7.3 Truncations

7.3 Truncations

In §3.7 (

http://planetmath.org/37propositionaltruncation

) we introduced the propositional

truncation

, which makes the “best approximation” of a type that is a mere
proposition, i.e. a

(

-

1

)

-type.
In §6.9 (

http://planetmath.org/69truncations

) we constructed this truncation as a

higher inductive type

, and gave one way to generalize it to a
0-truncation.
We now explain a better

generalization

of this, which

truncates

any type into an

n

-type for any

n

≥

-

2

; in classical

homotopy theory

this would be called its

n

th

Postnikov section

.

The idea is to make use of (

http://planetmath.org/72uniquenessofidentityproofsandhedbergstheorem# utoref

), which states that

A

is an

n

-type just when

Ω

n

+

1

⁢

(

A

,

a

)

is

contractible

for
all

a

:

A

, and Lemma 6.5.4 (

http://planetmath.org/65suspensions#Thmprelem3

), which implies that

Ω

n

+

1

⁢

(

A

,

a

)

≃

𝖬𝖺𝗉

*

⁢

(

𝕊

n

+

1

,

(

A

,

a

)

)

,

where

𝕊

n

+

1

is equipped with some

basepoint

which we may as well call

𝖻𝖺𝗌𝖾

.
However,

contractibility

of

𝖬𝖺𝗉

*

⁢

(

𝕊

n

+

1

,

(

A

,

a

)

)

is something that we can ensure directly by giving path constructors.

We might first of all try to define

∥

A

∥

n

to be generated by a

function

|

–

|

n

:

A

→

∥

A

∥

n

, together with for each

r

:

𝕊

n

+

1

→

∥

A

∥

n

and each

x

:

𝕊

n

+

1

, a path

s

r

⁢

(

x

)

:

r

⁢

(

x

)

=

r

⁢

(

𝖻𝖺𝗌𝖾

)

.
But this does not quite work, for the same reason that Remark 6.7.1 (

http://planetmath.org/67hubsandspokes#Thmprermk1

) fails.
Instead, we use the full “hub and spoke” construction as in §6.7 (

http://planetmath.org/67hubsandspokes

).

Thus, for

n

≥

-

1

, we take

∥

A

∥

n

to be the higher inductive type generated by:

•

a function

|

–

|

n

:

A

→

∥

A

∥

n

,

•

for each

r

:

𝕊

n

+

1

→

∥

A

∥

n

, a

hub

point

h

(

r

)

:

∥

A

∥

n

, and

•

for each

r

:

𝕊

n

+

1

→

∥

A

∥

n

and each

x

:

𝕊

n

+

1

, a

spoke

path

s

r

⁢

(

x

)

:

r

⁢

(

x

)

=

h

⁢

(

r

)

.

The

existence

of these constructors is now enough to show:

Lemma 7.3.1

.

∥

A

∥

n

is an

n

-type.

Proof.

By (

http://planetmath.org/72uniquenessofidentityproofsandhedbergstheorem# utoref

), it suffices to show that

Ω

n

+

1

(

∥

A

∥

n

,

b

)

is contractible for all

b

:

∥

A

∥

n

, which by
Lemma 6.5.4 (

http://planetmath.org/65suspensions#Thmprelem3

) is

equivalent

to

𝖬𝖺𝗉

*

(

𝕊

n

+

1

,

(

∥

A

∥

n

,

b

)

)

.

As center of

contraction

for the latter, we choose the function

c

b

:

𝕊

n

+

1

→

∥

A

∥

n

which is

constant

at

b

, together with

𝗋𝖾𝖿𝗅

b

:

c

b

⁢

(

𝖻𝖺𝗌𝖾

)

=

b

.

Now, an arbitrary element of

𝖬𝖺𝗉

*

(

𝕊

n

+

1

,

(

∥

A

∥

n

,

b

)

)

consists of a map

r

:

𝕊

n

+

1

→

∥

A

∥

n

together with a path

p

:

r

⁢

(

𝖻𝖺𝗌𝖾

)

=

b

.
By function extensionality, to show

r

=

c

b

it suffices to give, for each

x

:

𝕊

n

+

1

, a path

r

⁢

(

x

)

=

c

b

⁢

(

x

)

≡

b

.
We choose this to be the composite

s

r

⁢

(

x

)

⁢

\centerdot

⁢

s

r

⁢

(

𝖻𝖺𝗌𝖾

)

-

1

⁢

\centerdot

⁢

p

, where

s

r

⁢

(

x

)

is the spoke at

x

.

