The Royal Swedish Academy of Sciences(20)

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,

thatcoequalizesthekernelpair,whichmeansthat

z:C,v:D,w:D t{v/u}=t{w/u}:B

Thenwecanformthefactorizationoftthrough[u]asthewhereterm

z:C,y:[D] twhere[u]=y:B

Thesituationisdepictedinthefollowingdiagram:

(z:C,v:D,w:D) z,v

z,w (z:C,u:D)NNNNNNNN z,[u] NNNt

(C,y:[D])(B)ttttttt(tttwhere[u]=y)

Thetrianglecommutesbecause

twhere[u]=[u]=t

bytheβ-ruleforbrackettypes.Thefactorizationisunique,because[ ]isepic,stly,letusshowthatinSregularepisarestableunderpullbacks.SinceeveryarrowinSisisomorphictooneoftheform(5),everykernelpairisisomorphictooneoftheform(6)andsoeveryregularepiisisomorphictooneoftheform(7).Letusthencompute,withoutlossofgenerality,apullbackofsuchacoequalizeralonganarrowoftheform(5):

(y:B,z:C,u:D)_

y,z,[u] z,u (z:C,u:D) z,[u]

(y:B,z:C,v:[D]) z,v (C,[D])

Itisevidentthattheleft-handarrowisaregularepi,becauseitisoftheform(7).Itisalsoclearthatthediagramcommutes.Toseethatitisapull-back,observethatitappearsastheupperhalfofthefollowingcommutativediagram:

(y:B,z:C,u:D) y,z,[u] z,u (z:C,u:D) z,[u]

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