Hi,


Here is a page [about categories](http://ncatlab.org/nlab/show/nice+category+of+spaces) 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?



Where does Chu fit in with "Nice Categories of Spaces"?
(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.


