nLab theory of decidable objects



Type theory

Topos 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/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/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


topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




The theory of decidable objects is the logical theory 𝔻\mathbb{D} whose models in a coherent category are precisely the decidable objects.


The theory of decidable objects is the theory 𝔻\mathbb{D} over the signature with one sort and one binary relation ## besides equality with axioms (x#x) x(x#x)\vdash_x\perp and x,y((x#y)(x=y))\top\vdash_{x,y} ((x#y)\vee(x=y)).


  • For a topos \mathcal{E} the category of models Mod 𝔻()Mod_{\mathbb{D}}(\mathcal{E}) is the category of decidable objects in \mathcal{E}. The classifying topos Set[𝔻]Set[\mathbb{D}] for the theory of decidable objects is the functor category [FinSet mono,Set][FinSet_{mono},Set] where FinSet monoFinSet_{mono} is the category of finite sets and monomorphisms. Set[𝔻]Set[\mathbb{D}] is a locally decidable topos.

Infinite decidable objects

  • The theory of infinite decidable objects 𝔻 \mathbb{D}_\infty adds to 𝔻\mathbb{D} the axioms x 1,,x n(y) i=1 n(y#x i)\top\vdash_{x_1,\dots,x_n} (\exists y)\bigwedge_{i=1}^{n}(y#x_i) for all nn with (y)\top\vdash(\exists y)\top for n=0n=0. The models of 𝔻 \mathbb{D}_\infty are precisely the infinite decidable objects and its classifying topos Set[𝔻 ]Set[\mathbb{D}_\infty] is the Schanuel topos.


Last revised on October 10, 2020 at 09:30:55. See the history of this page for a list of all contributions to it.