Sorry if this isn't the right place for this, it hasn't gotten any answers on ME. I'm reading Lang's section on field theory and he stresses that, unlike typical "universal" constructions which are determined up to unique isomorphism, algebraic closures (and by extension, their Galois groups) are determined only up to automorphism (conjugation). It seems to me that there ought to be some interpretation of this in terms of bicategories (weak 2categories). This intuition is supported by the fact that 2cells are given by conjugation when we give Grp the structure of a 2category (view groups as 1object categories, get 2cells via natural transformations). Is there any such interpretation?

2$\begingroup$ If this helps, you should think of the group of automorphisms of a fixed algebraic closure K as analogous to the fundamental group of a space X based at a point *. The choice of algebraic closure is the choice of basepoint. And the "Galois group" of a nonalgebraically closed field is something like "the fundamental group of X" defined without reference to a basepoint  an object that is really defined only up to conjugation. $\endgroup$ – JSE Oct 23 '11 at 1:12

1$\begingroup$ @JSE: Continuing the analogy, is the concept of "Galois groupoid" then useful? $\endgroup$ – Yuri Sulyma Oct 23 '11 at 2:32

$\begingroup$ I don't have a concrete example, but I have seen papers using the groupoid language. In particular, morphisms in the Galois groupoid of $X$ are described by isomorphisms of fiber functors on the category of étale sheaves of finite sets on $X$. $\endgroup$ – S. Carnahan♦ Oct 23 '11 at 4:59
I don't see bicategories coming into this in a useful way, but I think what you have is a consequence of two more general facts:
The nonuniqueness of algebraic closures is a general fact about injective hulls  they are 'unique' up to nonunique isomorphism.
Every morphism in a groupoid yields an isomorphism of vertex groups by conjugation  if G is a groupoid and x is an object of G, then the vertex group at x is G(x,x), and if $f \colon x \to y$ is a morphism then $g \mapsto f^{1} g f$ is an isomorphism between G(x,x) and G(y,y).
With the objects satisfying a universal property the comparison isomorphisms between them are unique, so that the groupoid of objects satisfying the universal property is codiscrete, i.e. there is exactly one morphism between any two objects, so in particular the vertex groups of this groupoid are trivial. For an object A in a concrete category (or in a category with a chosen class of 'embeddings') there is a groupoid of injective hulls of A that is not in general codiscrete, and so it can have nontrivial vertex groups. But this groupoid, though not codiscrete, is still connected, so that each vertex group is (nonuniquely!) isomorphic to every other via conjugation by a morphism (necessarily invertible) of injective hulls.
Edit: The fundamental group of a space, as in JSE's analogy, bears much the same relationship with the fundamental groupoid of the space  in particular, $\pi_1$s at points in the same pathcomponent are isomorphic via conjugation in the same way.

1$\begingroup$ RE injective hull: Ah, that was the nLab page I vaguely remembered and couldn't find. Thanks! There is also either a comment at the ncategory cafe or an answer here on MO by Minhyong Kim about how the Galois group is much more analogous to the fundamental group in that there is a secret 'choice of basepoint' for an interpretation of that phrase that makes sense for field extensions. Sorry I can't find it in a hurry. $\endgroup$ – David Roberts Oct 23 '11 at 10:25

3$\begingroup$ Do you mean this one: mathoverflow.net/questions/546/… ? $\endgroup$ – Finn Lawler Oct 23 '11 at 19:01

$\begingroup$ Thanks, Finn. If I'd had a little more time before bed last night I would have tracked it down myself. $\endgroup$ – David Roberts Oct 24 '11 at 3:27

1$\begingroup$ I should have said: thanks for mentioning it, it's a fantastic piece. $\endgroup$ – Finn Lawler Oct 24 '11 at 10:32
Not necessarily. In the higher categorytheoretic setting one asks for a 'contractible space' of choices (space might mean simplicial set or ncategory) instead of uniqueness. The natural 2category one might define may not be the 'right' one to get such a collection of choices, and so one could define a 2category such that these things are unique in the appropriate sense, but this might just be cooked up to give that result and not of intrinsic interest. For example, one can define the 2category of fields where the underlying 1category is $Fields$, and there is a unique 2arrow between any two parallel 1arrows. This is clearly not an interesting 2category.
(Other examples of nonunique closures are given here: http://nlab.mathforge.org/nlab/show/completion#nonunique)