Let $V$ and $W$ be $k[G]$-modules for an algebraically closed field, $k$ and $G$ a group. Now I know that $V\otimes W$ has a $k[G]\otimes k[G]$-module structure, and because $k[G]$ is in fact a Hopf algebra, I can give $V\otimes W$ a $k[G]$-module structure using,
$$k[G] \xrightarrow{\Delta} k[G]\otimes k[G] \xrightarrow{\rho_V\otimes \rho_W} \mathrm{End}(V)\otimes\mathrm{End}(W)\xrightarrow{}\mathrm{End}(V\otimes W)$$
where $\Delta$ is $g \mapsto g\otimes g$ and $\rho_{V}$ is the map $k[G]\to\mathrm{End}(V)$ or identically for $W$.
I have heard this described as taking a pullback. In terms of diagrams, what is this pullback? I.e. we are taking the limit of which diagram of the form $A \xrightarrow{} B \xleftarrow{} C$?
Best Answer
The word "pullback" is sometimes used to refer to the map $f^* : \hom(Y,Z)\to \hom(X,Z)$ given by precomposition with $f: X\to Y$.
So in your case "pullback" just means $\Delta^*$, i.e. precomposition by the comultiplication.
The reason (or at least a reason) is that if you think of a function $g:Y\to Z$ as a sort of bundle over $Y$ (the fiber over $y$ being $g(y)$, so the elements of $Z$ are to be thought of as geometric or algebraic objets for instance) then the pullback bundle corresponds to $g\circ f$. There are ways in which this analogy is precise, others in which it is not.
In your situation, you're thinking of your tensor module as living "over $k[G]\otimes_k k[G]$" and you "pull it back" to $k[G]$ along the diagonal.
(Note : This conflicts with the geometric notion of pullback in the case of quasi-coherent sheaves over schemes which corresponds to an extension of scalars rather than a restriction such as here )