nLab class

Context

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axiom

Removing axioms

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Modalities, Closure and Reflection

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 UU, a class is an h-proposition in the context of a free variable x:Ux: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.

Examples

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:Setx:\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:Aa:A and b:Bb: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 UU be a universe. Then a class relative to UU is a subset CUC \subseteq U with a given injection i:CUi: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 U -small sets.

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

In dependent type theory

In dependent type theory, let (U,T)(U, T) be a univalent Tarski universe, let (Set U,Set T)(\mathrm{Set}_U, \mathrm{Set}_T) be the univalent h-groupoid of UU-small h-sets, let V UV_U be the material cumulative hierarchy higher inductive type relative to UU. Then a class is an endofunction C:Set USet UC:\mathrm{Set}_U \to \mathrm{Set}_U such that for all UU-small sets A:Set UA:\mathrm{Set}_U, (C(A))(C(A)) is a UU-small h-proposition, and a material class is a function C:V USet UC:V_U \to \mathrm{Set}_U such that for all material sets A:V UA:V_U, (C(A))(C(A)) is a UU-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 UU is a class relative to UU which is not a set. If classes are defined as subsets of UU with an injection into UU, 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, OrdOrd.

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 ClassClass. See also algebraic set theory.

 See also

References

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

A paper detailing one approach to the technical side of how classes appear in category theory (namely using Grothendieck universes) is

Last revised on November 19, 2022 at 19:01:51. See the history of this page for a list of all contributions to it.