Homotopy Type Theory integers > history (Rev #5)

Contents

Idea

The integers as familiar from school mathematics.

Definitions

The type of integers, denoted \mathbb{Z}, has several definitions as a higher inductive type.

Definition 1

The integers are defined as the higher inductive type generated by:

  • A function inj:2×inj : \mathbf{2} \times \mathbb{N} \rightarrow \mathbb{Z}.
  • An identity representing that positive and negative zero are equal: ν 0:inj(0,0)=inj(1,0)\nu_0: inj(0, 0) = inj(1, 0).

Definition 2

The integers are defined as the higher inductive type generated by:

  • A term 0:0 : \mathbb{Z}.
  • A function s:s : \mathbb{Z} \to \mathbb{Z}.
  • A function p 1:p_1 : \mathbb{Z} \to \mathbb{Z}.
  • A function p 2:p_2 : \mathbb{Z} \to \mathbb{Z}.
  • A dependent product of identities representing that p 1p_1 is a section of ss:
    σ: a:p 1(s(a))=a\sigma: \prod_{a:\mathbb{Z}} p_1(s(a)) = a
  • A dependent product of identities representing that p 2p_2 is a retracion of ss:
    ρ: a:s(p 2(a))=a\rho: \prod_{a:\mathbb{Z}} s(p_2(a)) = a

Definition 3

The integers are defined as the higher inductive type generated by:

  • A term 0:0 : \mathbb{Z}.
  • A function s:s : \mathbb{Z} \to \mathbb{Z}.
  • A function p:p : \mathbb{Z} \to \mathbb{Z}.
  • A dependent product of identities representing that pp is a section of ss:
    σ: a:p(s(a))=a\sigma: \prod_{a:\mathbb{Z}} p(s(a)) = a
  • A dependent product of identities representing that pp is a retracion of ss:
    ρ: a:s(p(a))=a\rho: \prod_{a:\mathbb{Z}} s(p(a)) = a
  • A dependent product of identities representing the coherence condition:
    κ: a:ap s(σ(a))=ρ(a)\kappa: \prod_{a:\mathbb{Z}} ap_s(\sigma(a)) = \rho(a)

Definition 4

The integers are defined as the higher inductive type generated by:

  • A term 0:0 : \mathbb{Z}.
  • A function s:s : \mathbb{Z} \to \mathbb{Z}.
  • A function n:n : \mathbb{Z} \to \mathbb{Z}.
  • An identity representing that zero and negative zero are equal: ν:n(0)=0\nu: n(0) = 0.
  • A dependent product of identities representing that negation is an involution:
    ι: a:n(n(a))=a\iota: \prod_{a:\mathbb{Z}} n(n(a)) = a
  • A dependent product of identities representing the coherence condition for the above:
    κ ι: a:ap n(ι(a))=ι(a)\kappa_\iota: \prod_{a:\mathbb{Z}} ap_n(\iota(a)) = \iota(a)
  • A dependent product of identities representing that nsnn \circ s \circ n is a section of ss:
    σ: a:n(s(n(s(a))))=a\sigma: \prod_{a:\mathbb{Z}} n(s(n(s(a)))) = a
  • A dependent product of identities representing that nsnn \circ s \circ n is a retracion of ss:
    ρ: a:s(n(s(n(a))))=a\rho: \prod_{a:\mathbb{Z}} s(n(s(n(a)))) = a
  • A dependent product of identities representing the coherence condition:
    κ: a:ap s(σ(a))=ρ(a)\kappa: \prod_{a:\mathbb{Z}} ap_s(\sigma(a)) = \rho(a)

Definition 5

The integers are defined as the higher inductive type generated by:

  • A function inj:×inj : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{Z}.
  • A dependent product of functions between identities representing that equivalent differences are equal:
    equivdiff: a: b: c: d:(a+d=c+b)(inj(a,b)=inj(c,d))equivdiff : \prod_{a:\mathbb{N}} \prod_{b:\mathbb{N}} \prod_{c:\mathbb{N}} \prod_{d:\mathbb{N}} (a + d = c + b) \to (inj(a,b) = inj(c,d))
  • A set-truncator
    τ 0: a: b:isProp(a=b)\tau_0: \prod_{a:\mathbb{Z}} \prod_{b:\mathbb{Z}} isProp(a=b)

Properties

TODO: Show that the integers are a ordered Heyting integral domain with decidable equality, decidable apartness, and decidable linear order, and that the integers are initial in the category of ordered Heyting integral domains.

See also

References

HoTT book

  • Thorsten Altinkirch, The Integers as a Higher Inductive Type, (abs:2007.00167)

  • Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson, Symmetry, (pdf)

Revision on February 26, 2022 at 21:01:12 by Anonymous?. See the history of this page for a list of all contributions to it.