Finally, we must show that when transported along this

equality

r

=

c

b

, the path

p

becomes

𝗋𝖾𝖿𝗅

b

.
By transport in path types, this means we need

(

s

r

⁢

(

𝖻𝖺𝗌𝖾

)

⁢

\centerdot

⁢

s

r

⁢

(

𝖻𝖺𝗌𝖾

)

-

1

⁢

\centerdot

⁢

p

)

-

1

⁢

\centerdot

⁢

p

=

𝗋𝖾𝖿𝗅

b

.

But this is immediate from path

operations

.
∎

(This construction fails for

n

=

-

2

, but in that case we can simply define

∥

A

∥

-

2

:

≡

𝟏

for all

A

.
From now on we assume

n

≥

-

1

.)

To show the desired

universal property

of the

n

-truncation, we need the

induction

principle.
We extract this from the constructors in the usual way; it says that given

P

:

∥

A

∥

n

→

𝒰

together with

•

For each

a

:

A

, an

element

g

(

a

)

:

P

(

|

a

|

n

)

,

•

For each

r

:

𝕊

n

+

1

→

∥

A

∥

n

and

r

′

:

∏

(

x

:

𝕊

n

+

1

)

P

⁢

(

r

⁢

(

x

)

)

, an element

h

′

⁢

(

r

,

r

′

)

:

P

⁢

(

h

⁢

(

r

)

)

,

•

For each

r

:

𝕊

n

+

1

→

∥

A

∥

n

and

r

′

:

∏

(

x

:

𝕊

n

+

1

)

P

⁢

(

r

⁢

(

x

)

)

, and each

x

:

𝕊

n

+

1

, a dependent path

r

′

⁢

(

x

)

=

s

r

⁢

(

x

)

P

h

′

⁢

(

r

,

r

′

)

,

there exists a

section

f

:

∏

(

x

:

∥

A

∥

n

)

P

⁢

(

x

)

with

f

(

|

a

|

n

)

≡

g

(

a

)

for all

a

:

A

.
To make this more useful, we reformulate it as follows.

Theorem 7.3.2

.

For any type family

P

:

∥

A

∥

n

→

U

such that each

P

⁢

(

x

)

is an

n

-type, and any function

g

:

∏

(

a

:

A

)

P

(

|

a

|

n

)

, there
exists a section

f

:

∏

(

x

:

∥

A

∥

n

)

P

⁢

(

x

)

such that

f

(

|

a

|

n

)

:

≡

g

(

a

)

for all

a

:

A

.

Proof.

It will suffice to

construct

the second and third data listed above, since

g

has exactly the type of the first datum.
Given

r

:

𝕊

n

+

1

→

∥

A

∥

n

and

r

′

:

∏

(

x

:

𝕊

n

+

1

)

P

⁢

(

r

⁢

(

x

)

)

, we have

h

(

r

)

:

∥

A

∥

n

and

s

r

:

∏

(

x

:

𝕊

n

+

1

)

(

r

(

x

)

=

h

(

r

)

)

.
Define

t

:

𝕊

n

+

1

→

P

⁢

(

h

⁢

(

r

)

)

by

t

(

x

)

:

≡

s

r

(

x

)

*

(

r

′

(

x

)

)

.
Then since

P

⁢

(

h

⁢

(

r

)

)

is

n

-truncated, there exists a point

u

:

P

⁢

(

h

⁢

(

r

)

)

and a contraction

v

:

∏

(

x

:

𝕊

n

+

1

)

(

t

(

x

)

=

u

)

.
Define

h

′

(

r

,

r

′

)

:

≡

u

, giving the second datum.
Then (recalling the definition of dependent paths),

v

has exactly the type required of the third datum.
∎

In particular, if

E

is some

n

-type, we can consider the constant family of types equal to

E

for every point of

A

.

Thus, every map

f

:

A

→

E

can be extended to a map

𝖾𝗑𝗍

(

f

)

:

∥

A

∥

n

→

E

defined by

𝖾𝗑𝗍

(

f

)

(

|

a

|

n

)

:

≡

f

(

a

)

; this is the

recursion

principle

for

∥

A

∥

n

.

