nLab subterminal object





An object UU in a category CC is subterminal or preterminal if any two morphisms with target UU and the same source are equal. In other words, UU is subterminal if for any object XX, there is at most one morphism XUX\to U.


An umbrella category is a nonempty category CC such that for every object XX in CC, there is at least one subterminal object TT such that C(X,T)C(X,T) is nonempty (hence being a singleton).


If CC has a terminal object 11, then UU is subterminal precisely if the unique morphism U1U \to 1 is monic, so that UU represents a subobject of 11; hence the name “sub-terminal.”

This is equivalent to the hypothesis that the cone given by identity morphisms UUUU \leftarrow U \rightarrow U is a product cone, or that some product U×UU \times U exists and the diagonal UU×UU \to U \times U is an isomorphism.

Therefore for a sheaf topos over a topological space the subterminal objects of the topos are the open subsets of the topological space. Accordingly, the subterminal objects in any topos are also called open objects (e.g. Johnstone 77, p. 94)

The classifying topos for subterminal objects (hence open objects) in toposes is the Sierpinski topos (see e.g. Johnstone 77, p. 117).


The subterminal objects in a topos can be viewed as its “external truth values.” For example, in the topos Sh(X)Sh(X) of sheaves on a topological space XX, the subterminal objects are precisely the open sets in XX.

The support of an object XX in a topos is the image U1U \hookrightarrow 1 of the unique map X1X \to 1. Any map UXU \to X is necessarily a section of XUX \to U.

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


  • Peter Johnstone, Topos theory, London Math. Soc. Monographs 10, Acad. Press 1977

  • Dieter Pumplün, Initial morphisms and monomorphisms, Manuscripta mathematica 32 (1980): 309-333.

Last revised on February 22, 2024 at 04:18:28. See the history of this page for a list of all contributions to it.