A C-system defined by a universe category

Vladimir Voevodsky

This is the third paper in a series. In it we construct a C-system CC(C,p) starting from a category C together with a morphism $p:\tilde{U} \to U$, a choice of pull-back squares based on $p$ for all morphisms to $U$ and a choice of a final object of C. Such a quadruple is called a universe category. We then define universe category functors and construct homomorphisms of C-systems CC(C,p) defined by universe category functors. In the sections before the last section we give, for any C-system CC, three different constructions of pairs ((C,p),H) where (C,p) is a universe category and $H : CC \to CC(C,p)$ is an isomorphism. In the last section we construct for any (set) category C with a choice of a final object and fiber products a C-system and an equivalence between C and the precategory underlying CC.

Keywords: contextual category, universe category, C-system

2010 MSC: 03F50, 03B15, 03G25

Theory and Applications of Categories, Vol. 30, 2015, No. 37, pp 1181-1214.

Published 2015-09-17.


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