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Chris, could you please elaborate the statement a bit? What is $A$ here? Any single object of category? What is a fixed point for a morphism in a cartesian closed category?
In fact, you can realize any homotopy type as a nerve of a category. Just take a closed under intersections cover $U$ of topological space $X$ by contractible subspaces. Then the nerve of corresponding poset is equivalent to $X$. On the other hand, groupoids can model only homotopy 1-types, i.e. spaces with $\pi_i (X)=0$ for $i>1$. To get unrestricted equivalence, one needs to consider higher categories and $\infty$-groupoids.
Dmitri, definitely they are 2-cateogries. I'm pretty sure that 1-categorically those algebraicity statements are false. E.g. associators correspond to associativity conditions in algebraic 2-theory, cartesian structure is encoded in adjunction between diagonal and multiplication, etc. But formally this doesn't seem to change anything, classical adjunction proofs can be repeated 2-categorically.
I doubt there can be any good categorification here. The problem is, positive-definite bilinear forms aren't very natural even in classical mathematics. They have no good analogues over most other fields or rings besides $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$. No finite characteristics, no p-adics, none over most extensions of $\mathbb{Q}$, and even over $\mathbb{R}$ they are not that important: indefinite products also matter much. So unless you start with something like Hilbert spaces, any good generalizations look unreasonable.
David, do you happen to know any good reference for Atiyah-Bott-Shapiro theorem? Their original paper doesn't really give any good proof, more of an observation that the groups coincide. Definitely not just a direct proof, as one could expect.
If $\mathbb{I}$ denotes $[0,1]$, then I don't know the answer, but this case is rather useless. For $\mathbb{I}$ a general topological space, the answer is no. Such topology can be given iff $X$ is locally compact. If you are fine with smaller subcategory of $\mathcal{T}op$, then there is a classical solution: work in a category of compactly generated hausdorff spaces and equip $Hom(X,Y)$ with compact-open topology.
@Charles: The 2-category of (1-)toposes looks much more like a category of locales (0-toposes) to me, and that one is certainly not a 1-topos, although many 1-toposes can be extracted from it, as reflective subcategories in overcategories (etale spaces). Also, $\mathbf{Topos}$ (and even $\mathbf{Topos}/S$) lack any good cartesian properties: most objects are not exponentiable, limits and colimits are generally difficult to describe explicitly, there is no way to classify subobjects and no good substitute for their class. It is not even well-powered! $\mathbf{Cat}$ is so much nicer.
