Homotopy Type Theory abelian group > history (Rev #7)

Definition

As a group

An abelian group or \mathbb{Z}-module consists of

  • A type GG,
  • A basepoint e:Ge:G
  • A binary operation μ:GGG\mu : G \to G \to G
  • A unary operation ι:GG\iota: G \to G
  • A contractible left unit identity
    c λ: (a:G)isContr(μ(e,a)=a)c_\lambda:\prod_{(a:G)} isContr(\mu(e,a)=a)
  • A contractible right unit identity
    c ρ: (a:G)isContr(μ(a,e)=a)c_\rho:\prod_{(a:G)} isContr(\mu(a,e)=a)
  • A contractible associative identity
    c α: (a:G) (b:G) (c:G)isContr(μ(μ(a,b),c)=μ(a,μ(b,c)))c_\alpha:\prod_{(a:G)} \prod_{(b:G)} \prod_{(c:G)} isContr(\mu(\mu(a, b),c)=\mu(a,\mu(b,c)))
  • A contractible left inverse identity
    c l: (a:G)isContr(μ(ι(a),a)=e)c_l:\prod_{(a:G)} isContr(\mu(\iota(a), a)=e)
  • A contractible right inverse identity
    c r: (a:G)isContr(μ(a,ι(a))=e)c_r:\prod_{(a:G)} isContr(\mu(a,\iota(a))=e)
  • A contractible commutative identity
    c κ: (a:A) (b:A)isContr(μ(a,b)=μ(b,a))c_\kappa:\prod_{(a:A)} \prod_{(b:A)} isContr(\mu(a, b)=\mu(b, a))
  • A 0-truncator
    τ 0: (a:G) (b:G)isProp(a=b)\tau_0: \prod_{(a:G)} \prod_{(b:G)} isProp(a=b)

As a module

An abelian group or \mathbb{Z}-module is a set SS with a term 0:S0:S and a binary function ()+():S×SS(-)+(-):S \times S \to S, and a left multiplicative \mathbb{Z}-action () l():×SS(-)\cdot_l(-):\mathbb{Z} \times S \to S, such that

a:S b:Sa+b=b+a\prod_{a:S} \prod_{b:S} a + b = b + a
a:S b:S c:Sa+(b+c)=(a+b)+c\prod_{a:S} \prod_{b:S} \prod_{c:S} a + (b + c) = (a + b) + c
a:S1 la=a\prod_{a:S} 1 \cdot_l a = a
a: b: c:S(ab) lc=a l(b lc)\prod_{a:\mathbb{Z}} \prod_{b:\mathbb{Z}} \prod_{c:S} (a \cdot b) \cdot_l c = a \cdot_l (b \cdot_l c)
a:S0 la=0\prod_{a:S} 0 \cdot_l a = 0
a: b:S c:Sa l(b+c)=a lb+a lc\prod_{a:\mathbb{Z}} \prod_{b:S} \prod_{c:S} a \cdot_l (b + c) = a \cdot_l b + a \cdot_l c
a: b: c:S(a+b) lc=a lc+b lc\prod_{a:\mathbb{Z}} \prod_{b:\mathbb{Z}} \prod_{c:S} (a + b) \cdot_l c = a \cdot_l c + b \cdot_l c

We define the functions :SS-:S \to S and () r():S×S(-)\cdot_r(-):S \times \mathbb{Z} \to S to be

a(1) la-a \coloneqq (-1) \cdot_l a
a rbb laa \cdot_r b \coloneqq b \cdot_l a

and SS is an abelian group and a \mathbb{Z}-bimodule

a=1 1a=(1+0) 1a=(1 1a)+(0 1a)=a+0a = 1 \cdot_1 a = (1 + 0) \cdot_1 a = (1 \cdot_1 a) + (0 \cdot_1 a) = a + 0
a=1 1a=(0+1) 1a=(0 1a)+(1 1a)=0+aa = 1 \cdot_1 a = (0 + 1) \cdot_1 a = (0 \cdot_1 a) + (1 \cdot_1 a) = 0 + a
0=0 1a=(1+1) 1a=(1 1a)+(1 1a)=a+a0 = 0 \cdot_1 a = (1 + -1) \cdot_1 a = (1 \cdot_1 a) + (-1 \cdot_1 a) = a + -a
0=0 1a=(1+1) 1a=(1 1a)+(1 1a)=a+a0 = 0 \cdot_1 a = (-1 + 1) \cdot_1 a = (-1 \cdot_1 a) + (1 \cdot_1 a) = -a + a

Examples

  • Every contractible magma (M,μ)(M, \mu) with a function f:MMf:M \to M is an abelian group.

  • The integers are an abelian group.

See also

Revision on April 23, 2022 at 15:41:28 by Anonymous?. See the history of this page for a list of all contributions to it.