nLab cofiber type

Context

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
propositionsetobjecttype
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

semantics

Contents

Definition

In dependent type theory with identity types, function types, and dependent function types, given a function f:ABf:A \to B, the cofiber type or cofibre type of ff is the higher inductive type generated by

  • a term y:cofib A,B(f)y:\mathrm{cofib}_{A, B}(f)

  • a function i:Bcofib A,B(f)i:B \to \mathrm{cofib}_{A, B}(f)

  • a dependent function

    glue: x:AId cofib A,B(f)(i(f(x)),y)\mathrm{glue}:\prod_{x:A} \mathrm{Id}_{\mathrm{cofib}_{A, B}(f)}(i(f(x)),y)

In essence, the cofiber type takes the image of the function f:ABf:A \to B and makes it contractible, but leaves the rest of the codomain untouched.

Induction rules

Formation rules for cofiber types:

ΓAtypeΓBtypeΓf:ABΓcofib A,B(f)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B}{\Gamma \vdash \mathrm{cofib}_{A, B}(f)}

Introduction rules for cofiber types:

ΓAtypeΓBtypeΓf:ABΓ*:cofib A,B(f)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B}{\Gamma \vdash *:\mathrm{cofib}_{A, B}(f)}
ΓAtypeΓBtypeΓf:ABΓi:Bcofib A,B(f)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B}{\Gamma \vdash i:B \to \mathrm{cofib}_{A, B}(f)}
ΓAtypeΓBtypeΓf:ABΓ: x:A*= cofib A,B(f)i(f(x))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B}{\Gamma \vdash \mathcal{g}:\prod_{x:A} * =_{\mathrm{cofib}_{A, B}(f)} i(f(x))}

Elimination rules for cofiber types:

ΓAtypeΓBtypeΓf:AB Γ,y:cofib A,B(f)C(y)typeΓc *:C(*)Γc i: y:BC(i(y)) Γc : x:Ac *= y:cofib A,B(f).C(y) (x)c i(f(x))Γind cofib A,B(f)(c *,c i,c ): y:cofib A,B(f)C(y)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\ \Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{y:B} C(i(y)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_i(f(x)) \end{array}}{\Gamma \vdash \mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g}):\prod_{y:\mathrm{cofib}_{A, B}(f)} C(y)}

Computation rules for cofiber types:

ΓAtypeΓBtypeΓf:AB Γ,y:cofib A,B(f)C(y)typeΓc *:C(*)Γc i: y:BC(i(y)) Γc : x:Ac *= y:cofib A,B(f).C(y) (x)c i(f(x))Γβ cofib A,B(f) *(c *,c i,c ):ind cofib A,B(f)(c *,c i,c )(*)= C(*)c *\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\ \Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{y:B} C(i(y)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_i(f(x)) \end{array}}{\Gamma \vdash \beta_{\mathrm{cofib}_{A, B}(f)}^{*}(c_*, c_i, c_\mathcal{g}):\mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g})(*) =_{C(*)} c_*}
ΓAtypeΓBtypeΓf:AB Γ,y:cofib A,B(f)C(y)typeΓc *:C(*)Γc i: y:BC(i(y)) Γc : x:Ac *= y:cofib A,B(f).C(y) (x)c i(f(x))Γβ cofib A,B(f) i(c *,c i,c ): y:Bind cofib A,B(f)(c *,c i,c )(i(y))= C(i(y))c i(y)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\ \Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{y:B} C(i(y)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_i(f(x)) \end{array}}{\Gamma \vdash \beta_{\mathrm{cofib}_{A, B}(f)}^{i}(c_*, c_i, c_\mathcal{g}):\prod_{y:B} \mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g})(i(y)) =_{C(i(y))} c_i(y)}
ΓAtypeΓBtypeΓf:AB Γ,y:cofib A,B(f)C(y)typeΓc *:C(*)Γc i: y:BC(i(y)) Γc : x:Ac *= y:cofib A,B(f).C(y) (x)c i(f(x))Γβ cofib A,B(f) (c *,c i,c ): x:Aapd y:cofib A,B(f).C(y)(ind cofib A,B(f)(c *,c i,c ),*,i(f(x)),(x))= y:cofib A,B(f).C(y) (x)c (x)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\ \Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{y:B} C(i(y)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_i(f(x)) \end{array}}{\Gamma \vdash \beta_{\mathrm{cofib}_{A, B}(f)}^{\mathcal{g}}(c_*, c_i, c_\mathcal{g}):\prod_{x:A} \mathrm{apd}_{y:\mathrm{cofib}_{A, B}(f).C(y)}(\mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g}), *, i(f(x)), \mathcal{g}(x)) =_{y:\mathrm{cofib}_{A, B}(f).C(y)}^{\mathcal{g}(x)} c_\mathcal{g}(x)}

Uniqueness rules for cofiber types:

ΓAtypeΓBtypeΓf:AB Γ,y:cofib A,B(f)C(y)typeΓc: y:cofib A,B(f)C(y)Γp:cofib A,B(f)Γη cofib A,B(f)(c,p):ind cofib A,B(f)(c *,c i,c )(p)= C(y)c(p)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:A \to B \\ \Gamma, y:\mathrm{cofib}_{A, B}(f) \vdash C(y) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{y:\mathrm{cofib}_{A, B}(f)} C(y) \quad \Gamma \vdash p:\mathrm{cofib}_{A, B}(f) \end{array}}{\Gamma \vdash \eta_{\mathrm{cofib}_{A, B}(f)}(c, p):\mathrm{ind}_{\mathrm{cofib}_{A, B}(f)}(c_*, c_i, c_\mathcal{g})(p) =_{C(y)} c(p)}

Examples

  • The cofiber type of the unique function out of the empty type to any type BB is the type B+1B + 1.

  • The cofiber type of a function out of the unit type to any type BB is equivalent to BB.

  • The cofiber type of the identity function on a type BB is the cone type of BB.

See also

References

  • Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study, 2013. (web, pdf)

Last revised on July 8, 2023 at 12:58:40. See the history of this page for a list of all contributions to it.