The induction principle also implies a uniqueness principle for functions of this form.
Namely, if

E

is an

n

-type and

g

,

g

′

:

∥

A

∥

n

→

E

are such
that

g

(

|

a

|

n

)

=

g

′

(

|

a

|

n

)

for every

a

:

A

, then

g

⁢

(

x

)

=

g

′

⁢

(

x

)

for all

x

:

∥

A

∥

n

, since the type

g

⁢

(

x

)

=

g

′

⁢

(

x

)

is an

n

-type.
Thus,

g

=

g

′

.
This yields the following universal property.

Lemma 7.3.3

(Universal property of truncations).

Let

n

≥

-

2

,

A

:

U

and

B

:

n

⁢

-

⁢

Type

. The following map is an

equivalence

:

{

(

∥

A

∥

n

→

B

)

⟶

(

A

→

B

)

g

⟼

g

∘

|

–

|

n

Proof.

Given that

B

is

n

-truncated, any

f

:

A

→

B

can be extended to a map

𝖾𝗑𝗍

(

f

)

:

∥

A

∥

n

→

B

.
The map

𝖾𝗑𝗍

⁢

(

f

)

∘

|

–

|

n

is equal to

f

, because for every

a

:

A

we have

𝖾𝗑𝗍

(

f

)

(

|

a

|

n

)

=

f

(

a

)

by definition.
And the map

𝖾𝗑𝗍

⁢

(

g

∘

|

–

|

n

)

is equal to

g

, because they both send

|

a

|

n

to

g

(

|

a

|

n

)

.
∎

In

categorical

language

, this says that the

n

-types form a

reflective subcategory

of the

category

of types.
(To state this fully precisely, one ought to use the language of

(

∞

,

1

)

-categories.)
In particular, this implies that the

n

-truncation is functorial:
given

f

:

A

→

B

, applying the recursion principle to the composite

A

→

𝑓

B

→

∥

B

∥

n

yields a map

∥

f

∥

n

:

∥

A

∥

n

→

∥

B

∥

n

.
By definition, we have a

homotopy

𝗇𝖺𝗍

n

f

:

∏

a

:

A

∥

f

∥

n

(

|

a

|

n

)

=

|

f

(

a

)

|

n

,

(7.3.4)

expressing

naturality

of the maps

|

–

|

n

.

Uniqueness implies functoriality laws such as

∥

g

∘

f

∥

n

=

∥

g

∥

n

∘

∥

f

∥

n

and

∥

𝗂𝖽

A

∥

n

=

𝗂𝖽

∥

A

∥

n

, with attendant coherence laws.
We also have higher functoriality, for

instance

:

Lemma 7.3.5

.

Given

f

,

g

:

A

→

B

and a homotopy

h

:

f

∼

g

, there is an induced homotopy

∥

h

∥

n

:

∥

f

∥

n

∼

∥

g

∥

n

such that the composite

\includegraphics

⁢

H

⁢

o

⁢

T

⁢

T

f

⁢

i

⁢

g

7

⁢

.3.1

.

p

⁢

n

⁢

g

(7.3.5)

is equal to

ap

|

–

|

n

⁢

(

h

⁢

(

a

)

)

.

Proof.

First, we indeed have a homotopy with components

𝖺𝗉

|

–

|

n

(

h

(

a

)

)

:

|

f

(

a

)

|

n

=

|

g

(

a

)

|

n

.
Composing on either sides with the paths

|

f

(

a

)

|

n

=

∥

f

∥

n

(

|

a

|

n

)

and

|

g

(

a

)

|

n

=

∥

g

∥

n

(

|

a

|

n

)

, which arise from the definitions of

∥

f

∥

n

and

∥

g

∥

n

, we obtain a homotopy

(

∥

f

∥

n

∘

|

–

|

n

)

∼

(

∥

g

∥

n

∘

|

–

|

n

)

, and hence an equality by function extensionality.
But since

(

–

∘

|

–

|

n

)

is an equivalence, there must be a path

∥

f

∥

n

=

∥

g

∥

n

inducing it, and the coherence laws for function extensionality imply (

7.3.5

).
∎

The following observation about reflective subcategories is also standard.

Corollary 7.3.7

.

A type

A

is an

n

-type if and only if

|

–

|

n

:

A

→

∥

A

∥

n

is an equivalence.

