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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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Another situation in which a comma category is a topos is where C = 1. This just says the product of two toposes is a topos. –  Tom Leinster Mar 11 '12 at 16:55
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1 Answer

up vote 11 down vote accepted

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:

Aurelio Carboni, Peter Johnstone, Connected limits, familial representability and Artin glueing. Mathematical Structures in Computer Science 5 (1995), 441-459. Corrigenda: Mathematical Structures in Computer Science 14 (2004), 185-187.

They attribute the result to:

Gavin Wraith, Artin glueing. Journal of Pure and Applied Algebra 4 (1974), 345-348.

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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Thanks, this result is actually even applicable to my case at hand! (Regarding "glueing": that's British English; my Ding dictionary for British English only contains "glueing", but apparently these days "gluing" can also be used in British English. But likely for "European" eyes "glueing" looks better (as it is the case for me), while likely for US eyes "gluing" looks better, I guess (like "colour" -> "color").) –  Oliver Kullmann Mar 11 '12 at 17:43
Ah, thanks. My Oxford English Dictionary does in fact list "glueing", but prefers "gluing". (I'd misremembered; I thought it didn't mention "glueing" at all.) But then, it also prefers "-ize" to "-ise", contrary to many writers of British English, so I guess it's non-traditional. Anyway, glad to help. –  Tom Leinster Mar 11 '12 at 17:46
Now what about the case that $F$ is not the identity functor (with Oliver's notation)? Or can we derive the general case by applying the special case twice? –  Martin Brandenburg Mar 11 '12 at 18:24
Tom, the result that Artin gluing along a pullback-preserving functor yields a topos can be derived as a corollary of the lex comonad theorem after replacing "lex" by "pullback-preserving". My understanding is that Bob Par\'e first noticed the pullback-preserving comonad theorem (maybe unpublished, but announced during Diaconescu's thesis defense), around 1973. (The slice result is of course an immediate corollary of the pb-preserving comonad result, without passing through Artin gluing first.) –  Todd Trimble Mar 11 '12 at 21:31
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