This is not a tensor: tensoring abelian groups over groups

Question

$\newcommand{\Cats}{\mathsf{Cats}}\newcommand{\MonCats}{\mathsf{MonCats}}\newcommand{\BrMonCats}{\mathsf{BrMonCats}}\newcommand{\SymMonCats}{\mathsf{SymMonCats}}\newcommand{\CMon}{\mathsf{CMon}}\newcommand{\Mon}{\mathsf{Mon}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\Ab}{\mathsf{Ab}}\newcommand{\Grp}{\mathsf{Grp}}\newcommand{\Sets}{\mathsf{Sets}}\newcommand{\eHom}{\mathbf{Hom}}\newcommand{\C}{\mathcal{C}}\newcommand{\V}{\mathcal{V}}\newcommand{\Obj}{\mathrm{Obj}}$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.

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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, $G\triangleleft H$ 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―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" 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".

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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, 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?

Answer 1

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

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

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

Note that a skew-action of a skew-monoidal category $\mathcal{V}$ on a category $\mathcal{C}$ is simply an oplax monoidal functor $\mathcal{V} \to \operatorname{Fun}(\mathcal{C},\mathcal{C})$.