Date: Mon, 3 Nov 1997 15:44:10 -0400 (AST) Subject: local maps of toposes are always UIAO Date: Mon, 03 Nov 1997 14:51:58 MEZ From: Thomas Streicher I'd like to know whether the following simple observation is well known. If F -| U : E -> S is a local map of toposes i.e. Gamma : Gl(F) -> Gl(Id_S) has a fibred right adjoint Nabla then U is full and faithful, i.e. one has the situation of a Unity and Identity of Adjoint Opposites in Lawvere's sense. Of course, UIAO entails that the geom. morph. is local. For the reverse direction the argument is as follows. If Gamma has a fibred right adjoint Nabla then Gamma preserves sums, i.e. is a cocartesian functor. Let FI ============ FI || | || | Ff be a cocartesian arrow in E / F || V FI -----------> FJ Ff then its image under Gamma is the left square in the diagram below i q I ----------> P ------>UFI || | | || | p pbk | UFf with q o i = eta_I || V V I ----------> J ------>UFJ f eta_J where i is an isomorphism as Gamma is cocartesian. That means eta_I I -------> UFI | | | | UFf is a pullback for all f : I -> J V V J -------> UFJ eta_J Choosing J = 1 we get that eta_I is an iso, i.e. eta is a natural iso. Thus, F and the right adjoint of U are both full and faithful. Of course, this argument doesn't go through when "local" is defined as U having an ordinary right adjoint. Thomas Streicher Date: Tue, 4 Nov 1997 08:19:03 -0400 (AST) Subject: Re: local maps of toposes are always UIAO Date: Tue, 4 Nov 1997 10:31:16 +0000 (GMT) From: Dr. P.T. Johnstone Thomas Streicher asked > I'd like to know whether the following simple observation is well known. > If F -| U : E -> S is a local map of toposes i.e. Gamma : Gl(F) -> Gl(Id_S) > has a fibred right adjoint Nabla then U is full and faithful, i.e. one has > the situation of a Unity and Identity of Adjoint Opposites in Lawvere's > sense. Yes, there is a simple proof of this fact in Proposition 1.4 of "Local maps of toposes" by Johnstone & Moerdijk (Proc. London Math. Soc. (3) 58 (1989), 281--305).