nLab pushout type



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


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




In homotopy type theory, a homotopy pushout type is a kind of higher inductive type whose categorical semantics is that of homotopy pushouts.


As a higher inductive type

The (homotopy) pushout of a span AfCgBA \xleftarrow{f} C \xrightarrow{g} B is the higher inductive type A f;g CBA \sqcup^{C}_{f ; g} B (or A CBA \sqcup^C B) generated by:

  • a function inl:AA CBinl : A \to A \sqcup^{C} B
  • a function inr:BA CBinr : B \to A \sqcup^{C} B
  • for each c:Cc:C a path glue(c):inl(f(c))=inr(g(c))glue(c) : inl (f (c)) = inr (g( c))

In Coq pseudocode:

Inductive hpushout {A B C : Type} (f : A -> B) (g : A -> C) : Type :=
| inl : B -> hpushout f g
| inr : C -> hpushout f g
| glue : forall (a : A), inl (f a) == inr (g a).

Inference rules

Explicitly, this means that the inference rules for pushout types are as follows (where we now label terms as shown in the following diagram):

type formation rule:

X,Y,Y :Type;f:XY,f :XY | || |YXY :Type \frac{ \begin{array}{l} X,\, Y,\, Y^' \,\colon\, Type \;; \;\; f \,\colon\, X \to Y \;, \;\; f^' \,\colon\, X \to Y^' \mathclap{\phantom{\vert_{\vert}}} \end{array} } { \mathclap{\phantom{\vert^{\vert}}} Y \underset{X}{\sqcup} Y^' \,\colon\, Type }

term introduction rules:

y:Y| || |cpr(y):YXY y :Y | || |cpr (y ):YXY x:X|| | |hmt(x):Id (YXY )(cpr(f(x)),cpr (f (x))) \frac{ y \,\colon\, Y \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} cpr(y) \,\colon\, Y \underset{X}{\sqcup} Y^' } \;\;\;\;\;\;\;\;\; \frac{ y^' \,\colon\, Y^' \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} cpr^'(y^') \,\colon\, Y \underset{X}{\sqcup} Y^' } \;\;\;\;\;\;\;\;\; \frac{ x \,\colon\, X \mathclap{\phantom{\vert}} }{ \mathclap{\phantom{\vert^{\vert^{\vert}}}} hmt(x) \,\colon\, Id_{\big(Y \underset{X}{\sqcup} Y^'\big)} \Big( cpr\big(f(x)\big) ,\, cpr^'\big(f^'(x)\big) \Big) }

term elimination rule:

y^:YXY P(y^):Type; cpr P:y:YP(cpr(y));cpr P :y :Y P(cpr (y )); hmt P:x:XId P(cpr (f (x)))(hmt(x) *(cpr P(f(x))),cpr P (f (x)))| |ind (P,cpr P,cpr P ,hmt P):y^:YXY P(y^) \frac{ \begin{array}{l} \hat{y} \,\colon\, Y \underset{X}{\sqcup} Y^' \;\vdash\; P(\hat{y}) \,\colon\, Type \;; \\ cpr_P \,\colon\, \underset{ y \colon Y }{\prod} P\big( cpr(y) \big) \;; \;\; cpr^'_P \,\colon\, \underset{ y^' \colon Y^' }{\prod} P\big( cpr^'(y^') \big) \;; \\ hmt_P \,\colon\, \underset{x \colon X}{\prod} Id_{P\Big( cpr^'\big(f^'(x)\big) \Big)} \bigg( hmt(x)_\ast \Big( cpr_P\big( f(x) \big) \Big) ,\, cpr^'_P\big( f^'(x) \big) \bigg) \end{array} }{ \mathclap{\phantom{\vert^{\vert}}} ind_{\big(P,\, cpr_P,\,cpr^'_P,\,hmt_P\big)} \,\colon\, \underset{ \hat{y} \,\colon\, Y \underset{X}{\sqcup} Y^' }{\prod} \, P(\hat{y}) }

computation rules:

y^:YXY P(y^):Type; cpr P:y:YP(cpr (y));cpr P :y :Y P(cpr (y )); hmt P:x:XId P(x)(hmt(x) *(cpr P(f(x))),cpr P (cpr (x)))| |ind (P,cpr P,cpr P ,hmt P)cpr = cpr P; | |ind (P,cpr P,cpr P ,hmt P)cpr = cpr P ; happly (ind (P,cpr P,cpr P ,hmt P))hmt = hmt P \frac{ \begin{array}{l} \hat{y} \,\colon\, Y \underset{X}{\sqcup} Y^' \;\vdash\; P(\hat{y}) \,\colon\, Type \;; \\ cpr_P \,\colon\, \underset{ y \colon Y }{\prod} P\big( cpr^'(y) \big) \;; \;\; cpr^'_P \,\colon\, \underset{ y^' \colon Y^' }{\prod} P\big( cpr^'(y^') \big) \;; \\ hmt_P \,\colon\, \underset{x \colon X}{\prod} Id_{P(x)} \bigg( hmt(x)_\ast \Big( cpr_P\big( f(x) \big) \Big) ,\, cpr^'_P\big( cpr^'(x) \big) \bigg) \end{array} }{ \begin{array}{rlc} \mathclap{\phantom{\vert^{\vert}}} ind_{\big(P,\, cpr_P,\,cpr^'_P,\,hmt_P\big)} \,\circ\, cpr &=& cpr_P \;; \\ \mathclap{\phantom{\vert^{\vert}}} ind_{\big(P,\, cpr_P,\,cpr^'_P,\,hmt_P\big)} \,\circ\, cpr^' &=& cpr^'_P \;; \\ happly_{\big( ind_{\big(P,\, cpr_P,\,cpr^'_P,\,hmt_P\big)} \big)} \,\circ\, hmt &=& hmt_P \end{array} }



See also


Early discussion of the idea of homotopy pushouts as higher inductive types:

Many references on higher inductive types mention pushout types and highlight their categorical semantics, but skip type-theoretic details, e.g.:

The full inference rules are stated in:

See also:

Formalization in the Coq proof assistant:

Last revised on May 25, 2023 at 15:55:56. See the history of this page for a list of all contributions to it.