$\newcommand{\Cats}{\mathsf{Cats}}\newcommand{\MonCats}{\mathsf{MonCats}}\newcommand{\BrMonCats}{\mathsf{BrMonCats}}\newcommand{\SymMonCats}{\mathsf{SymMonCats}}\newcommand{\CMon}{\mathsf{CMon}}\newcommand{\Mon}{\mathsf{Mon}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\Ab}{\mathsf{Ab}}\newcommand{\Grp}{\mathsf{Grp}}\newcommand{\Sets}{\mathsf{Sets}}\newcommand{\eHom}{\mathbf{Hom}}\newcommand{\C}{\mathcal{C}}\newcommand{\V}{\mathcal{V}}\newcommand{\Obj}{\mathrm{Obj}}$Let $\C$ be a $\V$-enriched category, let $V\in\Obj(\V)$, and let $A\in\Obj(\C)$.


* The **tensor of $V$ with $A$** (also called the **copower of $V$ with $A$**) is, if it exists, the object $V\odot A$ of $\C$ such that we have a $\V$-natural isomorphism
$$\eHom\_\C(V\odot A,-)\cong\eHom\_\V(V,\eHom\_\C(A,-)).$$
* Dually, the **cotensor of $V$ with $A$** (also called the **power of $V$ with $A$**) is, if it exists, the object $V\pitchfork A$ of $\C$ such that we have a $\V$-natural isomorphism
$$\eHom\_\C(-,V\pitchfork A)\cong\eHom\_\V(V,\eHom\_\C(-,A)).$$


Moreover, $\C$ is called **$\V$-co/tensored** if it has all co/tensors.


An example of these is given by any co/complete category $\mathcal{C}$, whose $\Sets$-co/tensors are given by
\begin{align\*}
X\odot A &\cong \coprod\_{x\in X}A,\\
X\pitchfork A &\cong \prod\_{x\in X}A.
\end{align\*}
Another example is given by the category $\Ab$:


* $\Ab$ is enriched over itself: given $A,B\in\Obj(\Ab)$, we have an abelian group $\eHom\_\Ab(A,B)$ whose product $(f,g)\mapsto f\*g$ is obtained via pointwise multiplication, i.e. by defining $[f\*g](a)=f(a)g(a)$. This relies crucially on the commutativity of $B$, which ensures that $f\*g$ is again a morphism of groups:
\begin{align\*}
[f\*g](ab) &= f(ab)g(ab)\\
&= f(a)\color{red}{f(b)}\color{blue}{g(a)}g(b)\\
&= f(a)\color{blue}{g(a)}\color{red}{f(b)}g(b)\\
&= [f\*g](a)[f\*g](b).
\end{align\*}
* $\Ab$ is tensored over itself via the tensor product of abelian groups $(A,B)\mapsto A\otimes\_\mathbb{Z}B$;
* $\Ab$ is cotensored over itself via the internal $\eHom$ given above, $(A,B)\mapsto \eHom\_\Ab(A,B)$.


Now, $\Ab$ is not enriched over $\Grp$, as there is no sensible tensor product for the latter; however it is "faux co/tensored" over it, as we have isomorphisms
\begin{align\*}
\eHom\_\Ab(G^\mathrm{ab}\otimes\_\Z A,B) &\cong \eHom\_\Grp(G,\eHom\_\Ab(A,B)),\\
\eHom\_\Ab(A,\eHom\_\Grp(G,B)) &\cong \eHom\_\Grp(G,\eHom\_\Ab(A,B)),
\end{align\*}
so $G“\odot”A=G^\mathrm{ab}\otimes\_\Z A$ and $G“\pitchfork”A=\eHom\_\Grp(G,B)$.


This situation is not exclusive to $\Grp$ and $\Ab$: it also occurs with $\Mon$ and $\CMon$, with $\BrMonCats$ and $\SymMonCats$, and seems more generally to occur with pairs of the form $(\Mon(\C),\CMon(\C))$ for $\C$ a symmetric monoidal category.




