I am not an expert on elementary topos (meaning by this to work with the internal language in a Grothendiek topos) and I rather be told than struggle with the following: Consider …

The idea that stacks can be identified with groupoids internal to the base site $S$ up to what is variously called essential/Morita equivalence is well known. The basic idea is tha …

My question is long on background and motivation, and almost but not quite answered over at the nLab. I'll write up a bunch before asking my question (feel free to skip to the end …

I'm working out of Sheaves in geometry and logic, for reference. There is a characterisation of flat functors $A:C \to Set$ as those such that the Grothendieck construction $\int_ …

By a theorem of Ehresmann, topological and Lie categories (by which I mean categories internal to $Top$ and $Diff$ respectively, with the condition that the source and target in th …