nLab theory of presheaf type

Contents

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

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

Theories of presheaf type though being geometric theories of a particular “simple” and tractable type are yet ubiquitous in the sense that every geometric theory is a quotient theory? of some theory of presheaf type.

Definition

Definition

A geometric theory 𝕋\mathbb{T} is of presheaf type if its classifying topos Set[𝕋]Set[\mathbb{T}] is equivalent to a presheaf topos.

Examples

  • Any cartesian theory 𝕋\mathbb{T} being (modulo neglectable size issues1) classified by Set 𝕋Mod fp(Set)Set^{\mathbb{T}-Mod_{fp}(Set)} is of presheaf type with 𝕋-Mod fp(Set)\mathbb{T}\text{-}Mod_{fp}(Set) the category of finitely presentable 𝕋\mathbb{T}-models in SetSet.

  • More concretely, e.g. the theory of objects or the theory of intervals are of presheaf type being classified by the object classifier respectively by the topos of simplicial sets.

  • The inconsistent theory with axiom \top\vdash\bot is of presheaf type since it is classified by the initial Grothendieck topos 1Pr()\mathbf{1}\simeq Pr(\emptyset), the presheaf topos on the empty category.

Properties

Proposition

For a category \mathcal{M} the following are equivalent:

  • \mathcal{M} is finitely accessible.

  • Pts(Set 𝒞)\mathcal{M}\simeq Pts(Set^{\mathcal{C}}) the category of points of some presheaf topos.

  • 𝕋-Mod(Set)\mathcal{M}\simeq \mathbb{T}\text{-}Mod(Set) for some theory 𝕋\mathbb{T} of presheaf type.

  • Ind-𝒞\mathcal{M}\simeq Ind \text{-}\mathcal{C} for some small category 𝒞\mathcal{C}.

Cf. Beke (2004, p.923) and the references given there. In fact, these equivalences are mostly (direct consequences of) classical results in the theory of accessible categories or Grothendieck toposes.

Note that despite the above equivalences the finite accessibility of 𝕋-Mod(Set)\mathbb{T}\text{-}Mod(Set) does not imply that 𝕋\mathbb{T} itself is of presheaf type! One sees this already in case 𝕋-Mod(Set)=\mathbb{T}\text{-}Mod(Set)=\emptyset since there famously are non trivial (Boolean sheaf) toposes lacking points (“non empty generalized spaces without points”) yet up to Morita equivalence the only theory of presheaf type corresponding to the finite accessibility of the empty category is the inconsistent theory.

The following proposition shows in which sense theories of presheaf type are still determined by their (finitely presentable) models in SetSet:

Proposition

A geometric theory 𝕋\mathbb{T} is of presheaf type iff (modulo neglectable size issues)

Set[𝕋][𝕋-Mod fp(Set),Set].Set[\mathbb{T}]\simeq [\mathbb{T}\text{-}Mod_{fp}(Set),Set]\, .

Proof. “\Rightarrow”:

(Cf. Caramello 2018, pp.198f)

By assumption Set[𝕋][𝒞,Set]Set[\mathbb{T}]\simeq [\mathcal{C}, Set]. Since [𝒞,Set][𝒞^,Set][\mathcal{C},Set]\simeq [\hat{\mathcal{C}}, Set] (by Johnstone 2002, p.10) we can assume that 𝒞\mathcal{C} is Cauchy complete.

We have:

  • Ind-𝒞Flat(𝒞 op,Set)Ind\text{-}\mathcal{C}\simeq Flat(\mathcal{C}^{op}, Set) (by Johnstone 2002, p.723, or Caramello 2018, p.198)

  • (Ind-𝒞) fp𝒞(Ind\text{-}\mathcal{C})_{fp}\simeq \mathcal{C} (by Johnstone 2002 4.2.2.(iii), p.724)

  • 𝕋-Mod(Set)Flat(𝒞 op,Set)\mathbb{T}\text{-}Mod(Set)\simeq Flat(\mathcal{C}^{op}, Set) (from Diaconescu’s theorem).

Whence (Ind-𝒞) fp(𝕋-Mod(Set)) fp=𝕋-Mod fp(Set)𝒞(Ind\text{-}\mathcal{C})_{fp}\simeq (\mathbb{T}\text{-}Mod (Set))_{fp} =\mathbb{T}\text{-}Mod_{fp}(Set)\simeq \mathcal{C} and, accordingly, [𝒞,Set]][𝕋-Mod fp(Set),Set][\mathcal{C},Set]]\simeq [\mathbb{T}\text{-}Mod_{fp}(Set),Set]. \qed

In other words, theories of presheaf type are precisely those geometric theories 𝕋\mathbb{T} such that their classifying toposes Set[𝕋]Set[\mathbb{T}] can be represented as presheaf toposes Set 𝕋-Mod fp(Set)Set^{\mathbb{T}\text{-}Mod_{fp}(Set)}.

This implies e.g. that any consistent theory of presheaf type has models in SetSet.

References


  1. This means here (and in the following) that the essentially small category 𝕋-Mod fp(Set)\mathbb{T}\text{-}Mod_{fp}(Set) has to be replaced by a skeleton.

Last revised on April 23, 2023 at 18:25:14. See the history of this page for a list of all contributions to it.