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When you think about it the right way the idea is fairly simple : Here the "Free topos" means the "object classifier", that is the classifying topos of the theory with just one sort (one type) and no …

No. Take $S$ to be Sets, then for any set $s$, $f^* s \times x$ is the coproduct of $s$-copies of $X$, and a complemented subobject of $f^* s \times x$ is the same as an $s$ indexed collection of com …

If you are willing to accept internal argument instead of purely categorical (external) one, a very good reference for this is Palmgren and Vickers' paper: " Partial Horn Logic and cartesian categorie …

In some of my works I need to prove some results within the internal logic of categories with not much structures (like pretoposes or even just categories with finite limits). The kind of things I wan …

There is on the nLab page "Grothendieck topos" a part about the theory of Grothendieck toposes in weak foundation. It claims that the equivalence for a category between the Giraud's axioms and being …

I would like to explain why I think the answer is no, but of course there is no way to prove this, and probably some way to use some geometric insight when talking about elementary toposes. My main p …

I am not sure I understand what is your question, but I think the following observation should answer it: The classifying topos of an algebraic theory is the topos of presheaves over the opposite cat …