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I've been reading through Lurie's book on higher topos theory, where he develops the theory of $(\infty,1)$-toposes, which leads me to the following question: Is there any sort of higher topos theory on the more general $\omega$-categories, where we don't require all higher morphisms to be invertible?

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(∞,1)-categories are a lot more like 1-categories than 2-categories are. Like Tom says you should start by trying to make sense of "2-topos". – Reid Barton Nov 14 '09 at 0:27
up vote 8 down vote accepted

The short answer is no. Even 2-toposes are poorly understood -- we don't know what the right definition is. For higher dimensions, including $\infty$, we definitely don't have the answers.

Just as the primordial example of a (1-)topos is $\mathbf{Set}$ (the 1-category of sets and functions), the primordial example of a 2-topos should be $\mathbf{Cat}$ (the 2-category of categories, functors and natural transformations).

Mark Weber has done some work on 2-toposes, building on earlier ideas of Ross Street. But I think Mark is quite open about the tentative nature of this so far.

There was a good discussion of the current state of 2-toposes (and more generally n-toposes) at the $n$-Category Café last year:

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Why shouldn't the primordial example of a 2-topos be $\mathbf{Topos}$ (the 2-category of toposes, logical morphisms, and natural transformations)? – Charles Rezk Nov 14 '09 at 0:03
Goodness, I was reading your paper (book?) Higher Operads, Higher Categories not two seconds before coming here to check for an answer. Is there any easy way to get an equivalent theory to quasicategories from omega categories the way they were defined in the above paper? – Harry Gindi Nov 14 '09 at 0:17
Charles, you could of course try it... But consider stacks, interpreted as something like "sheaves of categories". Just as a typical example of a topos is a category of sheaves, one might wish a typical example of a 2-topos to be a 2-category of stacks. So a particular example of a 2-topos should be the 2-category of stacks on the 1-point space, which is Cat. That's one point of view, anyway. – Tom Leinster Nov 14 '09 at 0:47
fpqc, the ignorance of humanity on higher categories should never be underestimated. The answer to almost any intelligent question is "no one knows precisely (though many people have made guesses)". That said: if you take an infty/omega category in just about any sense (including that of my book) then you can say what it means for a k-morphism to be (weakly) invertible. A quasicat is something like an (infty, 1)-cat, i.e. an infty-cat in which all k-morphisms for k>1 are invertible. But proving any kind of equivalence, or even saying what it MEANS, are big challenges. – Tom Leinster Nov 14 '09 at 0:52
To amplify what Tom says above: I would also think the archetypical example of a 2-topos should be Cat, the category of categories. The primordial example of an (oo,2)-topos should be (oo,1)-Cat, the (oo,2)-category of (oo,1)-categories, being the collection of "2-stacks" on the point. Wouldn't you agree with that, Charles? If I may be so blunt: you are the Charles Rezk whose thoughts on this are cited to prominently in Jacob Lurie's book, I suppose? – Urs Schreiber Nov 16 '09 at 11:51

One should keep in mind that Jacob Lurie's book "only" (if I may use this word) discusses the (oo,1)-version of Grothendieck toposes/category of sheaves: the (oo,1)-toposes in Jacob Lurie's book are (oo,1)-categories of (oo,1)-sheaves/of oo-stacks.

This is less general than the "elementary (oo,1)-toposes" that one would eventually want to see, but it already goes a long way -- and it is more accessible.

Similarly, while a general theory of n-toposes for higher n is largely missing, there is a bit more known about (oo,n)-sheaves, i.e. of oo-stacks which are presheaves with values not just in oo-groupoids but in (oo,n)-categories.

For instance Ross Street once proposed a notion of descent for strict-omega-category-valued presheaves. Using a result by Verity this may be regarded as presenting descent for strict oo-groupoid valued presheaves, but I'd expect that with the required care exercised it goes further than that (and this seems to be what Street had in mind, though I can't tell that for sure).

But a more developed general theory for descent of (oo,n)-category valued presheaves is developed notably in Hirschowitz-Simpson's Descente pour les n-champs. This yields at least part of a theory of Grothendieck-style n-toposes.

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Mike Shulman has been thinking about n-toposes in general and 2-toposes in particular.

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