foundations

Examples

# Contents

## Idea

In set theory, a class is a proposition or truth value in the context of a set free variable. In dependent type theory with a type universe $U$, a class is an h-proposition in the context of a free variable $x:U$.

In set-level dependent type theory with a separate set or type judgment but no type universes, there are no set/type free variables. However, one could nevertheless interpret a class in dependent type theory as a generalized modal operator on sets/types which takes sets to h-propositions by the propositions as some types interpretation. If we use the propositions as types interpretation of dependent type theory, then a class in any dependent type theory is just a generalized modal operator on sets/types. The generalized modal operators here are not required to be either idempotent or monadic.

One could internalize the notion of class inside of the foundations, and this internal notion of class is used to address size issues in foundations, and in particular, are used in category theory for defining various locally small categories and large categories.

## Definitions

There are multiple ways to define a class in different foundations of mathematics. Let us work in natural deduction.

• Suppose we are given an unsorted set theory as logic over type theory whose objects are sets. A class $C$ is a proposition or truth value in the context of a free variable $A$.

$\Gamma, A \vert \Phi \vdash C \; \mathrm{prop}$
• Suppose we are given a simply sorted set theory or dependently sorted set theory as logic over type theory with a sort $\mathrm{Set}$ of sets. A class $C$ is a proposition or truth value in the context of a free variable for a set $A:\mathrm{Set}$.

$\Gamma, A:\mathrm{Set} \vert \Phi \vdash C \; \mathrm{prop}$
• Suppose we are given an unsorted set theory as logic over type theory whose objects are “probable sets”, with a predicate $\mathrm{isSet}(A)$ saying whether a “probable set” $A$ is a set. A class $C$ is a proposition or truth value in the context of a free variable $A$ and the true proposition $\mathrm{isSet}(A) \; \mathrm{true}$.

$\Gamma, A \vert \Phi, \mathrm{isSet}(A) \; \mathrm{true} \vdash C \; \mathrm{prop}$
• Suppose we are given a simply sorted set theory or dependently sorted set theory as logic over type theory with a sort $\mathrm{ProbSet}$ of “probable sets” with a predicate $\mathrm{isSet}(A)$ on $\mathrm{ProbSet}$ saying whether a term $A:\mathrm{ProbSet}$ is a set. A class $C$ is a proposition or truth value in the context of a free variable for a probable set $A:\mathrm{ProbSet}$ and the true proposition $\mathrm{isSet}(A) \; \mathrm{true}$.

$\Gamma, A:\mathrm{ProbSet} \vert \Phi, \mathrm{isSet}(A) \; \mathrm{true} \vdash C \; \mathrm{prop}$
• Suppose we are given a dependent type theory with a type universes $\mathcal{U}$ and a separate type judgment, as well as a type of all propositions $\Omega$. Then a class $C$ is a proposition in the context of a type $A:\mathcal{U}$

$\Gamma, A:\mathcal{U} \vdash C:\Omega$
• Suppose we are given a dependent type theory with a type universes $\mathcal{U}$ and a separate type judgment, and are using the propositions as some types interpretation of dependent type theory. Then a class $C$ is a type in the context of a type $A:\mathcal{U}$

$\Gamma, A:\mathcal{U} \vdash C \; \mathrm{type}$

with the rule

$\frac{\Gamma, A:\mathcal{U} \vdash C \; \mathrm{type} \quad \Gamma \vdash B:\mathcal{U}}{\Gamma \vdash p^C(B):\mathrm{isProp}(C[B/A])}$
• Suppose we are given a dependent type theory with a type universes $\mathcal{U}$ and a separate type judgment, and are using the propositions as types interpretation of dependent type theory. Then a class $C$ is a type in the context of a type $A:\mathcal{U}$

$\Gamma, A:\mathcal{U} \vdash C \; \mathrm{type}$
• Suppose we are given a dependent type theory with a sequence of Russell universes $\mathcal{U}_i$ but no separate type judgment, and are using the propositions as some types interpretation of dependent type theory. Then a class $C$ is an h-proposition in the successor universe $\mathcal{U}_{s_\mathcal{N}(i)}$ in the context of a type $A:\mathcal{U}_i$

