Sorry for the basic question but I can't find a simple, straight answer.

Cathain a théann tú abhaile?

What is the function of "a" in that sentence? Obviously it lenites the following word.

Is it the equivalent of "that"?

Thanks.

The relative here, I think, for the direct, is just a historical 'do' added in to make speaking easier 'sin é an fear (a/do) chonaic mé'

It's the direct relative particle. Yes, a literal translation would be "that" - but of course you wouldn't use "that" at all when asking such a question in English, which makes it confusing. Bearn's sin é an fear a chonaic mé is a good example of the same particle in use in a declarative sentence (although I don't think he's right about its reason for being there.)

The direct relative is used after question words "cad", "cathain", "cé", "conas":
Cad a chonaic tú?
Cé a chonaic tú?
Cathain a théann tú abhaile?
Conas atá tú?

There's also an indirect relative particle spelled "a", but instead of leniting it eclipses. We don't write this particle after "cá" but you can see the eclipsis (and the dependent verb form) which are its signature:
Cá bhfuil tú?
Most 'question phrases' like "cén t-am", "cén chaoi", "cén fáth" use the indirect relative.
Cén t-am a dtéann tú abhaile?
Cén fáth a ndeachaigh tú abhaile?
Cén chaoi a ndeachaigh tú abhaile?

'But "conas" and "cén chaoi" mean the same thing!' I hear you cry. Yep. Any reason why one takes the direct relative and one the indirect relative are shrouded in etymological mist - which you can either attempt to penetrate, or you can print up a list and start learning them off by heart.

In my defense! (clears throat) On Lar's site we find this:

[ 1 ] the direct relative particle is actually unnecessary. It is a relatively new development, earlier there was only the lenition of the verb as an indicator of the relative without a particle or pronoun. (an fear chuireas = the man who lies)
a came to be from the earlier very common preterite particle do, that was used also in other tenses by irregular verbs. Eventually they wereable to be heard even in relative clauses, became accepted as the relativium and were applied also to other verbs: (an fear do chuireas =the man who lies), which would serve nicely as a possible "cause" of the lenition of the verbs. It then shrank to thecontemporary a (an fear a chuireas = the man who lies).

Maybe the man himself could drop by to explain, but like Batman, his appearances are apparently at random, tho grammatical etymology is a bit like the bat signal to him

quote:

Sorry for the basic question but I can't find a simple, straight answer.

Cathain a théann tú abhaile?

What is the function of "a" in that sentence? Obviously it lenites the following word.

Is it the equivalent of "that"?

Yes. It is a relative particle like "that" in English.

But why do we need a relative particle in a simple question?
Because of the strict Irish word order: VSO, verb, subject, object, other constituents.
Well, a wh-word like cé, cad, céard is subject or object, cathain is an adverb and so an "other constituent".
They are normally not allowed to come first in Irish word order. But on the other hand they must be first.
So we need a trick: The sentence is divided into two clauses:
1)the wh-word like cathain (in fact it is a copula clause withe the copula being understood: cathain = Is when)
2)a relative clause like "a théann tú abhaile"
In both clauses VSO word order is perfectly secured.

Lars

Putting back the implied copula that Lars mentions, you could literally translate Cathain a théann tú abhaile? as "*When is it that you go home?" (as a grammatical exercise only, of course; the proper translation is "When do you go home?")

Well put! My objection was to the idea that it was somehow grammatically optional, "added in to make speaking easier."

Classical Irish - more or less a codification of 12th-century spoken language - used "an fear chuireas" as described above. Some time in the next 300 years, the vernacular changed to "an fear do chuireas"; we don't know when, but the earliest written examples are from the 15th century so it must have been sometime before that. (Usually neologisms like this were adopted by religious authors first, as their uneducated audience became progressively less likely to understand full-on by-the-book Classical Irish.)

The classical standard finally collapsed around the mid-17th century (it had become so archaic as to be entirely dependent on the bardscoileanna for transmission, so effectively lasted only one generation longer than they did) and the prevalent form immediately following its collapse - presumably reflecting the prevalent vernacular form at that time - was "an fear do chuireas."

The substitution of "a" for "do" began shortly thereafter; there are a few examples in the late 17th century, and it becomes widespread in the 18th. However, Irish writers were a conservative lot and you'll still find the classical "an fear chuireas" recommended right alongside "an fear a chuireas" and "an fear do chuireas", right up into the Revival era. For what it's worth, this seems to have been a measure of the individual author's conservatism or otherwise, rather than of dialect.

"added in to make speaking easier."

Shock, horror -an unfounded opinion on a forum! May the Lord preserve us!

"when is that you go home?" would be common in Hiberno English

Thank you all for answering this thoroughly!