A comparison theorem for two types of single universes
Marta Bunge – 12 September 2001
Background A well-known method for constructing a category which contains both the open and the closed subspaces of a given space (or the open and the closed subtoposes of a given topos) is to perform a collage or glueing along a fringe functor.The resulting category inherits properties of those one starts with (e.g., a topos, a presheaf topos [CJ]). Another method is to consider the locally closed inclusions into a given space X. These agree with the UFL inclusions and are precisely the exponentiable subspaces of X [N]. This method applies also to small categories and to toposes [BN]. Twisting and Glueing In the context of an admissible KZ-doctrine M on a 2-category K, we now assume that the canonical action of the discrete opfibrations over an object E on the discrete fibrations over E (to be described) admits a right adjoint d (thought of as density or interior). Define a category Tw(A,B) of twisted morphisms of bifibrations (for M) from A to B. Assume that K has a terminal object T. We now prove the following new result. Theorem. There exists an equivalence of categories between Tw(E) and the category obtained by glueing along d. Some applications of this theorem will be given. The concrete example of the symmetric monad is dealt with in [BFu] by other methods not available in the general context. A comparison theorem of single universes for discrete fibrations and discrete opfibrations in the usual sense. In [BFi] we observed that for A any set and M(A) the free monoid on A, letting P(A) consist of the poset of post-extensions in the path-category M(A)#, there is a canonical inclusion of presheaves on the latter into the category UFL/M(A), and that this inclusion admits a right adjoint (unfolding). We now prove the following new result. Theorem. Let B be a category which admits a duration functor into a a commutative monoid T satisfying the cancellation properties that appear in [Law] (and which together have the consequence that every morphism in B is both an epi and a mono, e.g., T is taken to be the additive monoid of natural numbers or that of the non-negative reals). Then there is an inclusion of the category Tw(P(B)) into the category UFL/P(B) . We remark that whereas the former is a topos (in fact a presheaf topos) the latter need not even be a topos [J], except in the case where B also satisfies the fill-in property [BN] [BFi]. The above result extends to the case of an arbitrary (locally connected) topos E, with the notion of a UFL geometric morphism introduced in [BN]. Open questions These will be posed at the lecture. References [BFi] M.Bunge and M.Fiore, Unique factorization lifting functors and categories of linearly-controlled processes, Math. Str.Comp.Sci. 10 (2000) 137-163. [BFu] M.Bunge and J.Funk, On a bicomma object condition for KZ-doctrines, J. Pure Appl.Alg.143(1999) 69-105. [BN] M.Bunge and S.B.Niefield, Exponentiability and single universes, J. Pure Appl. Algebra 148-3 (2000) 217-250. [CJ] A.Carboni and P.T.Johnstone, Connected limits, familial representability and Artin glueing, Math. Str. Comp. Sci.5 (1993) 441-459. [J] P.T.Johnstone, A note of discrete Conduche fibrations, Theory and Application of categories 5 (1999) 1-11. [Law] F.W. Lawvere, State categories and response functors, Unpublished manuscript 1986. [N] S.B.Niefield, Cartesian inclusions: locales and toposes, Comm. Alg. 9 (16) (1981) 1639-1671.