$\Xi \vert \Gamma, A:\mathcal{U}_i \vdash C:\mathrm{Prop}_{\mathcal{U}_{s_\mathcal{N}(i)}}$
• Suppose we are given a dependent type theory with a sequence of Russell universes $\mathcal{U}_i$ but no separate type judgment, and are using the propositions as types interpretation of dependent type theory. Then a class $C$ is a type in the successor universe $\mathcal{U}_{s_\mathcal{N}(i)}$ in the context of a type $A:\mathcal{U}_i$

$\Xi \vert \Gamma, A:\mathcal{U}_i \vdash C:\mathcal{U}_{s_\mathcal{N}(i)}$
• Suppose we are given a dependent type theory with a separate type judgment but no universes. Then a class $C$ is a generalized modal operator on the type theory, which isn’t required to be either idempotent or monadic, such that given a type $A$, the type $C(A)$ is an h-proposition.

$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash C(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash p_A^C:\mathrm{isProp}(C(A))}$
• Suppose we are given a dependent type theory with a separate type judgment but no type universes, and are using the propositions as types interpretation of dependent type theory. Then a class $C$ is a generalized modal operator on the type theory which isn’t required to be either idempotent or monadic.

$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash C(A) \; \mathrm{type}}$

## Internal classes

It is possible to internalize the notion of class inside of the foundations itself. Classes are either primitive, such as in class theory, or a derived concept from a notion of universe in the foundations, such as in set theory or type theory. These internal classes are useful to address size issues in foundations, and in particular, are used in category theory for defining various locally small categories and large categories.

### Classes as primitive

There are foundational theories called class theories where classes are primitives, rather than propositions in the context of a free variable $x:\mathrm{Set}$. This is similar to dependently sorted allegorical set theories, where relations are primitives, rather than propositions in the context of free variables $a:A$ and $b:B$.

### Classes as derived from universes

#### In set theory

There are many notions of universe in set theory, including Grothendieck universes, well-pointed Heyting pretopoi, and well-pointed division allegories.

Let $U$ be a universe. Then a class relative to $U$ is a subset $C \subseteq U$ with a given injection $i:C \hookrightarrow U$. If one has choice, any subset comes with a given injection via the axiom of choice. Thus, by this definition, it is a injective family of $U$-small sets.

Equivalently, a class $C$ relative to $U$ is an endofunction $C:U \to U$ such that for all $U$-small sets $A \in U$, $C(A)$ is a subsingleton subset.

#### In dependent type theory

In dependent type theory, let $(U, T)$ be a univalent Tarski universe, let $(\mathrm{Set}_U, \mathrm{Set}_T)$ be the univalent h-groupoid of $U$-small h-sets, let $V_U$ be the material cumulative hierarchy higher inductive type relative to $U$. Then a class is an endofunction $C:\mathrm{Set}_U \to \mathrm{Set}_U$ such that for all $U$-small sets $A:\mathrm{Set}_U$, $(C(A))$ is a $U$-small h-proposition, and a material class is a function $C:V_U \to \mathrm{Set}_U$ such that for all material sets $A:V_U$, $(C(A))$ is a $U$-small h-proposition.

### Proper classes

A proper class is a class which is not a set. What a set is differs from foundations to foundations.

What not being a set means depends upon the foundation; in material set theory, one would use the property of not being equal to any sets, while in structural set theory, one would use the property of not being in bijection with any sets.

Given a notion of universe, a proper class relative to $U$ is a class relative to $U$ which is not a set. If classes are defined as subsets of $U$ with an injection into $U$, then a proper class is a class which is not a singleton subset.

In the context of the global axiom of choice, a proper class is a class which can be put in bijection with the class of all ordinals, $Ord$.

## Category of classes

The category of classes is closed under all large colimits and small limits. See the linked article for more information and precise definitions.

Just as an elementary topos is an axiomatization of basic properties of the category Set, a category with class structure is an axiomatization of basic properties of the category $Class$. See also algebraic set theory.

For the definition of a material class relative to a universe $U$ in homotopy type theory, see section 10.5.3 of: