nLab Russell universe

Context

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
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity 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 set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
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

Universes

Contents

Idea

Russell universes or universes à la Russell are types whose terms are types. In type theories without a separate type judgment AtypeA \; \mathrm{type}, only typing judgments a:Aa:A, what would have been type judgments are represented by typing judgments that AA is a term of a Russell universe UU, A:UA:U. Russell universes without a separate type judgment are used in many places in type theory, including in the HoTT book, in Coq, and in Agda.

Formal definition

Without a separate type judgment

One formal definition of a type theory with Russell universes is as follows:

The type theory has judgments

  • Γctx\Gamma \; \mathrm{ctx}, that Γ\Gamma is a context

  • iindexi \; \mathrm{index}, that ii is a universe index,

  • ϕprop\phi \; \mathrm{prop}, that ϕ\phi is a proposition,

  • ϕtrue\phi \; \mathrm{true}, that ϕ\phi is a true proposition,

and consists of the formal signature and inference rules of first-order Heyting arithmetic or Peano arithmetic. These rules ensure that there are an infinite number of indices, which are linearly ordered with strict order <\lt and upwardly unbounded, where i<s(i)i \lt s(i) is true for all indices ii.

Now, we introduce the typing judgment a:Aa:A, which says that aa is a term of the type AA. Instead of type judgments, we introduce a special kind of type called a Russell universe or universe à la Russell, whose terms are the types themselves. Russell universes are formalized with the following rules:

ΓiindexΓU i:U s(i)ΓiindexΓA:U iΓA:U s(i)cumulΓiindexΓA:U iΓLift i,j(A):U s(i)lifting\frac{\Gamma \vdash i \; \mathrm{index}}{\Gamma \vdash U_i:U_{s(i)}} \qquad \frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A:U_i}{\Gamma \vdash A:U_{s(i)}}\mathrm{cumul} \qquad \frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A:U_i}{\Gamma \vdash \mathrm{Lift}_{i, j}(A):U_{s(i)}}\mathrm{lifting}

In addition, we have rules for contexts which state that one could add typing judgments to the list of contexts:

ΓiindexΓA:U i(Γ,a:A)ctx\frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A:U_i}{(\Gamma, a:A) \; \mathrm{ctx}}

With a type judgment for each universe

One could also define Russell universes à la Coquand, in that the type theory has a type judgment for each universe UU. Using the dependent type theory with no separate type judgment, instead of having only one term judgment a:Aa:A, for index ii and type A:U iA:U_i, we instead have an infinite number of type judgments, one type judgment Atype iA \; \mathrm{type}_i for every index ii, indicating that AA is a type with level ii, in addition to the term judgments a:Aa:A. Then, one has the following rules for Russell universes à la Coquand:

ΓiindexΓU itype s(i)ΓiindexΓAtype iΓAtype s(i)cumulΓiindexΓAtype iΓLift(A)type s(i)lifting\frac{\Gamma \vdash i \; \mathrm{index}}{\Gamma \vdash U_i \; \mathrm{type}_{s(i)}} \qquad \frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A \; \mathrm{type}_i}{\Gamma \vdash A \; \mathrm{type}_{s(i)}}\mathrm{cumul} \qquad \frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A \; \mathrm{type}_i}{\Gamma \vdash \mathrm{Lift}(A) \; \mathrm{type}_{s(i)}}\mathrm{lifting}
ΓiindexΓAtype iΓA:U iΓiindexΓA:U iΓAtype i\frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A \; \mathrm{type}_i}{\Gamma \vdash A:U_i} \qquad \frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A:U_i}{\Gamma \vdash A \; \mathrm{type}_i}

In addition, we have rules for contexts which state that one could add typing judgments to the list of contexts:

ΓiindexΓAtype i(Γ,a:A)ctx\frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A \; \mathrm{type}_i}{(\Gamma, a:A) \; \mathrm{ctx}}

One could derive from these rules the above rules for Russell universes and context extension

ΓiindexΓU i:U s(i)ΓiindexΓA:U iΓA:U s(i)cumulΓiindexΓA:U iΓLift(A):U s(i)lifting\frac{\Gamma \vdash i \; \mathrm{index}}{\Gamma \vdash U_i:U_{s(i)}} \qquad \frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A:U_i}{\Gamma \vdash A:U_{s(i)}}\mathrm{cumul} \qquad \frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A:U_i}{\Gamma \vdash \mathrm{Lift}(A):U_{s(i)}}\mathrm{lifting}
ΓiindexΓA:U i(Γ,a:A)ctx\frac{\Gamma \vdash i \; \mathrm{index} \quad \Gamma \vdash A:U_i}{(\Gamma, a:A) \; \mathrm{ctx}}

