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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?

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Well, your are correct re: 1. More interestingly, every topos is just a quasi-topos where all bimorphisms are isomorphisms. I don't know the answer to 2 off the top of my head, though! –  Malice Vidrine Feb 15 at 20:31
    
Page 120 in Elephant.1 seems to suggest that you "always" get regular monics, even when you start with different classes. Sorry I can't be of more help! –  tetrapharmakon Feb 15 at 23:18
    
@tetrapharmakon: can you expand on that a little? - what do you mean by 'always' getting regular monics? –  Mozibur Ullah Feb 15 at 23:38
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Let $\mathcal{M}$ be a class of (mono)morphisms. In the presence of an $\mathcal{M}$-subobject classifier $\Omega$, every $\mathcal{M}$-morphism can be expressed as the equaliser of some pair of maps with codomain $\Omega$; thus, every morphism in $\mathcal{M}$ must be a regular monomorphism. –  Zhen Lin Feb 16 at 0:21
    
This is not what you asked, but maybe interesting nevertheless: You can characterize regular open subsets of a topological space $X$ as such subobjects of the terminal object in the category of sheaves on $X$ which are closed with respect to the double negation Lawvere-Tierney topology. –  Ingo Blechschmidt Feb 17 at 12:59

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