Why is the 13th (or the 6th) of a Cm chord A, not Ab?
When a 13th is written in a chord name, this always refers to the major 13th, which is the same as a major 6th - in this case an A natural. This is one of the conventions of how chords symbols are written. It may seem a little odd that the 6th or 13th of a minor chord is major, but there are a number of situations like this. For instance, 7ths are always flattened (minor) sevenths, when only a "7" is written, whether this is on a major or minor chord; if you want a major 7th, you have to write "maj7". (You're probably familiar with this, and wouldn't even think twice about it.)
This is simply the way these chords are written.
But, this has advantages; it means that we always know what interval a 13th (or 7th or 9th or anything else) is going to add to a chord, no matter what the other notes of the chord are. In particular, we know what interval to add even if the tonality or modality of surrounding chords, other parts of the chord itself, or written parts (eg. bass line or melody), might suggest a different interval.
I can see two reasons why you might think a Cm13 chord has an Ab; let me deal with both:
Lastly, it is worth mentioning where this misunderstanding arises; through trying to work out what intervals are in a chord by using a supposed mode/scale containing these notes, rather than choosing a mode to use which fits the chords (the harmony). The second way is how we actually play. The chord symbols tell us exactly which notes the composer wants in the harmony at any particular point in a piece/song, we then choose a mode for improvisation or creating a melody, bass or harmony line to fit the chord, not the other way around. For instance, Cm13 suggests the notes C Eb G Bb D (F) A (as a 13th chord is built using tertian harmony - sorry if you don't know anything about that yet!), this in turn would suggest that C Dorian would be a good scale to use with it.
To sum up. When reading a chord, don't worry about the key the piece/song is in, and which mode or scale this might suggest, or the character of the surrounding harmony. The chord you are reading might not relate closely to these anyway! Instead, follow some simple conventions for working out what the intervals in that chord are. The following intervals are presumed to be major unless adjusted with a flat or sharp: 2, 6, 9, 13. (Be aware, though, that the flat or sharp adjustment is relative to the major interval - eg. a 6 on E is G#, so a b6 is G natural!) Thirds are notated in the first bit of the chord itself: so C something-or-other is major, with a major third; Cm something-or-other is minor, with a minor third. Fourths, fifths and 11ths are presumed to be perfect unless adjusted. 7ths are the weird ones; they are presumed to be flattened (minor) unless marked as major - this is because the most common use of 7th chords traditionally is on dominant chords, and 7ths on dominant chords are minor - again, this is a whole other area of discussion and probably merits some study too (this post is useful, but there is also loads on the web…)
You should probably read up on how tertian harmony works, to fully understand which notes you usually put in 9th, 11th and 13th chords, and which ones can be safely left out (very important for guitarists!)
This is one of the weird bits with chords. With a maj.6 on C, it's C E G A. With a min.6 on C it's C Eb G A. It's the chord that's minor, not the 6th interval.The minor 6th interval is Ab rather than G# anyway. So for 6th or 13th, which is effectively and in reality the same note - possibly different octave - seems to depend on one's persuasion - its going to be A natural.It would never have been G in any case, as G is the 5th or 12th , and that is in the original triad.
On the subject of notes in scales, in the melodic minor,(classical) ascending, the 6th note IS A natural.
This is not an exception or something illogical. There are many scales whose 1, 3, and 5 form a minor triad. Some of these have a major 6, some a minor 6. A C minor chord could, for example, appear in the key of B flat, where an A natural occurs in the key signature. People reading chord symbols want to be able to connect the symbol with a specific chord, so we use specific rules about what the symbols mean. The rule is that in a 13th chord, the 13 (6) is major.
There's the closely related notation Cm6. Note that if you interpreted Cm6 as having an Ab in it, you'd get C Eb G Ab, which would sound to most people like an Ab maj7 chord, not a C minor triad.