nLab syntactic site



Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory


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
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




For 𝕋\mathbb{T} a theory, the syntactic site of a syntactic category 𝒞 𝕋\mathcal{C}_{\mathbb{T}} is the structure of a site on 𝒞 𝕋\mathcal{C}_{\mathbb{T}} such that geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}}) into the sheaf topos over the syntactic site are equivalent to models for the theory 𝕋\mathbb{T} in \mathcal{E}, hence such that Sh(𝒞 𝕋)Sh(\mathcal{C}_{\mathbb{T}}) is the classifying topos for 𝕋\mathbb{T}.


For 𝕋\mathbb{T} a theory and 𝒞 𝕋\mathcal{C}_{\mathbb{T}} its syntactic category, we define coverages JJ on 𝒞 𝕋\mathcal{C}_{\mathbb{T}}. These depend on which type of theory 𝕋\mathbb{T} is (or is regarded to be).




For 𝒯\mathcal{T} a cartesian theory, regular theory, etc. and 𝒞 𝕋\mathcal{C}_{\mathbb{T}} its syntactic site, according to def. , we have

  • For 𝕋\mathbb{T} a cartesian theory, left exact functors 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} are equivalent to geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})

    𝕋Model()Func ×(𝒞 𝕋,)Topos(,Sh(𝒞 𝕋)). \mathbb{T}-Model(\mathcal{E}) \simeq Func_\times(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.
  • For 𝕋\mathbb{T} a regular theory, regular functors 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} are equivalent to geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})

    𝕋Model()RegFunc(𝒞 𝕋,)Topos(,Sh(𝒞 𝕋)). \mathbb{T}-Model(\mathcal{E}) \simeq RegFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.
  • For 𝕋\mathbb{T} a coherent theory, coherent functors 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} are equivalent to geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})

    𝕋Model()CohFunc(𝒞 𝕋,)Topos(,Sh(𝒞 𝕋)). \mathbb{T}-Model(\mathcal{E}) \simeq CohFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.
  • For 𝕋\mathbb{T} a geometric theory, geometric functors 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} are equivalent to geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})

    𝕋Model()GeomFunc(𝒞 𝕋,)Topos(,Sh(𝒞 𝕋)). \mathbb{T}-Model(\mathcal{E}) \simeq GeomFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.

In each case the equivalence of categories Topos(,Sh(𝒞 𝕋))𝕋Model()Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \stackrel{\simeq}{\to} \mathbb{T}-Model(\mathcal{E}) is given by sending a geometric morphism f:Sh(𝒞 𝕋)f : \mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}}) to the precomposition of its inverse image f *f^* with the Yoneda embedding jj and sheafification LL:

f(𝒞 𝕋jPSh(𝒞 𝕋)LSh(𝒞 𝕋)f *). f \;\; \mapsto \;\; ( \mathcal{C}_\mathbb{T} \stackrel{j}{\to} PSh(\mathcal{C}_{\mathbb{T}}) \stackrel{L}{\to} Sh(\mathcal{C}_{\mathbb{T}}) \stackrel{f^*}{\to} \mathcal{E} ) \,.

This appears as (Johnstone), theorem 3.1.1, 3.1.4, 3.1.9, 3.1.12.


For cartesian theories this is the statement of Diaconescu's theorem.

The other cases follow from this by using this discussion at classifying topos, which says that geometric morphism Sh(𝒞)\mathcal{E} \to Sh(\mathcal{C}) are equivalent to morphisms of sites 𝒞\mathcal{C} \to \mathcal{E} (for the canonical coverage on \mathcal{E}). This means that in addition to preserving finite limits, as in Diaconescu's theorem, these functors also send covers in 𝒞\mathcal{C} to epimorphisms in \mathcal{E}.

In the cases at hand this last condition means precisely that 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} is a regular functor or coherent functor etc., respectively.


Section D3.1 of

Last revised on February 27, 2019 at 11:17:00. See the history of this page for a list of all contributions to it.