---


A second related point is that one may use the $\Sets$-co/tensoring of $\Grp$ together with the forgetful functor $|{-}|\colon\Grp\to\Sets$ to $\Sets$ to define "half-tensor products" $\triangleleft$ and $\triangleright$, given by
\begin{align\*}
G\triangleleft H &= |H|\odot G,\\
&\cong \coprod\_{h\in H}G,\\
G\triangleright H &= |G|\odot H,\\
&\cong \coprod\_{g\in G}H.
\end{align\*}
As noted [here](https://math.stackexchange.com/questions/616582), $G\triangleleft H$ is the free group on symbols $a\otimes b$ quotiented by the left distributivity relations $(a+b)\otimes c\sim a\otimes c+b\otimes c$, and similarly for $\triangleright$.


Because of this last point, while monoids in $(\Ab,\otimes\_\Z,\Z)$ are rings, the "monoids" in $(\Grp,\triangleleft,?)$ should be [near-rings](https://en.wikipedia.org/wiki/Near-ring)―rings with not necessarily commutative addition and only the left distributive law. The problem, however, is that $\triangleleft$ doesn't give $\Grp$ a monoidal category structure: it seems to form at best some variant of the notion of a "[lax monoidal category](https://ncatlab.org/nlab/show/lax+monoidal+category)" on it.


As with the faux co/tensors above, this kind of tensor product seems to occur also in many other contexts, including $\MonCats$ with $\Cats$-tensors or perhaps $\BrMonCats$ with "faux $\MonCats$-tensors".




---


This is a rather strange situation: we have these very natural "faux co/tensors", but the usual category-theoretic notions don't quite capture them. [Over Zulip](https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/is.20Ab.20tensored.20over.20Grp.3F), Reid Barton suggested grouping the categories $\Sets$, $\Grp$, $\Ab$, $\Ab$, $\ldots$ ($=(\Grp\_{\mathbb{E}\_n}(\Sets))\_{n\in\mathbb{N}}$) into an "$\mathbb{N}$-graded monoidal category", but again the units and associators seem to be problematic...


So---shortly---what exactly is going on here?


* How should we view these "faux co/tensors" of $\Mon\_{\mathbb{E}\_{n}}(\C)$ by $\Mon\_{\mathbb{E}\_{\leq n-1}}(\C)$?
* What exactly are the "faux monoidal structures" $\triangleleft$ and $\triangleright$ on $\Grp$, whose monoids are supposed to recover near-rings?



This is not a tensor: tensoring abelian groups over groups
The one line answer is that the category $\mathsf{Ab}$ of abelian groups is enriched over the **skew-monoidal category** $\mathsf{Gp}$ of groups, and that this "faux-tensor" defines a **skew-action** of the skew-monoidal category $\mathsf{Gp}$ on $\mathsf{Ab}$.


A *skew-monoidal structure* on a category $\mathcal{C}$ consists of a "tensor product" functor $\boxtimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$, a "unit" object $I \in \mathcal{C}$, and "associativity and unit constraint" natural transformations $\alpha \colon (X \boxtimes Y) \boxtimes Z \to X \boxtimes (Y \boxtimes Z)$, $\lambda \colon I \boxtimes X \to X$, and $\rho \colon X \to X \boxtimes I$, satisfying the original five coherence axioms of Mac Lane. The important point is that these associativity and unit constraints are *not required to be invertible*. This notion was introduced by Szlachányi in his paper



> 
> Kornél Szlachányi. Skew-monoidal categories and bialgebroids. Adv. Math. 231 (2012), no. 3-4, 1694--1730. <https://doi.org/10.1016/j.aim.2012.06.027>
> 
> 
> 


and has been much studied since, especially by the Australian school of category theory.


