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Why is an article not used before the noun in sentences such as the one below?

See you in (the, a) court.

Generally, you use a zero article with abstract nouns:

I fell in love. (not a or the love)

In your example, court is being used as an abstract noun representing the legal system and its activities in general. Note the difference between court (zero article) and courtroom (article):

The judge is in the courtroom this morning. (Specifies her location, but not her activity - maybe she's just sitting there catching up on paperwork.)

The judge is in court this morning. (Specifies that she is actively presiding over one or more legal procedures.)

Likewise, you can create the same distinction with your example:

See you in the courtroom.

This says that there's a place (the courtroom) where we will both be in the future, but does not specify what we will be doing: we could be involved in the trial or in the jury or even just spectators.

See you in court.

This says that we will be directly involved in a legal trial, usually as opposing parties on either side (although a judge might also say this to one of the litigants), and we will go through the process of holding the trial.

A few people have pointed out in the comments that you can use articles in front of what seem to be abstract nouns. This tends to be used when you are narrowing down the meaning to one specific instance of that abstraction. On another site, a frequent ELL contributor (FumbleFingers) wrote this:

[using] the focusses on a specific instance of something, whereas a implies one among many possibilities.

Examples:

The court ruled in favor of the defense. (one specific court)

To continue, we need access to a court of law. (one among many possibilities)

See you in court. (not specifically one instance; just the general abstract concept of formal legal proceedings and maybe I'm willing to drag you from venue to venue and through multiple appeals)

Or to use abstract noun "argument", which was raised in the comments:

This is not relevant to the argument. (the specific on we're having)

They were having an argument. (one argument at a particular time)

We can resolve this through argument. (the abstract concept of adversarial discussion, not necessarily confined to one single discussion)

And as a digression, you do see "the/a court" for other definitions of the noun court where it is a definite location. When speaking about a tennis court, for example, you could hear someone say:

See you on the court [to play a match of tennis].

In this case, court is being used to convey a concept (justice) rather than the building of the court. When using a noun as a concept, the article is often omitted, but not always.

This is analogous to 'going to church', 'going to school' and a host of other nouns. When saying "I'm going to church" you're implying that you're going to pray. The actual building isn't important. On the other hand, if you're going to a church to pick up some paperwork, you would generally say 'going to the church'.

With a hospital, I believe 'going to hospital' is said in the UK, whereas we say 'going to the hospital' in the U.S. However, even is the US, we would say 'he's going into surgery'. Going into the surgery would imply going for some purpose other than surgery.

It is interesting that we say 'going to church', 'going to synagogue' but I've never heard 'going to mosque'.