There are the following Nakayama style lemmata:


1. (the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m\_1, \ldots, m\_n$ generate $M$ modulo $I$, where $I \subset \mathrm{Jac}(R)$, then they generate $M$.
2. (the graded Nakayama lemma) See [How to memorise (understand) Nakayama's lemma and its corollaries?](https://mathoverflow.net/questions/61446/how-to-memorise-understand-nakayamas-lemma-and-its-corollaries).
3. (the filtered Nakayama lemma) See [How to memorise (understand) Nakayama's lemma and its corollaries?](https://mathoverflow.net/questions/61446/how-to-memorise-understand-nakayamas-lemma-and-its-corollaries).
4. (the topological Nakayama lemma, see [Neukirch, Schmidt, Wingberg], Cohomology of Number Fields, (5.2.18)): Let $\mathcal{O}$ be a commutative local ring complete in the $\mathfrak{m}$-adic topology with finite residue field of characteristic $p$. Assume $G$ is a pro-$p$-group and $M$ is a compact $\mathcal{O}[[G]]$-module. If $M/\mathfrak{m}$ is a finitely generated $\mathcal{O}[[G]]$-module, so is $M$.
5. (Burnside's basis theorem, see also <http://groupprops.subwiki.org/wiki/Burnside%27s_basis_theorem>) Let $G$ be a group such that its Frattini subgroup $\Phi(G)$ is finitely generated. Then a subset of $G$ generates $G$ iff it generates it modulo $\Phi(G)$.


[tbc]


Now my question is: Is there a common categorical version, like there is a categorical generalisation of Baer's criterion (In a suitable abelian catgory, an object $I$ is injective iff for all subobjects $U$ of a generator $G$ and morphisms $U \to I$ there is a lift $G \to I$)?



a categorical Nakayama lemma?
Let me describe a common generalization of Nakayama's lemmas and Burnside's basis theorem which may shed some light here. Let $G$ be a group and $P$ a set of endomorphisms of $G$. A $P$-subgroup will be a subgroup of $G$ which is closed under acting by elements of $P$. We'll call $\mathbb{Fr}\_P(G)$ the "$P$-Frattini subgroup of $G$", defined as the intersection of all maximal $P$-subgroups. Note the following special cases:


* When $P$ is empty then $\mathbb{Fr}\_P(G)$ is the Frattini subgroup of $G$.
* When $R$ is a ring, $G$ its additive group and $P$ its multiplicative semigroup then $\mathbb{Fr}\_P(G)$ is $J(R)$, the Jacobson radical of $R$.
* When $P$ is as above, and $G$ is an $R$-module then $\mathbb{Fr}\_P(G)$ contains $J(R)G$.


Let's denote the smallest $P$-subgroup containing a set $S$ by $\langle S\rangle\_P$, and call an element $x\in G$, a non-generator if $G=\langle S,x \rangle\_P$ always implies $G=\langle S\rangle\_P$. We have the following theorem:



> 
> The set of non-generators of $G$ is precisely $\mathbb{Fr}\_P(G)$.
> 
> 
> 


By taking $P$ empty we obtain Burnside's basis theorem. By taking $G$ to be an $R$-module and $P$ to be the multiplicative semigroup of $R$ we recover Nakayama's lemma. If $R$ is a graded ring and we take $P$ to be the semigroup of elements of positive degree and $G$ to be a graded $R$-module, we recover the graded version of Nakayama's lemma, something similar should hold for the filtered version. Surely someone has taken up this point of view (Which I learned from Gruenberg's "Cohomological topics in group theory") to define a Frattini object for a large class of categories? 


