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Let $(G_\lambda)_{\lambda\in L}$ be a family of $\mathbf{Z}$-modules. Let $\phi:\prod_{\lambda\in L}G_\lambda\rightarrow\mathbf{Z}^{(\prod_{\lambda\in L} G_\lambda)},\,x\mapsto e_x:=(\delta_x(x'))_{x'\in\prod_\lambda G_\lambda}$, be the canonical injection. Let $C$ be the sub-$\mathbf{Z}$-module of $\mathbf{Z}^{(\prod_{\lambda\in L} G_\lambda)}$ generated by elements of the form $$e_{x+y,(z_\lambda)_{\lambda\ne\mu}}-e_{x,(z_\lambda)_{\lambda\ne\mu}}-e_{y,(z_\lambda)_{\lambda\ne\mu}}$$ for $\mu\in L$, $x,y\in G_\mu$, and $z\in\prod_{\lambda\ne\mu}G_\lambda$. Write $\bigotimes_{\lambda\in L}G_{\lambda}:=\mathbf{Z}^{(\prod_{\lambda\in L} G_\lambda)}/C$ and let $\pi:\mathbf{Z}^{(\prod_{\lambda\in L} G_\lambda)}\rightarrow\bigotimes_{\lambda\in L}G_{\lambda}$ be the canonical surjection. Then the mapping $$\pi\circ\phi:\prod_{\lambda\in L}G_\lambda\rightarrow\bigotimes_{\lambda\in L}G_\lambda$$ is $\mathbf{Z}$-multilinear (definition). The $\mathbf{Z}$-module is called the tensor product (over $\mathbf{Z}$) of the family $(G_\lambda)_{\lambda\in L}$ of $\mathbf{Z}$-modules. For $x\in\prod_{\lambda\in L} G_\lambda$, write $\bigotimes_{\lambda\in L}x_\lambda:=\pi(\phi(x))$.

Let $(H_\lambda)_{\lambda\in L}$ be another family of $\mathbf{Z}$-modules and $(v_\lambda:G_\lambda\rightarrow H_\lambda)_{\lambda\in L}$ a family of $\mathbf{Z}$-linear mappings. Then there exists a unique $\mathbf{Z}$-linear mapping $$\bigotimes_{\lambda\in L}v_\lambda:\bigotimes_{\lambda\in L}G_\lambda\rightarrow\bigotimes_{\lambda\in L}H_\lambda$$ such that $\left(\bigotimes_{\lambda\in L}v_\lambda\right)\left(\bigotimes_{\lambda\in L}x_\lambda\right)=\bigotimes_{\lambda\in L}v_{\lambda}(x_{\lambda})$ for all $x\in\prod_{\lambda\in L}G_\lambda$.

In particular, let $\mu\in L$ and $\theta$ be an endomorphism of $G_\mu$. We denote by $\tilde{\theta}$ the endomorphism of $\bigotimes_{\lambda\in L}G_\lambda$ equal to $\bigotimes_{\lambda\in L}v'_{\lambda}$ where $v'_\mu=\theta$ and $v'_\lambda=1_{G_\lambda}$ for $\lambda\ne\mu$.

Now, suppose we are given a set $\Omega$, a mapping $c:\Omega\rightarrow L\times L,\,\omega\mapsto(\rho(\omega),\sigma(\omega))$ and, for all $\omega\in\Omega$, an endomorphism $p_\omega$ of $G_{\rho(\omega)}$ and an endomorphism $q_\omega$ of $G_{\sigma(\omega)}$; there correspond to them two endomorphisms $\tilde{p}_\omega$ and $\tilde{q}_\omega$ of $\bigotimes_{\lambda\in L}G_\lambda$. Set $$\bigotimes_{(c,p,q)}G_\lambda:=\left(\bigotimes_{\lambda\in L}G_\lambda\right)/\left(\sum_{\omega\in\Omega}\text{Im}(\tilde{p}_\omega-\tilde{q}_\omega)\right)$$ and let $\psi:\bigotimes_{\lambda\in L}G_\lambda\rightarrow\bigotimes_{(c,p,q)}G_\lambda$ be the canonical surjection. Then the mapping $$\varphi_{(c,p,q)}:=\psi\circ\pi\circ\phi:\prod_{\lambda\in L}G_\lambda\rightarrow\bigotimes_{(c,p,q)}G_\lambda$$ is $\mathbf{Z}$-multilinear.

Ok.–This construction seems rather involved, especially if you take into account that one still has to develop the notions of "associativity" and "commutativity"..Is there a way to clean this up? (Perhaps using some category theory?) If not, can someone suggest alternative constructions that don't sacrifice generality (e.g. by restricting to finite cases, etc.)?

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There is indeed a "less involved" construction using categorical terms. Given a family of modules $(G_\lambda)_{\lambda\in L}$, there is a functor which sends a module $M$ to the set of multilinear maps from $\prod_{\lambda}G_\lambda$ to $M$ and which sends a linear map $f:M\to N$ to the function which sends a multilinear map $g:\prod_\lambda G_\lambda\to M$ to the multilinear map $fg:\prod_\lambda G_\lambda\to N$. Then the tensor product $\otimes_\lambda G_\lambda$ can simply be defined to be a representing object of this functor. In terms of a universal property, this means that the tensor product is a module $\otimes_\lambda G_\lambda$ together with a multilinear map $\varphi:\prod_\lambda G_\lambda\to\otimes_\lambda G_\lambda$ such that for any module $M$, any multilinear map $\psi:\prod_\lambda G_\lambda\to M$ factors uniquely through $\varphi$ as $\psi=\overline{\psi}\varphi$ for some $\overline{\psi}:\otimes_\lambda G_\lambda\to M$.

