My favorite application is the generalization to semi-abelian categories of the fact that for every short exact sequence
$$\require{AMScd}\begin{CD} 0 @>>>K@>{k}>> A@>{p}>>B @>>>0\end{CD}\tag{1}\label{1} $$
such that $p$ is a split epimorphism, $A$ is isomorphic to $B\ltimes_\xi K$. To obtain this generalization, we need to define actions and semi-direct products in a semi-abelian category $\mathcal{C}$ first, and this is done by constructing a monad on $\mathcal{C}$. All this is done in the paper "Internal objects actions", by Francis Borceux, George Janelidze and Max Kelly.
Let me first recall the definition of semi-abelian category : a category $\mathcal{C}$ is said to be semi-abelian if
- it is pointed
- it is Barr-exact
- it is protomodular, which in this context is equivalent to the validity of the (split) Short Five Lemma
- it has binary coproducts.
Now let me first define the category $Pt(B)$ of points over an object $B$ of $\mathcal{C}$. Its objects are the triples $(A,p,s)$ where $A$ is an object of $\mathcal{C}$ and $p:A\to B$ and $s:B\to A$ are such that $ps=1_B$, and a morphism $(A,p,s)\to (A',p',s')$ is a morphism $f:A\to A'$ such that $p'f=p$ and $fs=s'$.
Define moreover two functors :
- the kernel functor $Ker:Pt(B)\to \mathcal{C}$ maps an object $(A,p,s)$ to $Ker(p)$ and a morphism $f:(A,p,s)\to (A',p',s')$ to its restriction to the kernels (which is well defined since $p'f=p$).
- the functor $B+\_$ maps an object $X$ to $(B+X,[1_B,0],i_B)$, where $i_B$ is the canonical injection $B\to B+X$, and $[1_B,0]:B+X\to B$ is the arrow induced by the identiy $1_B:B\to B$ and the zero map $0:X\to B$, and maps $f:X\to X'$ to $1_B+f:B+X\to B+X'$.
These functors form an adjunction $B+\_ \dashv Ker$, and the resulting monad is denoted $B\flat\_ $. Since we need to know the unit and counit, I will explain a little bit how it works : an arrow $(B+X,[1_B,0],i_B)\to (A,p,s)$ in $Pt(B)$ is an arrow $f:B+X\to A$ and thus it is determined by $fi_B$ and $fi_X$. But the definition of $Pt(B)$ implies that $fi_B=s$ and $pfi_X=[1_B,0]i_X=0$, so $fi_X$ is determined by its factorisation $\hat{f}X\to Ker (p)$. With this reasoning, we find that the $X$-component of the unit of this adjunction is the factorisation $X\to B\flat X=Ker([1_B,0]:B+X\to B)$ of $i_X:X\to B+X$, and the $(A,p,s)$-component of the counit is $[s,\ker (p)]:B+Ker(p)\to A$.
Now an action of $B$ on $X$ in $\mathcal{C}$ is simply a $B\flat\_$-algebra structure on $X$, i.e. a map $\xi : B\flat X\to X$. This already tells you how to define, for a split short exact sequence \eqref{1}, an action of $B$ on $Ker(p)$: it suffices to apply the comparison functor $Pt(B)\to \mathcal{C}^{B\flat\_}$, which in this case amounts to taking the restriction to $K$ of $[s,k]:B+K\to A$. Now we still need to define semi-direct products; for this, we use the left adjoint to the comparison functor. In general, this left adjoint is obtained for a $T$-algebra $(X,h)$ by taking the coequalizer of $F(h)$ and $\epsilon_{FX}$; in this case, given an action $\xi :B\flat X\to X$, the semi-direct product is thus obtained as the coequalizer of $1_B+\xi $ and $[i_B,\kappa_{B,X}]:B+B\flat X\to B+X$. Note that the comparison functor is defined on $Pt(B)$, so we have to take the coequaliser in $Pt(B)$, but this is just a coequalizer in $\mathcal{C}$. Thus the semi-direct product has a projection $\pi_B:B\ltimes_\xi X\to X$, which has a section $s_B:B\to B\ltimes_\xi X$.
Now the monadicity of the adjunction $B+\_ \dashv Ker$ (which I will prove after this) tells you exactly that for every split epimorphism $p:A\to B$, $(A,p,s)$ is isomorphic to $(B\ltimes_\xi Ker(p),\pi_B,s_B)$, and for every action $\xi : B\flat X\to X$, $Ker(\pi_B)=X$. In other words, every short exact sequence \eqref{1} with $p$ split is isomorphic to one of the form
$$\require{AMScd}\begin{CD} 0 @>>>K@>{}>> B\ltimes K@>{\pi_B}>>B @>>>0.\end{CD} $$
Now why exactly is the adjunction $B+\_ \dashv Ker$ monadic? By (a variant of) Beck's Monadicity theorem, it suffices to check that $Pt(B)$ has coequalizers of reflexive pairs, and that the functor $Ker$ preserves them and reflects isomorphisms. But it's easy to show that $Pt(B)$ has coequalizers of reflexive pairs if $\mathcal{C}$ does, and this is true if $\mathcal{C}$ is semi-abelian : indeed, given a pair of arrows $u_1,u_2:X\to Y$ with a common section in $\mathcal{C}$, then you can take the image of $(u_1,u_2):X\to Y\times Y$, since $\mathcal{C}$ is regular. This defines a relation on $Y$, which is reflective since $u_1$ and $u_2$ have a common section. Now $\mathcal{C}$ is protomodular, and thus also a Mal'tsev category, thus this relation is an equivalence relation, and by exactness it is thus a kernel pair; thus it has a coequalizer, which is then also a coequalizer of $u_1$ and $u_2$. The functor $Ker$ preserves kernel pairs (as any adjoint), but also regular epimorphisms, because these are stable under pullbacks in $\mathcal{C}$, and thus it preserves the construction given above of the coequalizer of a reflexive pair. Lastly, it reflects isomorphisms by protomodularity (as mentioned above).
