I am wondering if prestacks over a fixed category, or stacks over a fixed site, form an (interesting) category, equipped with prestacks morphisms. First of all, do they form a proper class or a set?
Also, since sheaf categories on a topological space form a topos, I would like to know if there is a similar (and similarly powerful) notion for stacks, which in many senses are parallel to sheaves. So: do stacks over a fixed site form a topos? And if not, do they form an interesting category?
My question is formulated for topological stacks. Anyway, also answers in the context of algebraic stacks are welcome.
Thank you in advance. Sorry if the question is too vague, but my tools do not allow me to formulate more precise statements.