theory of objects



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/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient 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


Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




The theory of objects is the logical theory whose models in a category 𝒞\mathcal{C} are precisely the objects of 𝒞\mathcal{C}. From an algebraic perspective one can think of it as the theory of an algebra with no operations.


The theory of objects 𝕆\mathbb{O} is the theory with no axioms over the signature with a single type and no primitive symbols except equality.


Of course, 𝕆\mathbb{O} is a geometric theory, and as models for 𝕆\mathbb{O} in a topos \mathcal{E} correspond to objects of \mathcal{E}, we can use its classifying topos to get a representation of the objects of \mathcal{E}.


The classifying topos for the theory of objects is the presheaf topos [FinSet,Set][FinSet, Set] over the opposite category of the category FinSet of finite sets.

This follows from the fact that 𝕆\mathbb{O} is the algebraic theory of a 1-sorted algebra without operations whence its algebras are merely sets. The classifying topos for algebraic theories is quite generally given by the presheaf topos on the opposite of the category of finitely presentable models in our case just the finite sets.

For a direct proof see at classifying topos for the theory of objects. For base toposes 𝒮\mathcal{S} other than SetSet, it is a theorem due to Andreas Blass that the theory of objects has a classifying topos precisely if 𝒮\mathcal{S} has a natural numbers object.

In the syntax-free approach to geometric theories of Johnstone (2002, I B4.2) the theory of objects corresponds to the forgetful functor sending an 𝒮\mathcal{S}-topos to its underlying category. (See at geometric theory the section on the functorial definition.)

  • A step up on the ladder of logical complexity is the theory of inhabited objects 𝕆 \mathbb{O}_\exists that adds to 𝕆\mathbb{O} the existential axiom (x)\top\vdash(\exists x)\top. Its classifying topos Set[𝕆 ]Set[\mathbb{O}_\exists] is the functor category [FinSet ,Set][FinSet_\exists, Set] with FinSet FinSet_\exists the category of finite nonempty sets. It has the property that every topos \mathcal{E} admits a localic morphism to Set[𝕆 ]Set[\mathbb{O}_\exists].1
  • When viewed as algebraic theory the initial algebra (in SetSet) is the empty set which satisfies (x)(\exists x)\top\vdash\bot. Adding the sequent to the theory of objects one obtains the theory of empty objects 𝕆 \mathbb{O}_\emptyset, its models are the initial objects. Its classifying topos is SetSet with \emptyset as generic object. Whence the duality between quotient theories and subtoposes implies that Set[𝕆]Set[\mathbb{O}] contains Set[𝕆 ]Set[\mathbb{O}_\emptyset]. The pattern that the classifying topos of an algebraic theory contains a copy of SetSet as a subtopos classifying Th(𝔄 0)Th(\mathfrak{A}_0) i.e. the theory of all geometric sequents holding at the initial algebra 𝔄 0\mathfrak{A}_0 in Set is valid more generally (see at geometric type theory for another example).

  • If instead of an additional axiom one adds a single constant symbol to the signature of 𝕆\mathbb{O} one obtains the theory of pointed objects 𝕆 *\mathbb{O}_\ast i.e. the empty theory relative to the signature with a single sort and a single constant. Its models are pointed objects and its classifying topos is [FinSet *,Set][FinSet_\ast,Set]. (See the discussion&references at classifying topos for the theory of objects.)

  • The theory of morphisms 𝕆 2\mathbb{O}^2 is the theory with no sequents over the signature with two sort symbols O 0O^0, O 1O^1 and a function symbol f O:O 0O 1f_O:O^0\to O^1. It is the theory of 𝕆 \mathbb{O} -model homomorphisms and its models in a Grothendieck topos \mathcal{E} are the morphisms of \mathcal{E}. It is the dual theory of the theory classified by the Sierpinski topos Set 2Set^2, or in other words, its classifying topos is Set[𝕆 2]=Set[𝕆] Set 2Set[\mathbb{O}^2]=Set[\mathbb{O}]^{Set^2}.


Sections B4.2, D3.2 of

  1. cf. Johnstone (2002 II, p.773) and Joyal-Tierney (1984). For some further information on FinSet FinSet_\exists see the references at generic interval.

Last revised on June 19, 2020 at 14:09:56. See the history of this page for a list of all contributions to it.