The Royal Swedish Academy of Sciences(12)

Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content,

thecompositiong f,whereasforacompletelywater-tightinterpretationequalityisrequired.

Thereareseveralstandardwaysofresolvingthisproblem,mostnotablybyinterpretingthetypetheoryinasuitable beredcategory[Jac99],andthenapplyingtechnicalresultspertainingtothese[Hof95].Wedonotwishtoobscuremattersbyemployingsuchtechnicaldevices.Theinterestedreadermayeithertranslateourpresentationintoasuitable beredsetting,orassumesomeotherremedy,suchasmakingacoherentchoiceofpullbacks.(ForthesyntacticcategoryinSection3,suchpullbackscanbechosensimplyassubstitutions.)

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