The Royal Swedish Academy of Sciences(24)

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,

di erentorders,asindicatedinthefollowingdiagram.

(Γ,[(Γ,A,B)AB])<<<<<<<<<<<< <<AB [AB]<<(Γ,A,[B])MMM<<MMM<MMM<<<MMM<< MMM<<[B]AMMM< (Γ)(Γ,[[B]])A[A[B]]

Sincethetwowaysaroundthediagramfrom(Γ,A,B)to(Γ)arebothregularepi–monofactorizationsofthesamearrow,byuniquenessofimageswehave:

[A[B]]=[AB]

ForequalitytypesEqA,theeliminationrule,

Γ e:EqA(a,b)

Γ e=r(a):EqA(a,b)

impliesthat(EqA(a,b))2=EqA(a,b),whence:

[EqA(a,b)]=EqA(a,b).

Togetherwith[1]=1,thatsummarizesthepropertiesof[ ]onitsown.Thingsbecomemoreinterestinginthepresenceofothertype-formingoperations,

0,A+B,AB,A→B,¬A,

where¬AstandsforA→0.

For nitesumsweget

[0]=0,[A+B]=[[A]+[B]].

byanargumentsimilartothatfor. Forwehave(A[B])2=A[B]2=A[B],sothat

[A[B]]=A[B],

andoneseeseasilythat

[AB]≤A[B].

19(9)

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