Proof.

“If” follows from

closure

of

n

-types under equivalence.
On the other hand, if

A

is an

n

-type, we can define

𝖾𝗑𝗍

(

𝗂𝖽

A

)

:

∥

A

∥

n

→

A

.
Then we have

𝖾𝗑𝗍

⁢

(

𝗂𝖽

A

)

∘

|

–

|

n

=

𝗂𝖽

A

:

A

→

A

by
definition. In order to prove that

|

–

|

n

∘

𝖾𝗑𝗍

⁢

(

𝗂𝖽

A

)

=

𝗂𝖽

∥

A

∥

n

, we only need to prove
that

|

–

|

n

∘

𝖾𝗑𝗍

⁢

(

𝗂𝖽

A

)

∘

|

–

|

n

=

𝗂𝖽

∥

A

∥

n

∘

|

–

|

n

.
This is again true:

\includegraphics

HoTT_fig_7.3.2.png

∎

The category of

n

-types also has some special properties not possessed by all reflective subcategories.
For instance, the reflector

∥

-

∥

n

preserves finite

products

.

Theorem 7.3.8

.

For any types

A

and

B

, the

induced map

∥

A

×

B

∥

n

→

∥

A

∥

n

×

∥

B

∥

n

is an equivalence.

Proof.

It suffices to show that

∥

A

∥

n

×

∥

B

∥

n

has the same universal property as

∥

A

×

B

∥

n

.
Thus, let

C

be an

n

-type; we have

(

∥

A

∥

n

×

∥

B

∥

n

→

C

)

=

(

∥

A

∥

n

→

(

∥

B

∥

n

→

C

)

)

=

(

∥

A

∥

n

→

(

B

→

C

)

)

=

(

A

→

(

B

→

C

)

)

=

(

A

×

B

→

C

)

using the universal properties of

∥

B

∥

n

and

∥

A

∥

n

, along with the fact that

B

→

C

is an

n

-type since

C

is.
It is straightforward to verify that this equivalence is given by composing with

|

–

|

n

×

|

–

|

n

, as needed.
∎

The following related fact about dependent sums is often useful.

Theorem 7.3.9

.

Let

P

:

A

→

U

be a family of types. Then there is an equivalence

∥

∑

x

:

A

∥

P

(

x

)

∥

n

∥

n

≃

∥

∑

x

:

A

P

(

x

)

∥

n

.

Proof.

We use the induction principle of

n

-truncation several times to construct
functions

φ

:

∥

∑

x

:

A

∥

P

(

x

)

∥

n

∥

n

→

∥

∑

x

:

A

P

(

x

)

∥

n

ψ

:

∥

∑

x

:

A

P

(

x

)

∥

n

→

∥

∑

x

:

A

∥

P

(

x

)

∥

n

∥

n

and homotopies

H

:

φ

∘

ψ

∼

𝗂𝖽

and

K

:

ψ

∘

φ

∼

𝗂𝖽

exhibiting them as

quasi-inverses

.
We define

φ

by setting

φ

(

|

(

x

,

|

u

|

n

)

|

n

)

:

≡

|

(

x

,

u

)

|

n

.
We define

ψ

by setting

ψ

(

|

(

x

,

u

)

|

n

)

:

≡

|

(

x

,

|

u

|

n

)

|

n

.
Then we define

H

(

|

(

x

,

u

)

|

n

)

:

≡

𝗋𝖾𝖿𝗅

|

(

x

,

u

)

|

n

and

K

(

|

(

x

,

|

u

|

n

)

|

n

)

:

≡

𝗋𝖾𝖿𝗅

|

(

x

,

|

u

|

n

)

|

n

.
∎

Corollary 7.3.10

.

If

A

is an

n

-type and

P

:

A

→

U

is any type family, then

∑

a

:

A

∥

P

(

a

)

∥

n

≃

∥

∑

a

:

A

P

(

a

)

∥

n

Proof.

