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@SebastianGoette : What I meant was that ringed spaces (or more generally ringed toposes) are just the right generalization to make sense of the phrase "being locally isomorphic to". Any fixed locally ringed space gives you a notion of "manifolds" modeled on that space.
What Anton means is the set of automorphisms of a single object in your groupoid. This will always be a group, for trivial reasons. In a given connected component all objects have isomorphic automorphism groups (again for trivial reasons). Furthermore, the $hom(x,y)$ sets for distinct $x,y$ in a connected components are $Aut(x)$-torsors, i.e. copies of $Aut(x)$ without a distinct identity element. Since the action groupoid of a group acting on itself is connected and has trivial $Aut(x)$, the groupoid is completely determined up to isomorphism by its cardinality.
Yes, but the $d_\infty$-norm also fails to be induced by a riemannian metric. (And, it also doesn't have all straight lines as geodesics). For every riemannian metric, geodesics are locally unique.
If your distance function comes from a metric structure, the Hopf Rinow-Theorem tells you that the minimizing curve between any two points (globally) is a geodesic. Since at least locally geodesics are unique, that minimizing curve has to be locally everywhere a line, hence is a line. I however do not know wether every distance function is induced by a metric structure.
My guess is that you get the universal abstract $\sigma-$algebra by adding formal complements of the elements in your $\sigma-$frame in a similar way to how you construct the Grothendieck group for a commutative monoid or localize a commutative ring. Note that the explicit construction isn't that important, you get the existence of the adjoint by the adjoint functor theorem (the forgetful functor preserves limits).
