Tensor product classifies cooperating arrows?

Question

In this MSE answer, it is explained maps out of the tensor product of (not necessarily commutative) $R$-algebras classifies pointwise commuting pairs of arrows from the factors in the following sense $$\mathrm{Hom}(A \otimes_R B, C) = \{ (g,h) \in \mathrm{Hom}(A,C) \times \mathrm{Hom}(B,C) : \forall a \in A, b \in B, g(a) \text{ and } h(b) \text{ commute}\}.$$

This notion of "pointwise commutation" is related to a more abstract notion - that of cooperating arrows, which appears in the book Mal'cev, Protomodular, Homological and Semi-Abelian Categories by Bourn and Borceux. The relevant details are below.

Definition. A unital category is a finitely complete pointed category such that the unit injections $\jmath_i:A_i\to A_1\times A_2$ given by $\jmath_1=(1,0),\jmath_2=(0,1)$ where $0,1$ are the zero and identity arrows are jointly extremally epimorphic.

Example. The unital algebraic varieties are unital categories.

Definition. In a unital category, arrows $A\overset{f}{\to}C\overset{g}{\leftarrow}B$ are said to cooperate if there's a unique arrow $\varphi_{f,g} : A\times B\to C$ such that $\varphi_{f,g}\circ\jmath_1 = f$ and $\varphi_{f,g}\circ\jmath_2 = g$.

Example. Monoid homomorphisms cooperate iff their images commute pointwise, i.e $f(x)g(y)=g(y)f(x)$.

Thus, the non-commutative tensor product classifies pairs of ring-homomorphisms which cooperate as multiplicative monoid homomorphisms. On the other hand, the category of unital rings is not unital because it has no zero object, though the presence of a multiplicative unit is of crucial importance in the argument in the linked answer. I am confused.

1. What is the right way to organize this information and relate the symmetric monoidal structure with the notion of cooperation?

In this comment to the same answer, it is written:

Actually this is more than an analogy. Take any symmetric monoidal category, then the tensor product of algebra objects makes sense and satisfies the universal property that it classifies "commuting" morphisms of algebras. For $R$-modules this gives the tensor product of $R$-algebras, for sets this gives the (tensor) product of monoids (for example, groups).

2. What is the meaning of "tensor product of algebra objects", and what is the notion of "commuting"? Is it related to cooperating?

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Added. Just trying to organize the information involved.

• I think we're starting with a (symmetric?) monoidal $(\mathsf C,\otimes)$, e.g $(\mathsf{Set},\times)$ and $(\mathsf{Ab},\otimes,\mathbb Z)$.
• Then, we want a tensor product of internal monoids, call it $\boxtimes$.
• I guess we want $\boxtimes$ to come with injections into it, so that we may analogously define cooperation of internal monoid morphisms: internal monoid morphisms $A\overset{f}{\to}C\overset{g}{\leftarrow}B$ are said to cooperate if there's a unique arrow $A\boxtimes B\to C$ making the obvious diagram with the injections commute.
• Then this $\boxtimes$ is actually characterized by $$\mathsf{Mon}(\mathsf C)(A\boxtimes B,C)\cong \left\{ (g,h)\in \mathsf{Mon}(\mathsf C)(A,C)\times \mathsf{Mon}(\mathsf C)(B,C) \mid g,h\text{ cooperate} \right\},$$ and it is this $\boxtimes$ which retrieves the tensor product of rings. Is that right?

Answer 1

Fix a symmetric monoidal category $(\mathcal C,\otimes,I)$. A unital magma structure on an object $A$ of $\mathcal C$ consists of moprhisms $I\xrightarrow{e_A}A$, $A\otimes A\xrightarrow{\mu_A}A$ so that both composites $I\otimes A\cong A\cong A\otimes I\underset{e_A}{\overset{e_A}{\rightrightarrows}} A\otimes A\xrightarrow{\mu_A}A$ are identity morphisms. A morphism of unital magma structures is a morphism $A_1\xrightarrow{f}A_2$ so that $I\xrightarrow{e_{A_2}}A_2$ factors as $I\xrightarrow{e_{A_1}}A_1\xrightarrow{f}A_2$ and the composites $A_1\otimes A_1\xrightarrow{f\otimes f}A_2\otimes A_2\xrightarrow{\mu_{A_2}}A_2$ and $A_1\otimes A_1\xrightarrow{\mu_{A_1}}A_1\xrightarrow{f}A_2$ are equal.

An important special unital magma object (in fact a monoid object) is the unit $I$ of the category with the natural isomorphism $I\cong I\cong I$ furnished by the unit component of the unitors of the monoidal category. In particular, the units $I\xrightarrow{e_A}A$ are themselves morphisms of unital magma structures, so $I$ is the initial object of the category of unital magma structures. (For example, $\mathbb Z$, the unit object of the monoidal category of abelian groups, is the initial object in the category of rings).

Now, given a unital magma structure on $A_1$ and $A_2$, there is a canonical unital magma structure on $A_1\otimes A_2$ with units $I\cong I\otimes I\xrightarrow{e_{A_1}\otimes e_{A_2}}A_1\otimes A_2$ and multiplication $(A_1\otimes A_2)\otimes(A_1\otimes A_2)\cong (A_1\otimes A_1)\otimes(A_2\otimes A_2)\xrightarrow{\mu_{A_1}\otimes\mu_{A_2}}A_1\otimes A_2$ (notice the use of the symmetry isomorphism).

You can check that if $A_1\xrightarrow{f_1}B_1$ and $A_2\xrightarrow{f_2}B_2$ are morphisms of unital magma strctures, then so is $A_1\otimes A_2\xrightarrow{f_1\otimes f_2}A_1\otimes A_2$. In particular, the canonical morphisms $A_1\cong A_1\otimes I\xrightarrow{A_1\otimes e_{A_2}}A_1\otimes A_2\xleftarrow{e_{A_1}\otimes A_2}I\otimes A_2\cong A_2$ are morphisms of unital magma structures.

Furthermore, the associators and unitors of the monoidal category are also morphisms of unital magma structures. This means that $(\mathrm{UMagm}(\mathcal C),\otimes, I)$ itself has the structure of a monoidal category, which is furthermore a semi-cocartesian monoidal structure, i.e. the unit $I$ of this monoidal category is also an initial object.

In the case where $\mathcal C$ is the category of abelian groups, more is true: the natural maps $A_1\cong A_1\otimes I\xrightarrow{A_1\otimes e_{A_2}}A_1\otimes A_2\xleftarrow{e_{A_1}\otimes I}I\otimes A_2\cong A_2$ are joint extremal epimorphisms, i.e. they do not jointly factor through any subobject of $A_1\otimes A_2$ (the elementary way of seeing it is the standard proof: the inclusions are exactly the inclusions of $(a\otimes 1)$ and $(1\otimes b)$ which generate the tensor product). I think this might be true when $\mathcal C$ is the category of models (in Sets) of a commutative algebraic theory, but I'm not willing to commit to that right now. The point, however, is that the property of being unital (in Borceux and Bourn's sense) is a property of a semi-cocartesian monoidal structure on a category, though I imagine the well-studied case is only what they cover: the semi-cocartesian monoidal structure of a finite products category with a zero object.