10.1.4 Set is a Π⁢𝖶-pretopos

10.1.4 Set is a

Π

⁢

𝖶

-pretopos

The notion of a

Π

⁢

W

-pretopos

— that is, a locally cartesian closed category
with

disjoint

finite

coproducts

,

effective equivalence relations

, and initial algebras for

polynomial

endofunctors

— is intended as a “predicative”
notion of topos, i.e. a

category

of “

predicative sets

”, which can serve the purpose for constructive mathematics
that the usual

category of sets

does for classical
mathematics.

Typically, in constructive type theory, one resorts to an external construction of “setoids” — an exact completion — to obtain a category with such

closure properties

.
In particular, the well-behaved

quotients

are required for many constructions in mathematics that usually involve (non-constructive)

power sets

. It is noteworthy that

univalent foundations

provides these constructions

internally

(via

higher inductive types

), without requiring such external constructions. This

represents

a powerful advantage of our approach, as we shall see in subsequent examples.

Theorem 10.1.1

.

The category

S

⁢

e

⁢

t

is a

Π

⁢

W

-pretopos.

Proof.

We have an

initial object

𝟎

and finite, disjoint sums

A

+

B

. These are stable under

pullback

, simply because pullback has a

right adjoint

. Indeed,

𝒮

⁢

e

⁢

t

is locally cartesian closed, since for any map

f

:

A

→

B

between sets, the “fibrant replacement”

∑

(

a

:

A

)

f

⁢

(

a

)

=

b

is

equivalent

to

A

(over

B

), and we have dependent function types for the replacement.
We’ve just shown that

𝒮

⁢

e

⁢

t

is

regular

(

\autoref

thm:set_regular) and that quotients are

effective

(

\autoref

lem:sets_exact). We thus have a locally cartesian closed pretopos. Finally, since the

n

-types are

closed under

the formation of

W

-types by

\autoref

ex:ntypes-closed-under-wtypes, and by

\autoref

thm:w-hinit

W

-types are initial algebras for polynomial endofunctors, we see that

𝒮

⁢

e

⁢

t

is a

Π

⁢

𝖶

-pretopos.
∎

One naturally wonders what, if anything, prevents

𝒮

⁢

e

⁢

t

from being an (

elementary

) topos?
In

addition

to the

structure

already mentioned, a topos has a

subobject classifier

:
a pointed object classifying (

equivalence classes

of)

monomorphisms

. (In fact, in the presence of a subobject
classifier, things become somewhat simpler: one merely needs cartesian

closure

in order to get the

colimits

.)
In

homotopy type theory

, univalence implies that the type

𝖯𝗋𝗈𝗉

:

≡

∑

(

X

:

𝒰

)

𝗂𝗌𝖯𝗋𝗈𝗉

(

X

)

does classify monomorphisms (by an

argument

similar to

\autoref

sec:object-classification), but in general it is as large as the ambient

universe

𝒰

.
Thus, it is a “set” in the sense of being a

0

-type, but it is not “small” in the sense of being an object of

𝒰

, hence not an object of the category

𝒮

⁢

e

⁢

t

.
However, if we assume an appropriate form of propositional resizing (see

\autoref

subsec:prop-subsets), then we can find a small version of

𝖯𝗋𝗈𝗉

, so that

𝒮

⁢

e

⁢

t

becomes an

elementary topos

.

Theorem 10.1.2

.

If there is a type

Ω

:

U

of all mere propositions, then the category

S

⁢

e

⁢

t

U

is an elementary topos.

A

sufficient

condition for this is the

law of excluded middle

, in the “mere-propositional” form that we have called

𝖫𝖤𝖬

; for then we have

𝖯𝗋𝗈𝗉

=

𝟐

, which

is

small, and which then also classifies all mere propositions.
Moreover, in topos theory a well-known sufficient condition for

𝖫𝖤𝖬

is the

axiom of choice

, which is of course often assumed as an axiom in classical

set theory

.
In the next

section

, we briefly investigate the

relation

between these conditions in our setting.

Title

10.1.4 Set is a

Π

⁢

𝖶

-pretopos

\metatable

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