We will make the convention that exact categories be skeletally small.
Is this construction (used in a mathematical context) correct? There is something that strikes me as odd in that “be”. Should it be “are”? I know I’ve seen a similar way of using “be” like that, without conjugating the verb, but I’m not completely sure when it should be used.
I am not a linguist, so please understand that this is only the best I can do, and may be more long-winded than necessary.
Grammatical mood is the quality of a verb that conveys the writer’s attitude toward a subject. Verb moods indicate a state of being or reality. Commonly known moods are indicative (states reality), interrogative (states questioning), imperative (states command), conditional (indicating a conditional state that will cause something else to happen), the now uncommon subjunctive mood (indicating a hypothetical state, a state contrary to reality, such as a wish, a desire, command, recommendation, or an imaginary situation, etc.) The conditional mood has largely replaced the subjunctive in English.
The subjunctive clause can be a mandative subjunctive which is a clause following a mandative word (expressing a demand, requirement, request, recommendation or suggestion) and usually, but not always, begins with ‘that’ and contains a bare infinitive.
Sometimes the bare infinitive can be hard to spot unless it stands out. With inflected verbs such as to be, it is easy, as the inflected forms are ‘am, are, is’, whereas the bare form is ‘be’.
- I suggest that you be careful.
With other verbs, sometimes the bare infinitive is apparent only in the third person singular.
- It is important that he stay by your side.
The important word is the mandative word.
- We demand that he refund our money immediately.
- He insists that the Carrot Bisque be the first course for the celebration.
NB: that is not a necessary element in the mandative subjunctive, but the bare infinitive is:
It is important he stay by your side.
It is imperative he tell the truth.
I suggest he depart immediately.
- We will make the convention (that) exact categories be skeletally small.
is a correct use of the subjunctive, which is very common in mathematics today. It can also be seen in poetry, and earlier writings before the conditional came to be used commonly, e.g. the Bible.
Edited to add: I had a lot of help in making this answer correct. I am very appreciative of this and just want to acknowledge that.