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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.
@article{TAC_2015_30_a36,
author = {Vladimir Voevodsky},
title = {A {C-system} defined by a universe category},
journal = {Theory and applications of categories},
pages = {1181--1214},
publisher = {mathdoc},
volume = {30},
year = {2015},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2015_30_a36/}
}
Vladimir Voevodsky. A C-system defined by a universe category. Theory and applications of categories, Tome 30 (2015), pp. 1181-1214. http://geodesic.mathdoc.fr/item/TAC_2015_30_a36/