Hmm, that seems possible, I guess. (-: On Sat, May 15, 2010 at 11:39 PM, Peter Selinger <selinger@mathstat.dal.ca> wrote:

Michael Shulman wrote:

...

On Sat, May 15, 2010 at 4:09 PM, Peter Selinger <selinger@mathstat.dal.ca> wrote:

...

I don't think the term is meant to suggest that the category is compact, but rather that the closedness is of a form typical for compact things.

I gather that that's the idea, but that's not the way my idiolect works: to me, "compact closed category" means a category that is compact and closed -- since "compact" is not an adverb it can't modify "closed" but only "category."  If it were "compactly closed category" I could see the argument, maybe.

I agree with that. If "compact closed category" is right associative, then "compact" is just a property of closed categories (not quite the same as "compact and closed", because compact would be a property of closed categories, not of categories). If it's left associative, then an adverb should usually be used.

But I heard that Max Kelly had great command of the English language and was picky about grammar to a fault, even to the point of driving his coauthors crazy. So I will speculate some more, keeping in mind that I basically don't know what I am talking about.

It's actually possible, though relatively rare, for adjectives to modify other adjectives in English. Examples are: "dark blue", "long gone", "deep seated", "ill advised", "low lying", "high flying", "high strung", "soft boiled". One would often hyphenate these combinations, although it doesn't seem to be mandatory. What they seem to have in common is that the first adjective does not describe a *manner* of the second one. For example, "darkly blue" would mean blue in a dark manner, whereas "dark blue" just means a darker shade of blue. "Softly boiled" would mean that the boiling is done in a soft way, but "soft boiled" refers to the softness of the boilee. "Badly advised" would mean that the advising is done badly, but "ill advised" means that the advice is bad. Etc. I realize that there is not really a fixed rule here, because for example "positively advised" still refers to the advice and not the advising. But it seems to be grammatically possible to have such combinations. The question is whether "compact closed category" is an example of this.

Since it is a purely grammatical question, perhaps it is useful to figure it out based on grammatical usage, rather than intended meaning.  Consider the following examples:

(1) dark blue spot -> the spot is dark blue (2) faintly blue spot -> the spot is faintly blue (3) round blue spot -> the spot is round and blue (4) registered legal advisor -> the legal advisor is registered

In (1), an adjective modifies an adjective. In (2), an adverb modifies an adjective. In (3), two adjectives modify a noun. In (4), an adjective modifies a noun phrase consisting of an adjective and a noun. Note that in (4), neither "the advisor is registered legal" nor "the advisor is registered and legal" would work.

It seems to me that phrases such as "compact closed", "monoidal closed", "symmetric monoidal", etc, fit pattern (1) but not (2)-(4).

This still does not shed any light on the meaning of the phrase "compact closed". But perhaps it's a possible "idiolect".

-- Peter