IAPS of groups

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This article gives a basic definition in the following area: APS theory
View other basic definitions in APS theory |View terms related to APS theory |View facts related to APS theory
This article defines the notion of group object in the category of IAPSs|View other types of group objects

Definition

An IAPS of groups is an IAPS over the category of groups. More specifically an IAPS (G,\Phi) is the following data:

  • For each natural number n, a group denoted G_n
  • For each ordered pair (m,n) of natural numbers, an injective homomorphism \Phi_{m,n}:G_m \times G_n \to G_{m+n}

Satisfying the following compatibility conditions:

For g, h, k in G_m, G_n, G_p respectively:

\Phi_{m+n,p} (\Phi_{m,n}(g,h),k) = \Phi_{m,n+p} (g, \Phi_{n,p}(h,k)).

The above condition is termed an associativity condition.

We may assume G_0 as the trivial group and define \Phi_{m,0} and \Phi_{0,n} as trivial paddings.

Note that if we remove the condition of injectivity, we get an APS of groups.

Constructions

Sub-IAPS

Further information: sub-IAPS of groups

Let (G,\Phi) be an IAPS of groups. A sub-IAPS H associated to every n a subgroup H_n of G_n such that the image of H_m \times H_n under \Phi_{m,n} lies inside H_{m+n}. Note that this is the same as a sub-APS of groups because the injectivity condition comes for free.

Quotient IAPS

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