The "half-tensor products" of groups that you describe are part of a skew-monoidal structure on the category $\mathsf{Gp}$ of groups. This skew-monoidal structure is an instance of the family of examples described in Example 2.7 of my paper:



> 
> Alexander Campbell. Skew-enriched categories. Applied Categorical Structures 26 (2018), no. 3, 597--615. <https://doi.org/10.1007/s10485-017-9504-0>
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> 
> 


The tensor product $G \boxtimes H$ of two groups $G$ and $H$ is the group you denote by $G \triangleleft H$, i.e. the copower of $G$ by the underlying set of $H$. Note that group homomorphisms $G \boxtimes H \to K$ correspond to functions $G \times H \to K$ that are group homomorphisms in the first variable. The unit object is the free group on one generator, i.e. $\mathbb{Z}$. The associativity and unit constraints are a little more complicated to describe, but suffice it to say that they are not invertible.


This skew-monoidal structure on $\mathsf{Gp}$ is closed: the functor $- \boxtimes H$ has a right adjoint which sends a group $K$ to the group $[H,K]$ of all functions from $H$ to $K$ with the pointwise group structure; this group $[H,K]$ is the *internal hom* for this skew-monoidal structure on $\mathsf{Gp}$. Thus $\mathsf{Gp}$ is also a **skew-closed category** in the sense introduced by Ross Street in his paper:



> 
> Ross Street. Skew-closed categories. J. Pure Appl. Algebra 217 (2013), no. 6, 973--988. <https://doi.org/10.1016/j.jpaa.2012.09.020>
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Now, just as one can define categories enriched over monoidal categories, one can also define categories enriched over skew-monoidal categories. (In the terminology of my paper cited above, this is the same thing as a "left normal skew-enrichment" over the skew-monoidal category. Enrichment over skew-closed categories is defined in Street's paper cited above.)


We can define an enrichment of $\mathsf{Ab}$ over the above skew-monoidal structure on $\mathsf{Gp}$ as the change of base of the usual self-enrichment of $\mathsf{Ab}$ along the inclusion functor $\mathsf{Ab} \to \mathsf{Gp}$ equipped with the lax monoidal structure whose tensor constraint $A \boxtimes B \to A \otimes B$ is the homomorphism $U(B) \odot A \to A \otimes B$ whose component at an element $b \in B$ is $-\otimes b \colon A \to A \otimes B$.


Unpacking this, we have that, for each pair of abelian groups $A$ and $B$, the hom-group $\underline{\operatorname{Hom}}(A,B)$ is the usual group of group homomorphisms from $A$ to $B$, with its pointwise group structure, but where we have forgotten that it's abelian. For each triple of abelian groups $A$, $B$, and $C$, the composition homomorphism $\underline{\operatorname{Hom}}(B,C) \boxtimes \underline{\operatorname{Hom}}(A,B) \to \underline{\operatorname{Hom}}(A,C)$ corresponds to the usual composition function $\operatorname{Hom}(B,C) \times \operatorname{Hom}(A,B) \to \operatorname{Hom}(A,C)$, but where we have forgetten that it's a group homomorphism in the second variable. Similarly, the unit homomorphisms $\mathbb{Z} \to \underline{\operatorname{Hom}}(A,A)$ simply pick out the identity homomomorphisms.


(Note that this enrichment of $\mathsf{Ab}$ over $\mathsf{Gp}$ can also be seen an instance of Example 2.7 of my paper cited above, since the category of abelian groups is equivalent to the category of group objects in $\mathsf{Gp}$.)


As you've spelled out in your question, the hom-functor $\underline{\operatorname{Hom}} \colon \mathsf{Ab}^\mathrm{op} \times \mathsf{Ab} \to \mathsf{Gp}$ is part of a two-variable adjunction, and so there are defined tensoring and cotensoring operations of an abelian group by a group. In particular, the tensoring operation defines a **skew-action** of the skew-monoidal category $\mathsf{Gp}$ on the category $\mathsf{Ab}$, in the sense of the paper:



> 
> Stephen Lack and Ross Street. Skew-monoidal reflection and lifting theorems. Theory Appl. Categ. 30 (2015), Paper No. 28, 985--1000. <http://tac.mta.ca/tac/volumes/30/28/30-28abs.html>
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> 
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Note that a skew-action of a skew-monoidal category $\mathcal{V}$ on a category $\mathcal{C}$ is simply an oplax monoidal functor $\mathcal{V} \to \operatorname{Fun}(\mathcal{C},\mathcal{C})$.


