nLab structure identity principle



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




By a stucture identity principle one means a statement of the form that all “identifications” of mathematical structures of a given type are necessarily structure respecting identifications (see also structuralism).

The source of the exact term “structure identity principle” seems to be Aczel (2011, slide 7), but the idea will have occurred earlier (see also the principle of equivalence).

The literature on the subject arguably suffers from absence of global conventions on how to name different notions of “identification” (identity, equality, isomorphism, equivalence, …), which can be confusing when speaking about a principle that is all about the fine-print of identifying two nominally different notions of identifications… But the general idea is always the same: Faced with notions of “direct”- and of “structure preserving”-identifications, a structure-identity-principle asserts that they are suitably identical.

Concretely, Aczel (2011, p. 9) declares that the structure identity principle (SIP) in homotopy type theory is the univalence axiom: This of course identifies terms p:Id Type(A,B)p \,\colon\, Id_{Type}(A,B) of identification type between types A,B:TypeA,B \,\colon\, Type with functions that are type-theoretic equivalences f:ABf \,\colon\, A \stackrel{\sim}{\longrightarrow} B.

The latter is what Aczel (2011, p. 22) calls “isomorphisms”. While this matches the usage of the term “isomorphism” in abstract category theory to mean “invertible morphism”, to complete the notion of “structure identity” one may want a further argument to verify that such isomorphisms really are invertible *homo*-morphism with respect to some given mathematical structure (such as group structure etc.). Such enhancement of Aczel’s notion of SIP is considered in Coquand & Danielsson (2013) (who do not use the terminology “structure udentity principle”) and in UFP (2013, §9.8), Escardó (2019, §3.33.1) (who do).


To make this more concrete and more manifest, we may notice that:

  1. at least a large class of kinds of mathematical structures on a base type B:TypeB \,\colon\, Type are expressible as nothing but “telescopes” of dependent pairs of BB with an iteration of further dependent pair- and dependent function- and identification types (for examples, such as group structure, see here);

  2. the univalence axiom implies extensionality principles for all three of these type formations (so with regards to UFP13 this is now using §2, instead), namely dependent function extensionality (here) and its analog for dependent pairs (here):


Together this means exactly that

  • identifications of two structured types which both are iterated dependent function-, dependent pair- and identity-types of the same form

  • are equivalent to giving the corresponding dependent function- and dependent pairing of componenwise identifications…

    …and this is just what one wants to mean by “preserving” this structure as in the notion of homomorphisms.

For example, with the type of group structures defined as

then the above extensionality principles (1) imply that the type of identifications of groups is equivalently that of bijections of the underlying types and equipped with the group homomorphism-property:


Discussion in the context of the philosophy of structuralism:

Last revised on February 4, 2023 at 12:08:23. See the history of this page for a list of all contributions to it.