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One of the definitions of elementary topos in nLab says that a category is a topos if it has finite limits, it is cartesian closed and it has a subobject classifier.

However, Lawvere in the introduction to "Toposes, algebraic geometry and logic" says that a category is a topos if it is cartesian closed and it has a subobject classifier.

I think that perhaps you can derive finite limits from being cartesian closed and the subobject classifier. Is this the case? Where can I find such proof? Why nLab is making such assumption explicitely?

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    $\begingroup$ You can find the proof that a topos has equalizers on page 6-7 of the paper in your link. $\endgroup$ – Arnaud D. Jun 21 '18 at 9:19
  • $\begingroup$ I should have kept reading. Any comment about why nLab is doing such assumption explicit? $\endgroup$ – M. Learner Jun 21 '18 at 9:27
  • $\begingroup$ @M.Learner Probably because the nLab usually takes an intentional point of view. Elementary topoi are tought as the categories in which you can make sense of geometric theories: finite limits are a prerequisite to interprete the arities of such theories. The fact that you can derive these limits from other properties is quite nice but it is noise. In the same way that a group morphism is defined to be a semi group morphism $\varphi$ such that $\varphi(1)=1$ even if we can deduce it. $\endgroup$ – Pece Jun 21 '18 at 11:32
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    $\begingroup$ @M.Learner Well, the nLab only defines subobject classifiers for categories with finite limits... This might seem strange, but if you look carefully at the proof I mention in my first comment (which I didn't really do before posting said comment), you will see that in fact it assumes that pullbacks (at least along the truth map $1\to \Omega$) exist. $\endgroup$ – Arnaud D. Jun 21 '18 at 12:07

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