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What is an example of a functor $$F : \mathcal{C} \to \mathcal{D}$$ between two Grothendieck toposes which preserves colimits and finite products, but is not left exact (i.e., does not preserve pullbacks)?

I just assume that there is such an example, since otherwise the notion of an algebraic morphism between toposes would probably not include left-exactness. But I am not experienced enough in topos theory to see such an example.

My question is essentially why the forgetful strict $2$-functor from (toposes with algebraic morphisms) to (cocomplete symmetric monoidal categories with cocontinuous symmetric monoidal functors) is not fully faithful.

I already asked this on math.SE.

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For any small category $J$, the colimit functor $\mathsf{Set}^J \to \mathsf{Set}$ preserves colimits. It preserves finite limits if and only if $J$ is filtered and it preserves finite products if and only if $J$ is sifted. So we only need an example of a sifted but non-filtered category. One such example is $\Delta^{\mathrm{op}}$ in which case the colimit functor reduces to $\pi_0 \colon \mathsf{sSet} \to \mathsf{Set}$.

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  • $\begingroup$ This is a wonderful example. I have learned a lot from it. Thank you. $\endgroup$
    – HeinrichD
    Commented Sep 28, 2016 at 14:24
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    $\begingroup$ More generally, the "$\pi_0$" functor of any strongly connected topos is an example. $\endgroup$
    – Mike Shulman
    Commented Sep 29, 2016 at 3:12

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