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Setup: we consider the category of locally convex topological vector spaces with morphisms as continuous linear maps. This time, I'm explicitly allowing the axiom of choice (or at least the Hahn-Banach Theorem) - for reasons that will become obvious in a moment. The space of continuous linear maps from one LCTVS to another can be given a variety of structures as an LCTVS, for example:

  1. Uniform convergence on bounded sets (strong topology)
  2. Uniform convergence on relatively compact sets
  3. Uniform convergence on compact sets
  4. Uniform convergence on finite sets (simple topology)
  5. Pointwise convergence on each of the above (varieties of weak topologies)

These examples are all functorial in both source and target.

Question: Are there any other functorial topologies?

Background: One day, gathered round a table in the back room of the n-category cafe, Todd Trimble and I (with occasional help from others) were talking about monoidal structures on the category of LCTVS (we were meant to be talking about smooth spaces). We got quite far and found several such structures which differ essentially just up to topology. In particular, there's a monoidal structure (even symmetric) for each of the uniform topologies above. And if anyone comes up with another functorial topology based on uniform convergence, there'll be a monoidal structure for that. The discussion stalled a little towards the end and I'd quite like to finish and see if it's possible to classify all monoidal structures on LCTVS.

Incidentally, the HBT is important in this because it allows one to show that the unit of a monoidal structure on LCTVS has to be $\mathbb{R}$. The unit has to satisfy the property that $\dim \mathcal{L}(X) = 1$. With HBT, this is just $\mathbb{R}$. Without HBT, one could take $\mathbb{R} \oplus \ell^\infty/c_0$.

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1 Answer 1

If $V$ is a vector space, there is a least topology $\tau_V$ on $V$ which makes all linear maps from $V$ to any other vector space continuous (consider the initial topology from the collection of all linear maps from $V$ to all vector spaces which are quotients of $V$---this is a set of maps, which is nice, but it is not really needed) This topology is locally convex: it is the largest locally convex topology, in fact.

If you endow $L(V,W)$, the space of continuous linear maps between two lcvts $V$ and $W$, with the topology $\tau_{L(V,W)}$, then you get a functor.

(You can play this game in many ways: pick a set $\mathcal F$ of your favorite locally convex topological vector spaces — for example, let $\mathcal F=\{L^{2/3}(\mathbb R), \mathcal E'\}$, — and endow $L(V,W)$, for all lctvs $V$ and $W$, with the least topology $\tau_\mathcal F$ for which all linear maps from $L(V,W)$ to an element of $\mathcal F$ are continuous. This gives again a functor.) (Moreover, one can do the same with final topologies, of course)

Remark that these topologies are probably useless :)

Later: There is another source of examples, generalizing the uniform topologies. Suppose you have an assignment to each lctvs $V$ of a set $C(V)\subseteq\mathcal P(V)$ of subsets of $V$ which is functorial, in the sense that whenever $f:V\to W$ is a continuous linear map of lctvs then $\{f(A):A\in C(V)\}\subseteq C(W)$. Then there is functorial topology on $L(V,W)$ given by uniform convergence on sets of $C(V)$.

If you take $C(V)$ to be the set of bounded subsets of $V$, of compact subsets of $V$, of relatively compact subsets of $V$, of finite subsets of $V$, then you get your examples 1 to 4. But there are other choices for $C(V)$: the set of star shaped subsets of $V$ (ie, subsets $S\subseteq V$ such that there is a point $x\in S$ such that for all $y\in S$ the segment $[x,y]$ is contained in $S$; this over $\mathbb R$---over $\mathbb C$ there is a corresponding notion whose name escapes me now), say. There are others.

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Hmmm, I think I agree with the "useless" tag! Perhaps I should put in the requirements that the monoidal structure imposes to avoid this sort of answer. In particular, we want nice properties with respect to the evaluation map V x L(V,W) -> W. –  Loop Space Dec 11 '09 at 14:40
The edit is more interesting: that's where I thought the answer was going to lie. You've added "star shaped" to my list (can you give a definition just so I'm sure I've understood?), what else is there, I wonder? –  Loop Space Dec 11 '09 at 21:58
Being star-shaped is a convexity and connectedness condition: en.wikipedia.org/wiki/Star-shaped –  Harry Gindi Dec 11 '09 at 22:30
Star-shaped doesn't add anything, then. As the resulting topology is locally convex, we can always take convex hulls without changing the topology. So star-shaped just becomes convex, whereupon it'll be one of the other families depending on what boundedness condition one adds (there must be one to get a reasonable topology). Do you know of any other families of subsets? That's what I'd like to know! –  Loop Space Dec 15 '09 at 9:06
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