If

A

is an

n

-type, then the left-hand type above is already an

n

-type, hence equivalent to its

n

-truncation; thus this follows from

Theorem

7.3.9 (

http://planetmath.org/73truncations#Thmprethm3

).
∎

We can characterize the

path spaces

of a truncation using the same method that we used in §2.12 (

http://planetmath.org/212coproducts

),§2.13 (

http://planetmath.org/213naturalnumbers

) for

coproducts

and

natural numbers

(and which we will use in

http://planetmath.org/node/87582

Chapter 8 to calculate

homotopy groups

).
Unsurprisingly, the path spaces in the

(

n

+

1

)

-truncation of

A

are the

n

-truncations of the path spaces of

A

.
Indeed, for any

x

,

y

:

A

there is a

canonical map

f

:

∥

x

=

A

y

∥

n

→

(

|

x

|

n

+

1

=

∥

A

∥

n

+

1

|

y

|

n

+

1

)

(7.3.11)

defined by

f

(

|

p

|

n

)

:

≡

𝖺𝗉

|

–

|

n

+

1

(

p

)

.

This definition uses the recursion principle for

∥

–

∥

n

, which is correct because

∥

A

∥

n

+

1

is

(

n

+

1

)

-truncated, so that the
codomain of

f

is

n

-truncated.

Theorem 7.3.12

.

For any

A

and

x

,

y

:

A

and

n

≥

-

2

, the map (

7.3.11

) is an equivalence; thus we have

∥

x

=

A

y

∥

n

≃

(

|

x

|

n

+

1

=

∥

A

∥

n

+

1

|

y

|

n

+

1

)

.

Proof.

The proof is a simple application of the encode-decode method:
As in previous situations, we cannot directly define a quasi-inverse to the map (

7.3.11

) because there is no way to induct on an
equality between

|

x

|

n

+

1

and

|

y

|

n

+

1

.
Thus, instead we generalize its type, in order to have general elements of the type

∥

A

∥

n

+

1

instead of

|

x

|

n

+

1

and

|

y

|

n

+

1

.
Define

P

:

∥

A

∥

n

+

1

→

∥

A

∥

n

+

1

→

n

-

𝖳𝗒𝗉𝖾

by

P

(

|

x

|

n

+

1

,

|

y

|

n

+

1

)

:

≡

∥

x

=

A

y

∥

n

This definition is correct because

∥

x

=

A

y

∥

n

is

n

-truncated, and

n

⁢

-

⁢

𝖳𝗒𝗉𝖾

is

(

n

+

1

)

-truncated by
Theorem 7.1.11 (

http://planetmath.org/71definitionofntypes#Thmprethm7

).
Now for every

u

,

v

:

∥

A

∥

n

+

1

, there is a map

𝖾𝗇𝖼𝗈𝖽𝖾

:

P

(

u

,

v

)

→

(

u

=

∥

A

∥

n

+

1

v

)

defined for

u

=

|

x

|

n

+

1

and

v

=

|

y

|

n

+

1

and

p

:

x

=

y

by

𝖾𝗇𝖼𝗈𝖽𝖾

(

|

p

|

n

)

:

≡

𝖺𝗉

|

¯

|

n

+

1

(

p

)

.

Since the codomain of

𝖾𝗇𝖼𝗈𝖽𝖾

is

n

-truncated, it suffices to define it only for

u

and

v

of this form, and then it’s just the same
definition as before.
We also define a function

r

:

∏

u

:

∥

A

∥

n

+

1

P

⁢

(

u

,

u

)

by induction on

u

, where

r

(

|

x

|

n

+

1

)

:

≡

|

𝗋𝖾𝖿𝗅

x

|

n

.

Now we can define an inverse map

𝖽𝖾𝖼𝗈𝖽𝖾

:

(

u

=

∥

A

∥

n

+

1

v

)

→

P

(

u

,

v

)

by

𝖽𝖾𝖼𝗈𝖽𝖾

(

p

)

:

≡

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

v

↦

P

⁢

(

u

,

v

)

(

p

,

r

(

u

)

)

.

