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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
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
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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 models in SetSet:

Proposition

Let 𝕋\mathbb{T} be of presheaf type. Then (modulo neglectable size issues) Set[𝕋][𝕋-Mod fp(Set),Set]Set[\mathbb{T}]\simeq [\mathbb{T}\text{-}Mod_{fp}(Set),Set].

Proof. (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 particular, 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 December 8, 2020 at 13:46:52. See the history of this page for a list of all contributions to it.