With a single separate type judgment

If the type theory has a separate type judgment AtypeA \; \mathrm{type}, the rules for Russell universes become simpler, as one doesn’t have to assume infinitely many Russell universes and a second level to house the natural numbers type used to index the universes. Instead, we merely have

ΓctxΓUtypeΓA:UΓAtype\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash U \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:U}{\Gamma \vdash A \; \mathrm{type}}

Furthermore, the separate type judgment amounts to collapsing the two levels in the theory above into one level: Instead of defining the identity type and the type of natural numbers as external to the type theory, we instead define it as normal types internal to the type theory. Then the rules for the infinite tower of Russell universes are as follows:

Γi:ΓU(i):U(s(i))Γi:ΓA:U(i)ΓAtypeΓi:ΓA:U(i)ΓLift(i)(A):U(s(i))\frac{\Gamma \vdash i:\mathbb{N}}{\Gamma \vdash U(i):U(s(i))} \qquad \frac{\Gamma \vdash i:\mathbb{N} \quad \Gamma \vdash A:U(i)}{\Gamma \vdash A \; \mathrm{type}} \qquad \frac{\Gamma \vdash i:\mathbb{N} \quad \Gamma \vdash A:U(i)}{\Gamma \vdash \mathrm{Lift}(i)(A):U(s(i))}

We could also add rules which state that every type is an element of a Russell universe:

ΓAtypeΓlevel A:ΓAtypeΓA:U(level A)\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{level}_A:\mathbb{N}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A:U(\mathrm{level}_A)}

Then the first rule for the infinite tower becomes one of

Γi:Γlevel U(i)s(i):strictΓi:ΓdefLevelU(i):level U(i)= s(i)weak\frac{\Gamma \vdash i:\mathbb{N}}{\Gamma \vdash \mathrm{level}_{U(i)} \equiv s(i):\mathbb{N}}\mathrm{strict} \quad \frac{\Gamma \vdash i:\mathbb{N}}{\Gamma \vdash \mathrm{defLevelU}(i):\mathrm{level}_{U(i)} =_\mathbb{N} s(i)}\mathrm{weak}

This is how one would formally present a dependent type theory like the one in Book HoTT or the ones in Agda, Coq, Lean, without resorting to external layers.

However, these set of rules are incompatible with the statement that all dependent sum types exist, because the dependent sum type n:U(n)\sum_{n:\mathbb{N}} U(n) contains as elements pairs of every type in the type theory and the universe level the type is in. However, by the inference rules above, n:U(n)\sum_{n:\mathbb{N}} U(n) is in the universe U(level n:U(n))U(\mathrm{level}_{\sum_{n:\mathbb{N}} U(n)}) which means that by right projecting n:U(n)\sum_{n:\mathbb{N}} U(n), every type is in the universe U(level n:U(n))U(\mathrm{level}_{\sum_{n:\mathbb{N}} U(n)}) resulting in Girard's paradox which is contradictory.

Instead, usually the inference rules for dependent sum types require that the type family x:AB(x)x:A \vdash B(x) in the dependent sum type have the same universe level n:n:\mathbb{N}, x:AB(x):U(n)x:A \vdash B(x):U(n). Since none of the universes in the above diagram have the same universe level, the dependent sum type n:U(n)\sum_{n:\mathbb{N}} U(n) cannot be constructed. The same is true of the formers of any type which could construct dependent sum types, such as wide pushouts.

Similarly, these set of rules are incompatible with the statement that all sequential colimits exist, because one could then find the sequential colimit of the diagram

U(0)Lift(0)U(s(0))Lift(s(0))U(s(s(0)))Lift(s(s(0)))U(0) \overset{\mathrm{Lift}(0)}\to U(s(0)) \overset{\mathrm{Lift}(s(0))}\to U(s(s(0))) \overset{\mathrm{Lift}(s(s(0)))}\to \ldots

and one would get a type U U_\infty, which by definition of sequential colimits contains every type in the universe hierarchy. However, U U_\infty is in the universe U(level U )U(\mathrm{level}_{U_\infty}), which means that lifting U U_\infty from U(level U )U(\mathrm{level}_{U_\infty}) to U U_\infty via the sequential colimit results in Girard's paradox which is contradictory.

Instead, usually the inference rules for sequential colimits require that the types in the above diagram have the same universe level n:n:\mathbb{N}. Since none of the universes in the above diagram have the same universe level, the sequential colimit U U_\infty cannot be constructed. The same is true of the formers of any type which could construct sequential colimits, such as pushout types.

But none of this are problems in Book HoTT as the formers for every type have the same universe level as the type to be put in the universe.

See also

References

The notion is due to:

For more see the references at type universe.

Last revised on November 28, 2023 at 16:42:12. See the history of this page for a list of all contributions to it.