To show that the composite

(

u

=

∥

A

∥

n

+

1

v

)

→

𝖽𝖾𝖼𝗈𝖽𝖾

P

(

u

,

v

)

→

𝖾𝗇𝖼𝗈𝖽𝖾

(

u

=

∥

A

∥

n

+

1

v

)

is the

identity function

, by path induction it suffices to check it for

𝗋𝖾𝖿𝗅

u

:

u

=

u

, in which case what we need to know is that

𝖽𝖾𝖼𝗈𝖽𝖾

⁢

(

r

⁢

(

u

)

)

=

𝗋𝖾𝖿𝗅

u

.
But since this is an

n

-type, hence also an

(

n

+

1

)

-type, we may assume

u

≡

|

x

|

n

+

1

, in which case it follows by definition
of

r

and

𝖽𝖾𝖼𝗈𝖽𝖾

.
Finally, to show that

P

(

u

,

v

)

→

𝖾𝗇𝖼𝗈𝖽𝖾

(

u

=

∥

A

∥

n

+

1

v

)

→

𝖽𝖾𝖼𝗈𝖽𝖾

P

(

u

,

v

)

is the identity function, since this goal is again an

n

-type, we may assume that

u

=

|

x

|

n

+

1

and

v

=

|

y

|

n

+

1

and that we are
considering

|

p

|

n

:

P

(

|

x

|

n

+

1

,

|

y

|

n

+

1

)

for some

p

:

x

=

y

.
Then we have

𝖽𝖾𝖼𝗈𝖽𝖾

(

𝖾𝗇𝖼𝗈𝖽𝖾

(

|

p

|

n

)

)

=

𝖽𝖾𝖼𝗈𝖽𝖾

⁢

(

𝖺𝗉

|

¯

|

n

+

1

⁢

(

p

)

)

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

v

↦

P

(

|

x

|

n

+

1

,

v

)

(

𝖺𝗉

|

¯

|

n

+

1

(

p

)

,

|

𝗋𝖾𝖿𝗅

x

|

n

)

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

v

↦

∥

u

=

v

∥

n

(

p

,

|

𝗋𝖾𝖿𝗅

x

|

n

)

=

|

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

v

↦

(

u

=

v

)

(

p

,

𝗋𝖾𝖿𝗅

x

)

|

n

=

|

p

|

n

.

This

completes

the proof that

𝖾𝗇𝖼𝗈𝖽𝖾

and

𝖽𝖾𝖼𝗈𝖽𝖾

are quasi-inverses.
The stated result is then the special case where

u

=

|

x

|

n

+

1

and

v

=

|

y

|

n

+

1

.
∎

Corollary 7.3.13

.

Let

n

≥

-

2

and

(

A

,

a

)

be a pointed type. Then

∥

Ω

(

A

,

a

)

∥

n

=

Ω

(

∥

(

A

,

a

)

∥

n

+

1

)

Proof.

This is a special case of the previous lemma where

x

=

y

=

a

.
∎

Corollary 7.3.14

.

Let

n

≥

-

2

and

k

≥

0

and

(

A

,

a

)

a pointed type. Then

∥

Ω

k

(

A

,

a

)

∥

n

=

Ω

k

(

∥

(

A

,

a

)

∥

n

+

k

)

.

Proof.

By induction on

k

, using the recursive definition of

Ω

k

.
∎

We also observe that “truncations are cumulative”: if we truncate to an

n

-type and then to a

k

-type with

k

≤

n

, then we might as
well have truncated directly to a

k

-type.

Lemma 7.3.15

.

Let

k

,

n

≥

-

2

with

k

≤

n

and

A

:

U

. Then

∥

∥

A

∥

n

∥

k

=

∥

A

∥

k

.

Proof.

We define two maps

f

:

∥

∥

A

∥

n

∥

k

→

∥

A

∥

k

and

g

:

∥

A

∥

k

→

∥

∥

A

∥

n

∥

k

by

f

(

|

|

a

|

n

|

k

)

:

≡

|

a

|

k

and

g

(

|

a

|

k

)

:

≡

|

|

a

|

n

|

k

.

The map

f

is well-defined because

∥

A

∥

k

is

k

-truncated and also

n

-truncated (because

k

≤

n

), and the map

g

is well-defined because

∥

∥

A

∥

n

∥

k

is

k

-truncated.

The composition

f

∘

g

:

∥

A

∥

k

→

∥

A

∥

k

satisfies

(

f

∘

g

)

(

|

a

|

k

)

=

|

a

|

k

, hence

f

∘

g

=

𝗂𝖽

∥

A

∥

k

.
Similarly, we have

(

g

∘

f

)

(

|

|

a

|

n

|

k

)

=

|

|

a

|

n

|

k

and hence

g

∘

f

=

𝗂𝖽

∥

∥

A

∥

n

∥

k

.
∎

Title

7.3 Truncations

\metatable

Generated on Thu Feb 8 19:39:16 2018 by

LaTeXML