2024-10-18

This following is a very brief summary of what happened in class on 2024-10-18.

Today we asked:

Running through a thought experiment, we ultimately decided that the search for a bimodule $P$ that would "embody the spirit" of multiplying elements in $M$ with elements in $N$ would amount to search through all of the sets of "bilinear, $S$-balanced, $(R,T)$-set maps" $M\times N\to P$ (where here we are suppressing the forgetful functors to $\mathbf{\text{Set}}$). We then claimed that this collection of sets (one for each bimodule $P$) was the object function of a functor $F$ from the category of $(R,T)$-bimodules to the category of sets.

In this language, the "universal property" of our bimodule $M{\otimes }_{S}N$ is that it "represents" that functor; i.e., we have a natural isomorphism

Although we didn't prove this natural isomorphism, we did give the idea for how it works. The full details are available here.

We ended by covering some additional properties of our tensor product functor, and the idea of accidentally stumbling into the land of The 2-category of bimodules. Finally, we noted (without proof), that there is a second universal property of our tensor product construction, which reveals it to be a left adjoint of another functor.

Concepts

• Tensor Products III - Balanced Maps and a Universal Property of the Tensor Product
• Tensor Products IV - The Adjoint Property
• The 2-category of bimodules

References

• Mac Lane: Section 10.4