So I don't have to worry about formalities, in the following let $\mathscr{C}$ be a sufficiently nice category--at least nice enough so that the following definition makes sense. I believe concrete and locally small is enough, but I really am just thinking of tame examples like $\mathbf{Grp},\mathbf{Ring},R$-$\mathbf{Mod}$,$\mathbf{Top}$, etc.
Definition: Let us say that a homgeneity pair in $\mathscr{C}$ consists of a pair $(X,G)$ where $X$ is an object of $\mathscr{C}$ and $G$ is a subgroup of $\text{Aut}_\mathscr{C}(X)$ such that the usual action $G\curvearrowright X$ is transitive. Let us say that an object $X$ of $\mathscr{C}$ has a homogeneity principle if there exists a subgroup $G$ of $\text{Aut}_\mathscr{C}(X)$ such that $(X,G)$ is a homogeneity pair.
I have two questions. The first is simple:
- What are some examples of categories $\mathscr{C}$ for which we have a full description of the elements with a homogeneity principle?
EDIT: As Thomas Andrews points out below, there is obviously a big boo-boo with the below. Non-trivial groups can never have a homogeneity principle, because the identity element is uniquely different from other elements. The below shows that the only f.g. groups such that any two non-zero points look alike are the elementary abelian $p$-groups. All vector spaces have the same issue--everything looks the same, except the zero point. I will try to think about how to remedy this, to make it coherent.
For example, in the category $f.g.\mathbf{Grp}$ of finitely generated groups, I claim that the only objects with a homogeneity principle are the elementary abelian $p$-groups. First note that necessarily any such $G$ must be equal to any of its characteristic subgroups, and so in particular, $G=Z(G)$. Now, let us then note that any such $G$ cannot have a free part. Indeed, suppose that $G\cong \mathbb{Z}^r\times A$ where $A$ is some finite abelian group. Note that there is no element of $\text{Aut}(G)$ sending $(1,0,\ldots,0,0)$ to $(2,0,\ldots,0,0)$ since $2x=(1,0,\ldots,0,0)$ has no solution with $x\in G$. Thus, we see that $G$ is necessarily finite abelian. Note necessarily every element of $G$ has the same order. Immediately from this we see that if
$$G\cong (\mathbb{Z}/p_1^{e_1}\mathbb{Z})\times\cdots\times(\mathbb{Z}/\mathbb{Z}p_m^{e_m})$$
then $p_1^{e_1}=\cdots=p_m^{e_m}=p$. Thus, $G$ is necessarily an elementary abelian $p$-group. Conversely, note that if $G$ is an elementary abelian $p$-group then $\text{Aut}_{f.g.\mathbf{Grp}}(G)=\text{Aut}_{\mathbf{Vec}_{\mathbb{F}_p}}(G)$ from where the fact that $G$ has a homogeneity principle as an element of $f.g.\mathbf{Grp}$ follows from the following trivial observation:
Observation: For any field $k$, every object of $\mathbf{Vec}_k$ has a homogeneity principle.
Examples of objects with homogeneity principle in $\mathbf{Top}$ and $\mathbf{Man}_\infty$ are topological groups and Lie groups respectively. Note though that, in general, group objects of a category do not possess a homogeneity principle. For example, in $\mathbf{Grp}$ the group objects are just abelian groups, and so can't have a homogeneity principle in general (by the above). The problem for $\mathbf{Grp}$ is that the map $G\to G\times G:x\mapsto (g,x)$ is a morphism if and only if $g=e$.
So, can we classify all the elements of $\mathbf{Top}$ or $\mathbf{Man}_\infty$ with a homogeneity principle? What about for other common categories?
The second question is almost certainly too complicated to answer. For any category $\mathscr{C}$ as above, we can form a new category $\mathscr{D}$, the category of homogenous pairs. The objects of $\mathscr{D}$ consist of homogeneity pairs $(X,G)$ and the morphisms $(X,G)\to (Y,H)$ consist of pairs $(f,\varphi)$ where $f:X\to Y$ is a $\mathscr{C}$-morphism, and $\varphi:G\to H$ is a group map such that for all $x\in X$ and $g\in G$ we have that $f(gx)=\varphi(g)f(x)$ (i.e. $f$ and $\varphi$ intertwine the actions of $(X,G)$ and $(Y,H)$).
For example, if we let $\mathscr{D}$ be the category of homogeneity pairs for $\mathbf{Man}_\infty$, then the category $\mathbf{Lie}$ of Lie groups is naturally identifiable with a subcategory of $\mathscr{D}$. Indeed, we can identify $\mathbf{Lie}$ with the category of pairs $(G,G)$ and $(f,f)$ where $f$ is a Lie group map.
So the second question is:
- Are there any categories $\mathscr{C}$ for which we can find get a handle on the category of homogeneity pairs (e.g. find a more familiar category it is equivalent to)?
EDIT: Also, if anyone has any tag suggestions, that would be greatly appreciated!
A definition which might be a useful extension is to say pick a set of objects, $\mathcal Y$ and say an object $X$ is $\mathcal Y$-homogeneous if, for any $Y\in\mathcal Y$ the action of $\mathrm{Aut}(X)$ on the set of monomorphisms $Y\to X$ is transitive.
This definition has the advantage that is it categorical, at least for locally small categories.
It's not clear how we can get your idea with $\mathbf{Ab}$.
In $\mathbf{Vec}_k$ your notion of transitivity is actually $k$-transitivity - that is, $\mathcal Y=\{k\}$.
If the category is $\mathbf{Top}$, and we define $\mathcal Y=\{1\}$ then we can say $X$ is $1$-homogeneous in the normal sense. If $\mathcal Y=\{\mathbf n\}$ consists of the space of $n$ discrete points, then a space is $\mathbf n$-homogenous if $\mathrm{Aut}(X)$ is $n$-transitive. (A while back, I asked on this site for examples of spaces that are $1$-homogenous but not $2$-homogenous.)