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Identity types and weak factorization systems in Cauchy complete categories
Published online by Cambridge University Press: 29 March 2019
Abstract
It has been known that categorical interpretations of dependent type theory with Σ- and Id-types induce weak factorization systems. When one has a weak factorization system $({\cal L},{\cal R})$ on a category
$\mathbb{C}$ in hand, it is then natural to ask whether or not
$({\cal L},{\cal R})$ harbors an interpretation of dependent type theory with Σ- and Id- (and possibly Π-) types. Using the framework of display map categories to phrase this question more precisely, one would ask whether or not there exists a class
${\cal D}$ of morphisms of
$\mathbb{C}$ such that the retract closure of
${\cal D}$ is the class
${\cal R}$ and the pair
$(\mathbb{C},{\cal D})$ forms a display map category modeling Σ- and Id- (and possibly Π-) types. In this paper, we show, with the hypothesis that
$\cal{C}$ is Cauchy complete, that there exists such a class
$\cal{D}$ if and only if
$(\mathbb{C},\cal{R})$itself forms a display map category modeling Σ- and Id- (and possibly Π-) types. Thus, we reduce the search space of our original question from a potentially proper class to a singleton.
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- © Cambridge University Press 2019
Footnotes
This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-16-1-0212.
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