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I'm by no means an expert in either category theory or topos theory, but I'm trying to gain some perspective on traditional geometric ideas in this context.

Topos theory claims that it is a geometric theory as it is a generalisation of the idea of a sheaf (which is also dual to the idea of an etale bundle).

Is it possible to introduce the notion of curvature in this context?

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Take a look at synthetic differential geometry, developed by John L. Bell and others, in a category theoretical context and in particular toposes. –  user72694 Jul 31 '13 at 12:09
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The question confuses two notions: toposes as generalised spaces (as exemplified by sheaf toposes $\mathbf{Sh}(X)$ for a topological space $X$) and toposes as categories of spaces (as exemplified by simplicial sets). Which do you want to think about? –  Zhen Lin Jul 31 '13 at 17:34
    
@Lin: presumably its when I'm talking about cohesive toposes that you mention categories of space? I assmue it is - so I've removed it. –  Mozibur Ullah Jul 31 '13 at 19:22
    
Curvature isn't a property of a bare topological space, though. Before asking about toposes you may as well ask about notions of curvature that can be applied to more than just Riemannian manifolds. –  Zhen Lin Jul 31 '13 at 21:18
    
@Lin: Isn't just a smooth manifold enough? Once you have a connection, which is a horizontal sub-bundle of the 2nd tangent space, you can also define a curvature at least locally. This is why I was asking about cohesive toposes as they have a notion of smoothness. I recall reading somewhere on nlab, of bundles with connections in the context of higher gauge theory. So maybe that is the way to go. I'm really just looking for some orientation. –  Mozibur Ullah Jul 31 '13 at 21:34
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