The Royal Swedish Academy of Sciences(13)

Wenowproceedwiththeinterpretationofbrackettypes.AregularcategoryChasstableregularepi–monoimagefactorizations.Everyarrowf:A→Bcanbefactoreduniquelyuptoisomorphismasaregularepifollowedbyamono

Af

EEEEEEByyyyyy

Im(f)

Thefactorizationisobtainedbytakingthecoequalizerqofthekernelpair(π1,π2)off,asinthefollowingdiagram:

A×BAπ1

2ADDDDqDDfBzzzzzzi

Im(f)

Thearrowi:Im(f)→Bexistsandisuniquewithi q=fbecausefcoequalizesitsownkernelpair.Itcanmoreoverbeshownthatiisalwaysmonic.

Abrackettype

Γ Atype

Γ [A]type

isinterpretedastheimageof[[Γ A]]:

(Γ,A)[[Γ,[A]]]=Im(Γ A)OOOOOOOOOO[[Γ [A]]]Γ AOOOOOOO[ ](Γ)

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