attaching discs along their

boundaries

.
It is called a

CW complex

if the boundary of an

n

-dimensional disc is constrained to lie in the discs of

dimension

strictly

less than

n

(the

(

n

-

1

)

-

skeleton

).

Any finite CW complex can be presented as a

higher inductive type

, by turning

n

-dimensional discs into

n

-dimensional paths and partitioning the image of the

attaching map

into a source and a target, with each written as a composite of lower dimensional paths.
Our explicit

definitions

of

𝕊

1

and

𝕊

2

in §6.4 (

http://planetmath.org/64circlesandspheres

) had this form.

Another example is the torus

T

2

, which is generated by:

•

a point

b

:

T

2

,

•

a path

p

:

b

=

b

,

•

another path

q

:

b

=

b

, and

•

a 2-path

t

:

p

⁢

\centerdot

⁢

q

=

q

⁢

\centerdot

⁢

p

.

Perhaps the easiest way to see that this is a torus is to start with a

rectangle

, having four corners

a

,

b

,

c

,

d

, four edges

p

,

q

,

r

,

s

, and an

interior

which is manifestly a 2-path

t

from

p

⁢

\centerdot

⁢

q

to

r

⁢

\centerdot

⁢

s

:

\includegraphics

HoTT_fig_6.6.1.png

Now identify the edge

r

with

q

and the edge

s

with

p

, resulting in also identifying all four corners.
Topologically, this identification can be seen to produce a torus.

The

induction

principle for the torus is the trickiest of any we’ve written out so far.
Given

P

:

T

2

→

𝒰

, for a

section

∏

(

x

:

T

2

)

P

⁢

(

x

)

we require

•

a point

b

′

:

P

⁢

(

b

)

,

•

a path

p

′

:

b

′

=

p

P

b

′

,

•

a path

q

′

:

b

′

=

q

P

b

′

, and

•

a 2-path

t

′

between the “composites”

p

′

⁢

\centerdot

⁢

q

′

and

q

′

⁢

\centerdot

⁢

p

′

, lying over

t

.

In order to make sense of this last datum, we need a composition operation for dependent paths, but this is not hard to define.
Then the induction principle gives a

function

f

:

∏

(

x

:

T

2

)

P

⁢

(

x

)

such that

f

⁢

(

b

)

≡

b

′

and

𝖺𝗉𝖽

f

(

p

)

=

p

′

and

𝖺𝗉𝖽

f

(

q

)

=

q

′

and something like “

𝖺𝗉𝖽

f

2

(

t

)

=

t

′

”.
However, this is not well-typed as it stands, firstly because the equalities

𝖺𝗉𝖽

f

(

p

)

=

p

′

and

𝖺𝗉𝖽

f

(

q

)

=

q

′

are not judgmental, and secondly because

𝖺𝗉𝖽

f

only

preserves

path

concatenation

up to

homotopy

.
We leave the details to the reader (see

http://planetmath.org/node/87797

Exercise 6.1).

Of course, another definition of the torus is

T

2

:

≡

𝕊

1

×

𝕊

1

(in

http://planetmath.org/node/87800

Exercise 6.3 we ask the reader to verify the equivalence of the two).
The cell-complex definition, however, generalizes easily to other spaces without such descriptions, such as the

Klein bottle

, the

projective plane

, etc.
But it does get increasingly difficult to write down the induction principles, requiring us to define notions of dependent

n

-paths and of

𝖺𝗉𝖽

acting on

n

-paths.
Fortunately, once we have the spheres in hand, there is a way around this.

Title

6.6 Cell complexes

\metatable

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