If not, can one refer to any formal (written) prohibition of wearing caps for lecturers, in a North American university?
The cap may be text-less or have any non-offensive text/image.
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I very much doubt that you're going to get a positive answer, and a negative answer ("there is no such rule at any North American university") would be almost impossible. I've never heard of a dress code for faculty at a North American university (again, doesn't mean that one doesn't exist somewhere, even at an institution where I've been ...)
As support for the claim that this probably doesn't exist, I looked up the dress code at Brigham Young University, which is likely to have among the strictest/most conservative rules. The code applies to "students, staff, and faculty", but hats are not mentioned, either for men or women:
A clean and well-cared-for appearance should be maintained. Clothing is inappropriate when it is sleeveless, revealing, or form fitting. Shorts must be knee-length or longer. Hairstyles should be clean and neat, avoiding extreme styles or colors, and trimmed above the collar, leaving the ear uncovered. Sideburns should not extend below the earlobe or onto the cheek. If worn, moustaches should be neatly trimmed and may not extend beyond or below the corners of the mouth. Men are expected to be clean-shaven; beards are not acceptable. Earrings and other body piercing are not acceptable. Shoes should be worn in all public campus areas.
A clean and well-cared-for appearance should be maintained. Clothing is inappropriate when it is sleeveless, strapless, backless, or revealing; has slits above the knee; or is form fitting. Dresses, skirts, and shorts must be knee-length or longer. Hairstyles should be clean and neat, avoiding extremes in styles or colors. Excessive ear piercing (more than one per ear) and all other body piercing are not acceptable. Shoes should be worn in all public campus areas.
(I also looked at the dress code for Oral Roberts University, another religious university: the rules only apply to students, but again nothing is mentioned about hats for either men or women ...)
As mentioned by commenters, even if there were such a rule there would probably be an exception made for religious observance (Jewish kippot/yarmulkes, Sikh turbans, etc.)
(The only prohibition on hats I've encountered is that students are sometimes forbidden to wear hats with brims that project in front during exams — if they're wearing a baseball cap they have to turn the brim to the back, so that the proctors/invigilators can see where they're looking.)
This is entirely context dependent from my experience. Most universities are quite casual now. As others have noted, it is also dependent on what department you're in as to what the informal dress code is. Also, I expect seniority plays in. If you're a PhD teaching assistant you could easily get away with dressing more casually than a full professor.
There is no dress-code at my university nor at my previous universities. The only ones wearing caps are undergraduates who are rumored to (a) not had time to shower in the morning, (b) worried about bad hair, (c) be socially inept. ;)
Wearing a cap or a hat indoors is still considered bad etiquette and would elicit comments by colleagues. They might also assume a good reason for covering up ones head such as an injury - bandages or some other medical condition. (Edited: Or because of religious tenets etc.)