6.7 Hubs and spokes

6.7 Hubs and spokes

In

topology

, one usually speaks of building

CW complexes

by attaching

n

-dimensional discs along their

(

n

-

1

)

-dimensional

boundary

spheres.
However, another way to express this is by gluing in the

cone

on an

(

n

-

1

)

-dimensional sphere.
That is, we regard a disc as consisting of a cone point (or “hub”), with

meridians

(or “spokes”) connecting that point to every point on the boundary, continuously, as shown in Figure 6.3 (

http://planetmath.org/67hubsandspokes#S0.F3

).

\includegraphics

HoTT_fig_6.7.1.png

Figure 6.3:

A 2-disc made out of a hub and spokes

We can use this idea to express

higher inductive types

containing

n

-dimensional path-constructors for

n

>

1

in terms of ones containing only 1-dimensional path-constructors.
The point is that we can obtain an

n

-dimensional path as a

continuous

family of 1-dimensional paths parametrized by an

(

n

-

1

)

-dimensional object.
The simplest

(

n

-

1

)

-dimensional object to use is the

(

n

-

1

)

-sphere, although in some cases a different one may be preferable.
(Recall that we were able to define the spheres in §6.5 (

http://planetmath.org/65suspensions

) inductively using

suspensions

, which involve only 1-dimensional path constructors.
Indeed, suspension can also be regarded as an

instance

of this idea, since it involves a family of 1-dimensional paths parametrized by the type being suspended.)

For instance, the torus

T

2

from the previous

section

could be defined instead to be generated by:

•

a point

b

:

T

2

,

•

a path

p

:

b

=

b

,

•

another path

q

:

b

=

b

,

•

a point

h

:

T

2

, and

•

for each

x

:

𝕊

1

, a path

s

⁢

(

x

)

:

f

⁢

(

x

)

=

h

, where

f

:

𝕊

1

→

T

2

is defined by

f

(

𝖻𝖺𝗌𝖾

)

:

≡

b

and

f

(

𝗅𝗈𝗈𝗉

)

:=

p

\centerdot

q

\centerdot

p

-

1

\centerdot

q

-

1

.

The

induction

principle for this version of the torus says that 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

′

,

•

a point

h

′

:

P

⁢

(

h

)

, and

•

for each

x

:

𝕊

1

, a path

g

⁢

(

x

)

=

s

⁢

(

x

)

P

h

′

, where

g

:

∏

(

x

:

𝕊

1

)

P

⁢

(

f

⁢

(

x

)

)

is defined by

g

(

𝖻𝖺𝗌𝖾

)

:

≡

b

′

and

𝖺𝗉𝖽

g

(

𝗅𝗈𝗈𝗉

)

:=

p

′

\centerdot

q

′

\centerdot

(

p

′

)

-

1

\centerdot

(

q

′

)

-

1

.

Note that there is no need for dependent 2-paths or

𝖺𝗉𝖽

2

.
We leave it to the reader to write out the computation rules.

Remark 6.7.1

.

One might question the need for introducing the hub point

h

; why couldn’t we instead simply add paths continuously relating the boundary of the disc to a point

on

that boundary, as shown in Figure 6.4 (

http://planetmath.org/67hubsandspokes#S0.F5

)?
This does work, but not as well.
For if, given some

f

:

S

1

→

X

, we give a path constructor connecting each

f

⁢

(

x

)

to

f

⁢

(

base

)

, then what we end up with is more like the picture in Figure 6.5 (

http://planetmath.org/67hubsandspokes#S0.F5

) of a cone whose vertex is twisted around and glued to some point on its base.
The problem is that the specified path from

f

⁢

(

base

)

to itself may not be

reflexivity

.
We could add a 2-dimensional path constructor ensuring this, but using a separate hub avoids the need for any path constructors of

dimension

above

1

.

\includegraphics

HoTT_fig_6.7.2a.png

Figure 6.4:

Hubless spokes

\includegraphics

HoTT_fig_6.7.2b.png

Figure 6.5:

Hubless spokes, II

Remark 6.7.2

.

Note also that this “translation” of higher paths into 1-paths does not preserve judgmental computation rules for these paths, though it does preserve propositional ones.

Title

6.7 Hubs and spokes

\metatable

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