Hi,
Here is a page about categories of spaces with "Nice" properties. When we consider Chu spaces, I believe it was chosen to have certain properties that were "nice" at least to Pratt. How does Chu fit in with the notion of "nice" that is outlined in the NLab site above?
(h/t to Andrej Bauer for drawing my attention to this question.)
While I'm flattered to be the first name associated with Chu spaces, Ben, were justice to be done here, one should think first of Michael Barr and second of Jean-Yves Girard, who arrived at the appropriate abstraction of the Chu space notion independently in that order. (And Mike in turn would point to Mackey's thesis from the 1940's as considerably earlier yet.)
*-autonomous categories (to which Chu spaces are as presheaf categories are to toposes), and full (as opposed to intuitionistic) linear logic, share with Boolean algebras the virtue of being self-dual. This "nice" property is incompatible with the notion of "nice" that everyone from Steenrod to Todd Trimble has variously called "convenient" or "nice." In particular it is inconsistent for a category to be simultaneously cartesian closed and self-dual. (An inconsistent category is one that is a preordered set, i.e. no homset contains two distinct morphisms.)
There is little to distinguish toposes from abelian categories besides the difference that whereas the former has no arrow from 1 to 0 (unless equivalent to the category 1), the latter makes them isomorphic. This little difference is enough to make abelian categories "non-nice" in the sense of "Nice Categories of Spaces." Other than that small difference, toposes and abelian categories are essentially the same thing.
In the class of categories all of whose arrows from 1 to 0 are isomorphisms, why should the absence of such an arrow be a necessary condition for niceness?
Its presence is not a sufficient condition, witness Chu(Set,K) for |K| <= 1 (which is equivalent to Set + Set^op or Set x Set^op when K is empty or not respectively) which has an iso from 1 to 0 but is not nice. All other Chu categories lack any arrow from 1 to 0, let alone an iso, and are also not nice.
To paraphrase Richard Feynman, if you think quantum mechanics is nice then you don't understand quantum mechanics.
As one of the principal authors of the nLab page on "nice category of spaces" mentioned by the OP, let me quote:
Within the nLab, “nice category of spaces” is a general but inexact term referring to nice or convenient properties one would like a category of spaces to have for some purpose (“space” here connoting something along topological lines)
I say this to correct the impression of Vaughan Pratt (seeing that he mentions my name in this regard, which was a surprise) that 'nice' in the nLab sense implies cartesian closure; for that, we use the more specific terminology "convenient category of topological spaces", after Ronnie Brown and Norman Steenrod who made that phrase famous. 'Nice' in the nLab sense is applied in a much more liberal and inclusive sense, as ought to be clear from the quote. Indeed, a number of examples given on the nLab page are not cartesian closed at all. (Many of the examples given however do emphasize various exactness properties.)
Thus, there is no reason that categorical self-duality (or $\ast$-autonomous categorical structures) couldn't also be considered 'nice'. As the article says, 'nice' really only means nice for some purpose at hand.
However, the main thrust of the nLab article is toward "spaces" in a topological sense. Chu spaces are, to my mind, "spaces" much more general than that. It's okay to call them "spaces" of course -- but consider that classical Chu spaces also include structures like $\mathbb{Z}_2$-vector spaces which would be a stretch to consider specifically topological. So Chu spaces go a bit beyond the types of structures originally envisioned for that article. Thus, the discrepancy isn't really centered on the word 'nice', but more on the word 'space'.
If anyone wishes to discuss (or dispute!) the contents of an nLab article, the best and most proper place to do so would be on the nForum.