The downside to this categorical definition is that it is not immediately obvious that tensor products actually exist since not all functors are representable. To prove that tensor products exist with this definition, you would need to either give an explicit construction like Bourbaki did, or prove that the functor is representable using categorical methods, such as the representable functor theorem.

Edit: Here are some of the details on how to apply the representable functor theorem (or adjoint functor theorem if you prefer) to show that tensor products exist.

Let $R$ be a (small) commutative ring (such as $\mathbb{Z}$). Then it is well-known that $R$-Mod, the category of $R$-modules, is complete. Let $(G_\lambda)_{\lambda\in L}$ be a (small) collection of $R$-modules and let $F:R$-Mod$\to$Set be the functor described earlier. Explicitly, for $R$-modules $M$ and $N$ and map $f:M\to N$,

$$F(M):=\left\{\alpha:\prod_{\lambda\in L}G_\lambda\to M\mid\alpha\text{ is a multilinear map}\right\}$$

$$F(f)(\alpha):=f\alpha$$

It is relatively straight forward to show that $F$ preserves limits. We can do this by showing that the image of a limit in $R$-mod under $F$ is isomorphic to (aka in bijection with) that limit in Set. By a theorem of category theory, it is sufficient to show that $F$ preserves products and equalizers.

Given a (small) collection $(M_i)_{i\in I}$ of $R$-modules, let $\pi_j:\prod_{i\in I}M_i\to M_j$ be the projection homomorphism and define a function $\Phi:F(\prod_i M_i)\to\prod_iF(M_i)$ by $\Phi(h)=(\pi_ih)_{i\in I}$. Verify that $\Phi$ is a bijection such that $F(\pi_j)=pr_j\Phi$ where $pr_j:\prod_i F(M_i)\to M_j$ is the projection function and conclude that $F$ preserves products.

Next, given two maps $f,g:M\to N$ between $R$-modules, the equalizer of the $R$-linear maps is $\ker(f-g)$ together with the inclusion map $\iota:\ker(f-g)\hookrightarrow M$ whereas the equalizer of the set functions $F(f),F(g):F(M)\to F(N)$ is given by $Eq_{f,g}=\{\alpha\in F(M)\mid F(f)(\alpha)=F(g)(\alpha)\}$ together with the inclusion $\tau:Eq_{f,g}\hookrightarrow F(M)$. Then define a function $\Psi:F(\ker(f-g))\to Eq_{f,g}$ by $\Psi(h)=\iota h$. Verify that $\Psi$ is a bijection such that $F(\iota)=\tau\Psi$ and conclude that $F$ preserves equalizers.

Now that we know that $F$ preserves limits, we just need to check the solution set condition which will be just slightly technical. Let $Y$ be a (small) set. Let $\kappa$ be the cardinal number

$$\kappa=\max\{\vert Y\vert,\vert R\vert,\vert\prod_{\lambda\in L}G_\lambda\vert,\aleph_0\}$$

Let $S$ be the set of all $R$-modules whose underlying set is a cardinal number less than or equal to $\kappa$ (the fact that this is actually a (small) set follows from some ZFC axioms including the axiom of restricted comprehension and the axiom of the power set). Essentially, $S$ is obtained by looking at all cardinal numbers up to $\kappa$ and looking at the set of all $R$-module structures on those cardinal numbers. Now, let $I=\cup_{N\in S}F(N)^Y$ and for each function $i:Y\to F(N)$ for $N\in S$, let $f_i=i$ and $X_i=N$. Then $(X_i)_{i\in I}$ is a (small) indexed collection of sets together with a family of functions $(f_i:Y\to F(X_i))$. Let $M$ be an $R$-module and let $h:Y\to F(M)$ be a function. Let $N$ be the submodule of $M$ generated by the set

$$\bigcup_{y\in Y}\left(h(y)\left(\prod_{\lambda\in L}G_\lambda\right)\right)\subseteq M$$

Then it follows from elementary cardinal arithmetic that $N$ has cardinality at most $\kappa$. Consequently, there exists $N'\in S$ with $N'\cong N$. Let $t:N'\to M$ be the composition of the isomorphism $N'\to N$ with the inclusion $N\hookrightarrow M$. For $y\in Y$, let $i(y)$ be the function obtained by restricting the codomain of $h(y)$ to $N$ and composing it with the isomorphism $N\to N'$. Then $i:Y\to F(N')$ is in $I$ and $F(t)\circ f_i=h$. Thus, we conclude that $F$ satisfies the solution set condition.

So, by the representable function theorem, $F$ has a representing object, and by our definition, this object is the tensor product $\otimes_{\lambda\in L}G_\lambda$.

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  • $\begingroup$ Sorry to bother you.–Do you happen to know any books/notes/etc. that use an adjoint functor theorem to prove the existence of the finite generalized tensor product? This answer gives it for the binary case: math.stackexchange.com/questions/614461/… discusses. Is there a way to apply the same argument to to a finite family of modules? $\endgroup$
    – alf262
    Jun 20 '20 at 20:27
  • $\begingroup$ @alf262 Sorry for the late response. If you are only looking for finite tensor products, then you can easily apply induction on the binary tensor product which is proven to exist in that post. But we can also show that arbitrary tensor products using a similar method as in that post. I'll edit my answer to fill in some of the details. $\endgroup$
    – Anonymous
    Jun 25 '20 at 4:18
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    $\begingroup$ Wow–thank you so much! $\endgroup$
    – alf262
    Jun 26 '20 at 20:15

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