# Existence of tensor product via category theory

In my class of category theory, my professor stated (without prove it) that the existence of tensor products between modules over commutative rings follows from the following result: a category $\mathcal{C}$ contains a final object, the equalizer of any pair of arrows and the direct product of any pair of objects iff all finite limits exist in $\mathcal{C}$.

Of course, since the tensor product is an initial object, one should use the dual version of the above result. More precisely, fixed a commutative ring $A$, $M_1$ and $M_2$ modules over $A$, we consider $\mathcal{D}$ as the category of $A$-bilinear morphisms of the form $M_1\times M_2\stackrel{f}{\rightarrow} X$, where a morphism between $M_1\times M_2\stackrel{f}{\rightarrow} X$ and $M_1\times M_2\stackrel{g}{\rightarrow} X'$ is a morphism $h:X\rightarrow X'$ such that $hf=g$. Then, an initial object of $\mathcal{D}$ is the usual tensor product $M_1\times M_2\rightarrow M_1\otimes_A M_2$.

So, my question is: how to use the result above in order to prove the existence of tensor product (avoiding its usual construction)?

Thank you.

• Are you willing to accept the existence of the tensor product of Abelian groups? In that case, $M ⊗_A N$ is just the coequalizer of the two maps $M ⊗ A ⊗ N → M ⊗ N$ given by the $A$-action on $M$ and $N$. I don't know if the product over $ℤ$ can be constructed as a colimit in Ab. – user54748 Aug 30 '15 at 18:20
• That result is more or less irrelevant. What you need is an adjoint functor theorem. – Zhen Lin Aug 30 '15 at 18:50
• @ZhenLin I am aware of the adjoint functor theorem, the point is that it was stated to me that this irrelevant result can be used to prove the existence of tensor products, I don't know if it is really true. – Renan Maneli Mezabarba Aug 30 '15 at 19:05
• @user54748 If I can prove this, then I can accept this :) Anyway, I'll ask my professor. Thanks – Renan Maneli Mezabarba Aug 30 '15 at 19:08
• I'm sure you can prove it, it's just a rewording of the usual construction. – user54748 Aug 30 '15 at 19:33