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5 Reconstructing G from the skeletal category of finite right $G$-sets
 5.1 Reconstruction Tools
 5.2 Examples

5 Reconstructing G from the skeletal category of finite right $G$-sets

5.1 Reconstruction Tools

5.1-1 EndAsEqualizer
‣ EndAsEqualizer( C, HomC, ForgetfulFunctor, IndexSet )( function )

5.1-2 EndByLifts
‣ EndByLifts( C, HomC, ForgetfulFunctor, Objects )( function )

5.1-3 ReconstructTableOfMarks
‣ ReconstructTableOfMarks( C, MinimalGeneratingSet, Decompose )( function )

5.1-4 HomSkeletalFinRightGSets
‣ HomSkeletalFinRightGSets( S, T )( function )

Returns: a finite set (see FinSetsForCAP)

The finite set \mathrm{Hom}_{\mathrm{SkeletalFinGSets}}( S, T ).

5.1-5 ForgetfulFunctorSkeletalCategoryOfFiniteRightGSets
‣ ForgetfulFunctorSkeletalCategoryOfFiniteRightGSets( G )( attribute )

Returns: a functor SkeletalFinGSets \rightarrow SkeletalFinSets

The forgetful functor SkeletalFinGSets \rightarrow SkeletalFinSets.

5.1-6 ReconstructGroup
‣ ReconstructGroup( C, HomC, ForgetfulFunctor, GeneratingSet, EndImplementation )( function )

Returns: a group

The input is a CAP category C which is equivalent to the skeletal category of finite right G-sets for some group G, a function HomC computing homs in C (e.g. HomSkeletalFinRightGSets), a generating set of C, and a function computing ends (e.g. EndAsEqualizer or EndByLifts). The output is a group isomorphic to G.

5.2 Examples

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