Is there a topos-like category that classifies regular subobjects? 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?
 A: Oswald Wyler's Lecture Notes on Topoi and Quasitopoi suggests an interesting view on this question. Let me regurgitate some of his Chapter 1.
Define a subobject classifier for a class $\mathscr M$ of monomorphisms to be a monomorphism $\Omega\hookleftarrow T$ such that monomorphisms $X\hookleftarrow U\in\mathscr M$ are pullbacks of $\Omega\hookleftarrow T$ along unique characteristic morphisms $X\xrightarrow{\chi}\Omega$. Note the class $\mathscr M$ is stable under pullbacks whenever they exist.

The answer to the second question is that, under a reasonable assumption, $\mathscr M$ can only be the class of all regular monomorphisms. In detail:
The object $T$ is subterminal, and furthermore the unique morphism $X\to T$ exists if and only if the class $\mathscr M$ includes the identity morphism $X\xleftarrow{\mathrm{id}_X}X$ (why? because $X\xleftarrow{\mathrm{id}_X}X\xrightarrow{f}T$ is always pullback of $X\xrightarrow{f}T\hookrightarrow\Omega\hookleftarrow T$, hence the characteristic morphism $X\to\Omega$ of $X\xleftarrow{\mathrm{id}_X}X$ is the unique morphism $X\to\Omega$ that factors through $\Omega\hookleftarrow T$).
Hence, if $\mathscr M$ includes all identity morphisms, $T$ is a terminal object. But we need less: if $\mathscr M$ includes the identity morphism $\Omega\xleftarrow{\mathrm{id}_\Omega}\Omega$, then $\Omega\hookleftarrow T$ is a split monomorphism because a composite $T\hookrightarrow \Omega\to T$ exists and has to be the unique morphism $T\xrightarrow{\mathrm{id}_T}T$. Since split monomorphisms are regular monomorphisms, and regular monomorphisms are stable under pullbacks, it follows that $\mathscr M$ is a class of regular monomorphisms.
Hence, if you want a subobject classifier for a class $\mathscr M$ that contains all equalizers (this is the reasonable condition), $\mathscr M$ has to be precisely the class of regular monomorphisms (and $T$ is a terminal object since identity morphisms are equalizers). 

The answer to the first question is that a topos and a quasitopos both have a regular subobject classifier, but the classes of regular monomorphisms in topoi and quasitopoi are assumed to have a different extent.


*

*In a topos, regular monomorphisms exhaust all the monomorphisms

*In a quasitopos, regular monomorphisms exhaust all the strong monomorphisms (in particular, regular monomorphisms are closed under composition).


The latter condition is implied by requiring that every monomorphism has a regular mono-epi factorization, which in the presence of cokernel pairs of monomorphisms (which quasitopoi have by definition) is implied by requiring that regular monomorphisms are closed under composition. So from Wyler's point of view, topoi and quasitopoi have subobject classifiers for the same kind of class $\mathscr M$ of monomorphisms: a class closed under composition and containing all equalizers.
