Maths - Groups Objects

So far, we have defined a group as a set+structure, here we want to use the same structure but on top of other types of categories, so instead of defining a group in terms of its internal elements we define it in terms of its external properties, that is structure preserving mappings called functors (this is the category theory approach see this page).

To do this we take the structure in terms in the group identities:

and we put this in category theory terms:

We define:

and now define the identities in these terms:

Group Associativity

μ(a,μ(b,c)) = μ(μ(a,b),c)

μ(id×μ)(a,b,c) = μ(μ×id)(a,b,c)

μ(id×μ) = μ(μ×id)

group associativity
Group Identity μ(a,a-1) = e group identity
Group Inverse   group inverse

 


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Terminology and Notation

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