It is well known that for a topos C and an object x of C, the slice category C/x is also a topos ("topos" here as "elementary topos"). Now there is the more general concept of a comma category (F,G), with functors F:A->C, G:B->C. I believe that there are reasonable generalisations of the above fact, but I can't find anything in the literature. So my question: Which criterions are known for functors F, G, so that the comma category (F,G) is a topos?
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The Artin gluing of a functor $G\colon B \to C$ is the comma category $(1_C \Downarrow G)$ (or in the notation of the question, $(1_C, G)$). If $B$ and $C$ are toposes and $G$ preserves pullbacks, then its Artin gluing is also a topos. I learned this from:
They attribute the result to:
According to Carboni and Johnstone, Wraith proved it under the hypothesis that $G$ preserves all finite limits, but they add that it was observed very soon afterwards (by whom, they don't say) that it's enough to assume that $G$ preserves pullbacks. (Incidentally, I don't know what that extra "e" in "glueing" is doing there. My dictionary says that's wrong.) The strengthened version of Wraith's result includes the famous result on slices, since if $B = 1$ then a pullback-preserving functor $B \to C$ is just an object $c$ of $C$, and its Arting gluing is then $C/c$. |
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