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?
[Math] $\omega$-topos theory
ct.category-theoryhigher-category-theorytopos-theory
Related Solutions
About 1: Yes!
About 2: (Internal logic of Zariski topos) I don't think it has been done systematically. A glimpse of it is in Anders Kock, Universal projective geometry via topos theory, if I remember well, and certainly in some other places. But one point is that it is not at all easy to find formulas in the internal language which express what you have in mind. See my answer at "synthetic" reasoning applied to algebraic geometry
About 3:You can indeed glue all sorts of things:
Things fitting into the axiomatic framework of "geometric contexts": Look at the "master course on Algebraic stacks" here: http://perso.math.univ-toulouse.fr/btoen/videos-lecture-notes-etc/ This one is great reading to understand the functorial point of view on schemes and manifolds!
Commutative Monoid objects in good monoidal (model) categories: http://arxiv.org/abs/math/0509684
Commutative monads (here you can glue monoids, semirings and other algebraic structures mixing them all): http://arxiv.org/abs/0704.2030
In Shai Haran's "Non-Additive Geometry" you can even glue the monoids and semirings etc. with relations (although I wouldn't know why)
You can also glue things "up to homotopy instead" of strictly - this is roughly what Lurie's infinity-topoi are about, and also the model catgeory part of the 2nd point, or any oter approaches to derived algebraic geometry
(Edit in 2017) The PhD thesis by Zhen Lin Low is relevant. "The main purpose of this thesis is to give a unified account of this procedure of constructing a category of spaces built from local models and to study the general properties of such categories of spaces."
One of several good points of view on what a Grothendieck topology does, is to say it determines which colimits existing in your site should be preserved under the Yoneda embedding, i.e. what glueing takes already place among the affine objects. So, if you insist on glueing groups it could be a good idea to look e.g. for a topology which takes amalgamated products (for me this means glueing groups, you may want only selected such products, e.g. along injective maps) to pushouts of sheaves... Then feel free to develop a theory on this and send me a copy!
About 4: (Why don't people work with sheaves instead of schemes) They do. One situation where they do is when taking the quotient of a scheme by a group action. The coequalizer in the category of schemes is often too degenerate. One answer is taking the coequalizer in the category of sheaves, the "sheaf quotient" (but sometimes better answers are GIT quotients and stack quotients).
I suspect your confusion arises in part because homotopies of paths are continuous maps $I^2 \to X$, while 2-morphisms in $\pi_{\lt \infty} X$ are continuous maps $\Delta^2 \to X$. That is, 2-morphisms are not strictly the same as homotopies of paths.
The dictionary between the two structures is not too bad:
A 2-morphism in $\pi_{\lt \infty} X$ is a homotopy of paths, where either the beginning or the end is fixed (or perhaps it is a homotopy to or from a constant path).
A continuous map $I^2 \to X$ can be viewed as a composite of two 2-morphisms in $\pi_{\lt \infty} X$. You end up using the diagonal $(0,0) \to (1,1)$ in the square to separate the two 2-morphisms, because the simplicial structure of $\Delta^1 \times \Delta^1$ has a diagonal 1-simplex.
I personally find it rather magical that among the sixteen 2-simplices in $\Delta^1 \times \Delta^1$, a pair of them pops out as non-degenerate - it is a rewarding computation.
More generally, higher dimensional cubes, viewed as products of the simplicial set $\Delta^1$, have canonical decompositions into nondegenerate simplices. The corresponding higher homotopies are composites of homotopies with various pieces held constant.
I think this should account for the discrepancy you see in the number of face maps.
Best Answer
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: