What is an "inner isomorphism" between different groups?

Question

It is well known that if $X$ is a path-connected topological space containing points $x$ and $y$, then the fundamental groups $\pi_1(X,x)$ and $\pi_1(X,y)$ are isomorphic. Wikipedia makes the further claim that the two groups are not only identical up to isomorphism but "actually even up to inner isomorphism".

What does "inner isomorphism" mean here? I know what an inner automorphism is, but here the elements of the groups are different objects (homotopy classes of closed paths that begin and end at $x$, versus ditto at $y$), so automorphisms are definitely not on the table.

The obvious isomorphism(s) does arise by fixing a path $\alpha$ from $y$ to $x$ and considering the map $$[\gamma]\in\pi_1(X,x)\mapsto [\alpha+\gamma-\alpha]\in\pi_1(X,y)$$ which does look at bit like conjugation with $\alpha$ -- but (a) $\alpha$ lives outside either group so I'm not quite prepared to call that "inner", and (b) this seems to be very specific to the case of fundamental groups.

Is the remark in Wikipedia simply nonsense, or is there a relevant technical meaning of "inner isomorphism"? Googling didn't seem to uncover one.

Answer 1

The difference between "up to isomorphism" and "up to unique isomorphism" is that in the former case the isomorphism is not guaranteed to be unique. In general, if $X, Y$ are two isomorphic objects, two isomorphisms between them differ by an element of $\text{Aut}(Y)$.

"Up to inner isomorphism" can then be interpreted to mean the following (although I find it a strange way to put it). The only "meaningful" isomorphisms between fundamental groups at two basepoints are those that arise from paths between basepoints. Two such isomorphisms don't differ by an arbitrary element of the automorphism group; in fact they are guaranteed to differ by at worst an arbitrary element of the inner automorphism group.

Here is how I would prefer to say things. There is a very familiar category of groups and group homomorphisms $\text{Grp}$ that we all know and love, as well as a very familiar functor from pointed topological spaces to this category, namely the fundamental group functor.

There is also a 2-category of groups that is maybe less familiar: it's the category of groups, group homomorphisms, and natural transformations between group homomorphisms regarded as functors between one-object categories. Very explicitly, if $G, H$ are two groups, then a 2-morphism between two morphisms $f, g : G \to H$ is an element $h \in H$ such that $f(x) = h g(x) h^{-1}$.

This 2-category has a homotopy category $\text{Ho}(\text{Grp})$ which is not the category of groups and group homomorphisms; it's the category of groups and conjugacy classes of group homomorphisms. Now the claim is that

The fundamental group functor, viewed as a functor on path-connected but unpointed spaces, naturally takes values in $\text{Ho}(\text{Grp})$.

In fact $\text{Ho}(\text{Grp})$ is equivalent to the homotopy category of classifying spaces $BG$ where $G$ is discrete, and the above functor restricts to such an equivalence. The 2-category of groups is in turn equivalent to the homotopy 2-category of classifying spaces $BG$, continuous maps between them, and homotopy classes of homotopies between them. The ordinary category of groups is the homotopy category of pointed classifying spaces.

The punchline is that you can make sense of what a "group up to inner isomorphism" is: it's an object in $\text{Ho}(\text{Grp})$. Many groups in nature are in fact only groups up to inner isomorphism in this sense, such as the absolute Galois group of a field. (To make it a group you need to pick a "basepoint," or equivalently an algebraic closure.) This is one way of making precise the common intuition in number theory that in some sense the only meaningful information you can extract from Galois groups is conjugacy-invariant information, e.g. characters of representations.

For example, "underlying set" does not define a functor $\text{Ho}(\text{Grp}) \to \text{Set}$. An approximation is the functor $\text{Hom}(\mathbb{Z}, -)$, which returns the conjugacy classes.

Answer 2

This can be stated in a more rigorous way. In particular, for any two homotopy classes $[q], [p]$ from $x$ to $y$ (that is, they start at $x$ and end at $y$), they induce isomorphisms on the fundamental group. Namely, if $\ell$ is a loop at $x$, then $p^{-1}\ell p$ is a loop at $y$. (Similarly for $q$).

We can relate the isomorphisms as follows.

$$ [p^{-1} \ell p] = [p^{-1} q q^{-1} \ell q q^{-1} q p] = [p^{-1}q][q^{-1}\ell q][p^{-1}q]^{-1},$$ but where now we see that $[p^{-1}q]$ is a loop, and so is in the group. So they really are the same up to conjugation by an element of the group, and hence inner automorphism.

Answer 3

Let $G$ be any groupoid. There is an ${\rm Aut}$ functor $G\to{\rm Grp}$, so that any arrow $x\to y$ induces an isomorphism ${\rm Aut}(x)\to{\rm Aut}(y)$. Of course there exist pairs of group homomorphisms

$$\phi:{\rm Aut}(x)\to {\rm Aut}(y) \\ \psi:{\rm Aut}(y)\to{\rm Aut}(x)$$

such that $\psi\circ\phi\not\in{\rm Inn}({\rm Aut}(x))$ or $\phi\circ\psi\not\in{\rm Inn}({\rm Aut}(y))$ in general, but if $\phi$ and $\psi$ are images of morphisms $x\to y$ and $y\to x$ (respectively) in $G$ then $\psi\circ\phi$ and $\phi\circ\psi$ are both inner.

I believe the above is the idea behind "inner" isomorphisms. (Perhaps we could generalize it to any collection of isomorphisms of groups such that any automorphism composed from them is inner.) Here we are applying this idea to the fundamental groupoid $\Pi(X)$, which is a category whose objects are points $x\in X$ and whose morphisms are homotopy classes of paths $x\to y$.

Answer 4

"inner isomorphism" has no agreed meaning and I don't think it is a useful term to use in the wikipedia article. An inner automorphism $f: G \rightarrow G$ of a group is one that is given by conjugation by a group element ($f(x) = g^{-1}xg$, for some element $g$ of $G$).