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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
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

I realize this question is old, but I think it deserves a detailed response.

Generalization of curvature

First, I would like to point out that Zhen Lin is right to acknowledge that the concept of curvature is not something unique to Riemannian manifolds. In the differential geometric setting, the furthest generalization of curvature comes from the concept of a smooth vector bundle $p:E\to X$ equipped with connection $\nabla$. If the bundle has transition functions with values in some Lie group $G$, then we have an associated principal $G$ bundle $P\to X$. Defining a horizontal distribution of the tangent bundle $HP\subset TP$ (equivariant with respect to the $G$ action on $P$) gives a connection on $P$. Equivalently, this can be identified with a Lie algebra valued form $$\Theta\in \Omega^1(P,\mathfrak{g})$$ which transforms as $$R^*_g\Theta=Ad_{g^{-1}}\Theta\ \ \ g\in G.$$ The curvature of this connection is defined to be $$F=d\Theta+\Theta\wedge \Theta.$$ In the case where $X$ is a Riemannian manifold, the structure group of the tangent bundle $TX$ can be reduced to the orthogonal group $O(n)$ by imposing local trivializations which restrict to isometries on the fibers.

Classification of bundles

In the category of topological spaces, there is a classifying space $BG$ for principal $G$ bundles. More precisely, there is a universal principal $G$ bundle $$EG\to BG.$$ The maps $$P:X\to BG$$ are in bijective correspondence with principal $G$ bundles over $X$ (via pullback of $EG$ by $P$).

Since principal $O(n)$ bundles on $X$ are classified by maps into the classifying space $BO(n)$, the frame bundle for a Riemannian manifold can be identified with a map $$g:X\to BO(n)$$

Notice that in this context, we have said nothing about connections and curvatures. In fact, as far as I know, there is no topological space which classifies bundles with connection.

In a smooth, cohesive $\infty$ topos

Finally, to address your question on curvature. There is a cohesive $\infty$ topos which is the natural home for smooth manifolds and other smooth stacks.

Let $C$ be the category with objects cartesian spaces $\mathbb{R}^n$, $n\in \mathbb{N}\cup \{0\}$ and morphisms smooth maps. This category can be made into a site by taking covering families to be the good open covers of $\mathbb{R}^n$. The functor category $[C,sSet]$ is the category of smooth simplicial presheaves. It becomes a model category via the projective model structure (I won't go too much into model categories: just enough to be precise). The simplicial sheaves are given by formally (weakly) inverting the maps induced out of the homotopy colimit over the Cech nerves of good open covers. This gives a new model structure (Bousfield localization) on $[C,sSet]$. Passing to fibrant/cofibrant object gives an $\infty$ topos $Sh_{\infty}(C)$

In fact, one can show that this particular $\infty$ topos is cohesive. Formally, this means that there are adjunctions $({\rm \Pi} \dashv {\rm disc} \dashv \Gamma \dashv {\rm codisc})$ between $Sh_{\infty}(C)$ and $sSet$. In particular, the property of cohesion gaurantees the extra left adjoint $\Pi$.

In the category $Sh_{\infty}(C)$ there is an object $\mathbb{B}G_{\rm conn}$ which classifies principal $G$ bundles with connection. In fact, we have an equivalence of spaces $$\vert \Pi \mathbb{B}G_{\rm conn}\vert \simeq BG,$$ where $\vert \cdot \vert:sSet\to Top$ denotes the geometric realization functor.

One can embedd a manifold $X$ into the category $Sh_{\infty}(C)$. In particular, for a Riemannian manifold, its frame bundle equipped with the Levi-Civita connection is given by a map $$g:X\to \mathbb{B}O(n)_{\rm conn}.$$ Finally, there is a curvature morphism of simplicial sheaves $$R:\mathbb{B}G_{\rm conn}\to \Omega_{cl}^2(-,\mathfrak{g})$$ where the simplicial sheaf on the right is the $0$-truncated object given by embedding the sheaf of closed Lie algebra valued 2 forms into $Sh_{\infty}(C)$.

In the case of a Riemannian manifold, the curvature of the Levi-Civita connection is given by the composite morphism $$X\overset{g}{\to}\mathbb{B}O(n)_{\rm conn} \overset{R}{\to} \Omega^2_{\rm cl}(-,\mathfrak{o}(n)).$$ This entire discussion can be made to be intrinsic to the $\infty$ topos (using modality functors induced by the cohesive structure).

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