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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 2-categories). This intuition is supported by the fact that 2-cells are given by conjugation when we give Grp the structure of a 2-category (view groups as 1-object categories, get 2-cells via natural transformations). Is there any such interpretation?

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1 Answer 1

It may be a partial answer to your question:

Instead of functors $\mathcal A\to\mathcal B$ we can consider profunctors $\mathcal F:\mathcal A^{op}\times\mathcal B\to\mathcal{Set}$, best to consider the collage of $\mathcal F$, which is a category containing (isomprphic copy of) $\mathcal A$ and $\mathcal B$ and the set $\mathcal F(A,B)$ is considered as the set of (so called hetero-)morphisms from $A$ to $B$.

A profunctor is functorial iff every object $A\in\mathcal A$ has a reflection in $\mathcal B$ (i.e., a universal arrow among all $A\to\mathcal B$ arrows. This defines an $\mathcal A\to\mathcal B$ functor, up to nat.isomorphism. Dually, $\mathcal F$ is cofunctorial, if each $B$ has a coreflection in $\mathcal A$, determining a $\mathcal B\to\mathcal A$ functor. Adjunction means exactly both.

Functors arising by universal properties are functorial or cofunctorial profunctors, more rigorously. Categories with profunctors and profunctor morphisms (either natural transformation between $\mathcal A^{op}\times\mathcal B\to\mathcal{Set}$ functors, or simply functors between their collages, acting on $\mathcal A$ and $\mathcal B$ identically) do form a bicategory instead of a 2-category, probably that's what you are looking for.

For groups $G,H$, viewed as categories, a profunctor between them is just a biact, which consists of two commuting group actions on the same set ($\mathcal F(*_G,*_H)$), $G$ acts from left, $H$ acts from right.

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