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It has been known that categorical interpretations of dependent type theorywith Sigma- and Id-types induce weak factorization systems. When one has a weakfactorization system (L, R) on a category C in hand, it is then natural to askwhether or not (L, R) harbors an interpretation of dependent type theory withSigma- and Id- (and possibly Pi-) types. Using the framework of display mapcategories to phrase this question more precisely, one would ask whether or notthere exists a class D of morphisms of C such that the retract closure of D isthe class R and the pair (C, D) forms a display map category modeling Sigma-and Id- (and possibly Pi-) types. In this paper, we show, with the hypothesisthat C is Cauchy complete, that there exists such a class D if and only if(C,R) itself forms a display map category modeling Sigma- and Id- (and possiblyPi-) types. Thus, we reduce the search space of our original question from apotentially proper class to a singleton.

Identity types and weak factorization systems in Cauchy complete categories