# Meaning of topological tensor products in Frenkel-Gaitsgory

The appendix to http://arxiv.org/abs/math/0508382 by Frenkel & Gaitsgory (following an earlier work of Beilinson) describes three different monoidal structures, denoted by $\otimes^!,\otimes^*,$ and $\otimes^\to,$ on the category of topological vector spaces, which are used to define various kinds of structures and actions in this category. They are defined as follows:

1. If we write $V=\lim_i V^i$ and $W=\lim_i W^i$ for $V^i,W^i$ discrete, then $$V\otimes^!W:=\lim_{i,j} (V^i\otimes W^j)$$
2. If $W$ is discrete and $W=\bigcup_kW_k$ with $W_k$ finite-dimensional, then $$V\otimes^\to W:=\text{colim}_k (V\otimes W_k).$$ If we write $W=\lim_j W^j$ for $W^j$ discrete, then $$V\otimes^\to W:=\lim_j (V\otimes^\to W^j)$$
3. Finally, $V\otimes^* W$ is defined so that $\text{Hom}(V\otimes^*W,U)$, for $U$ discrete, is the set of bilinear continuous maps $V\times W\to U.$

What follows is a series of definitions that combine these operations in ways which I have never been able to keep straight in my head. For instance, topological algebras are defined with respect to the $\otimes^\to$ structure, their modules are defined with respect to the $\otimes^!$ structure, and topological Lie algebras are defined with respect to the $\otimes^*$ structure. Then various combinations of these are defined with respect to some combination of the monoidal structures.

I'm sure that there's some much better ways to think about these tensor products that makes clearer what each of them is good for and why certain kinds of structure ought to be defined with respect to certain of them. This is my question. In other words,

What is the right way to understand these three monoidal structures? How can I keep straight what each of them is good for?

Thing one is that if $\mathscr{C}$ is a category enriched over vector spaces, then the pro-category $\text{Pro}(\mathscr{C})$ is enriched over pro-vector spaces with the $\stackrel{\to}{\otimes}$ monoidal structure.
Thing two is a remark from Beilinson's paper. A topological algebra $A$ is an algebra with respect to $\stackrel{*}{\otimes}$. If the topology on $A$ has a base consisting of left, resp. two-sided ideals then $A$ is an algebra with respect to $\stackrel{\to}{\otimes}$, resp. $\stackrel{!}{\otimes}$. So e.g. the ring of functions on an affine formal scheme is a commutative algebra for $\stackrel{!}{\otimes}$.