A quasi-topos is a category that characterised by being finitely complete and finitely cocomplete that is also locally cartesian closed and has a strong sub-object classifier.
A topos is finitely complete, is cartesian closed and has a subobject classifier. From this it follows that it is also finitely cocomplete and locally cartesian closed. Then one can see the similarities between the definitions.
1.Is is correct to say that a topos can be defined exactly as a quasi-topos in the definition above but classifying any sub-object and not just strong ones?
2.There are other kinds of monics apart from the strong ones - split, normal, extremal & regular come to mind. Can we use the first definition (quasitopos), but using say a regular classifier? Or does it not produce anything useful?