Dear Thomas, How shall we wrap this up? 1. Do you agree that the structure I was aiming for was correct for my intended applications? As you say, the pseudo-pullbacks of bgms are already in the Elephant, though I think there are still 2-categorical details to be dealt with in order to make that indexed. I believe I have done that with a fibred 2-category along the lines of Buckley's paper. My original question was to see if anyone else had gone in a similar direction. 2. Why are algebraic and geometric reindexing given by pre- and post-composition? My reading is that "algebraic" is composition, "geometric" is pullback. 3. What's your understanding of "change of base" here? The prototype example is the codomain fibration, where the base change morphisms between slices are given by pullback. What that does within an elementary topos translates into what I was doing restricted to BTop, the fibre over S being just the slice (of bgms with codomain S). All the best, Steve.
On 4 Dec 2016, at 12:00, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:
Steve,
we are speaking about different things. Pullbacks of bgm's along arbitrary gm's are described sufficiently well [Joh77] and in the Elephant.
But what I have commented on was how to appropriately captured change of base when studying topose sover a base topos.
I am working on the "algebraic" side whereas you think on the "geometric" side. Reindexing in terms of the algebraic view is given by precomposition but reindexing in terms of the geometric view is postcomposition.
What I want to say is that pullbacks in Top are very different from the change of base for relative topos theory.
Thomas
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