3.7 Propositional truncation

3.7 Propositional truncation

The

propositional

truncation

, also called the

(

-

1

)

-truncation

,

bracket type

, or

squash type

, is an additional type former which “squashes” or “

truncates

” a type down to a mere proposition, forgetting all information

contained

in inhabitants of that type other than their

existence

.

More precisely, for any type

A

, there is a type

∥

A

∥

.
It has two constructors:

•

For any

a

:

A

we have

|

a

|

:

∥

A

∥

.

•

For any

x

,

y

:

∥

A

∥

, we have

x

=

y

.

The first constructor means that if

A

is inhabited, so is

∥

A

∥

.
The second ensures that

∥

A

∥

is a mere proposition; usually we leave the witness of this fact nameless.

The

recursion

principle of

∥

A

∥

says that:

•

If

B

is a mere proposition and we have

f

:

A

→

B

, then there is an induced

g

:

∥

A

∥

→

B

such that

g

(

|

a

|

)

≡

f

(

a

)

for all

a

:

A

.

In other words, any mere proposition which follows from (the inhabitedness of)

A

already follows from

∥

A

∥

.
Thus,

∥

A

∥

, as a mere proposition, contains no more information than the inhabitedness of

A

.
(There is also an

induction

principle for

∥

A

∥

, but it is not especially useful; see

http://planetmath.org/node/87806

Exercise 3.20.)

In

http://planetmath.org/node/87799

Exercise 3.17,

http://planetmath.org/node/87834

Exercise 3.18,§6.9 (

http://planetmath.org/69truncations

) we will describe some ways to construct

∥

A

∥

in terms of more general things.
For now, we simply assume it as an additional rule alongside those of

http://planetmath.org/node/87533

Chapter 1.

With the propositional truncation, we can extend the “logic of mere propositions” to cover

disjunction

and the

existential quantifier

.
Specifically,

∥

A

+

B

∥

is a mere propositional version of “

A

or

B

”, which does not “remember” the information of which disjunct is true.

The recursion principle of truncation implies that we can still do a case

analysis

on

∥

A

+

B

∥

when attempting to prove a mere proposition

.
That is, suppose we have an

assumption

u

:

∥

A

+

B

∥

and we are trying to prove a mere proposition

Q

.
In other words, we are trying to define an element of

∥

A

+

B

∥

→

Q

.
Since

Q

is a mere proposition, by the recursion principle for propositional truncation, it suffices to construct a

function

A

+

B

→

Q

.
But now we can use case analysis on

A

+

B

.

Similarly, for a type family

P

:

A

→

𝒰

, we can consider

∥

∑

(

x

:

A

)

P

⁢

(

x

)

∥

, which is a mere propositional version of “there exists an

x

:

A

such that

P

⁢

(

x

)

”.
As for disjunction, by combining the induction principles of truncation and

Σ

-types, if we have an assumption of type

∥

∑

(

x

:

A

)

P

⁢

(

x

)

∥

, we may introduce new assumptions

x

:

A

and

y

:

P

⁢

(

x

)

when attempting to prove a mere proposition

.
In other words, if we know that there exists some

x

:

A

such that

P

⁢

(

x

)

, but we don’t have a particular such

x

in hand, then we are free to make use of such an

x

as long as we aren’t trying to construct anything which might depend on the particular value of

x

.
Requiring the codomain to be a mere proposition expresses this independence of the result on the witness, since all possible inhabitants of such a type must be equal.

For the purposes of set-level mathematics in

http://planetmath.org/node/87585

Chapter 11,

http://planetmath.org/node/87584

Chapter 10,
where we deal mostly with sets and mere propositions, it is convenient to use the
traditional logical notations to refer only to “propositionally truncated logic”.

Definition 3.7.1

.

We define

traditional logical notation

using truncation as follows, where

P

and

Q

denote mere propositions (or families thereof):

⊤

:

≡

1

⊥

:

≡

0

P

∧

Q

:

≡

P

×

Q

P

⇒

Q

:

≡

P

→

Q

P

⇔

Q

:

≡

P

=

Q

¬

⁢

P

:

≡

P

→

𝟎

P

∨

Q

:

≡

∥

P

+

Q

∥

∀

(

x

:

A

)

.

P

(

x

)

:

≡

∏

x

:

A

P

(

x

)

∃

(

x

:

A

)

.

P

(

x

)

:

≡

∥

∑

x

:

A

P

(

x

)

∥

The notations

∧

and

∨

are also used in homotopy theory for the smash product and the wedge of pointed spaces, which we will introduce in

http://planetmath.org/node/87579

Chapter 6.
This technically creates a potential for conflict, but no confusion will generally arise.

Similarly, when discussing subsets as in §3.5 (

http://planetmath.org/35subsetsandpropositionalresizing

), we may use the traditional notation for

intersections

, unions, and

complements

:

{

x

:

A

|

P

(

x

)

}

∩

{

x

:

A

|

Q

(

x

)

}

:

≡

{

x

:

A

|

P

(

x

)

∧

Q

(

x

)

}

,

{

x

:

A

|

P

(

x

)

}

∪

{

x

:

A

|

Q

(

x

)

}

:

≡

{

x

:

A

|

P

(

x

)

∨

Q

(

x

)

}

,

A

∖

{

x

:

A

|

P

(

x

)

}

:

≡

{

x

:

A

|

¬

P

(

x

)

}

.

Of course, in the absence of

𝖫𝖤𝖬

, the latter are not “complements” in the usual sense: we may not have

B

∪

(

A

∖

B

)

=

A

.

Title

3.7 Propositional truncation

\metatable

Generated on Thu Feb 8 19:35:10 2018 by

LaTeXML