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Transferring (lifting and setting down) an object corresponds to one of the 39 melachot.

Which of the 39 does carrying correspond to? By carrying, I mean, for example, receiving an object from a person, walking in a public domain, and having it taken from one with all of this taking place in a public domain.

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I don't follow your distinction between carrying vs transferring. The only things prohibited in the category of the 39 that relates to this are: 1) picking something up in either private or public domain and putting it down in the other, or 2) picking something up in a public domain and putting it down at least four cubits away in the public domain. –  Double AA Jan 6 at 21:21
    
Is picking up in a public domain and setting down in a public domain prohibited? –  ast Jan 6 at 22:09
    
Only if it falls in one of the two categories above (ie only if you move it four cubits away) –  Double AA Jan 6 at 22:13
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1 Answer

Many rishonim (medieval rabbis) say that the removal of an object from its area of 4 Amot (cubits) by carrying it in a public domain (רה״ר) is the equivalent (or, it is a תולדה) of removing it from its רשות, domain.

This, for example, is the wording of the רמב״ן on .שבת דף עג:

ומעביר ארבע אמות ברה״ר היינו טעמא דלא תני ליה משום דתולדה דמוציא הוא, דכל ארבע אמות של אדם רשותו הוא וכשהוא מוציא חוץ לד״א שלו מוציא מיקרי

This is the wording of the בעל המאור at the beginning of פרק הזורק:

כל ד׳ אמות בר״ה גמרא גמירי לה וכולהו תולדות דרה״י לר״ה נינהו לפי שד׳ אמות של אדם בכל מקום קונות לו וכרשותו דמיין וכשמוציא חוצה להן בזורק או במעביר כמוציא מרה״י לר״ה דמי

And this is the wording of the ר״ן on .דף עג:

והמוציא מרשות לרשות. פי׳ והכנסה בכלל דתנא כל עקירת חפץ ממקומו הוצאה קרי לי׳ כדאמרינן בריש פ״ק דמכילתין, ואפשר דהיינו טעמא למעביר ארבע אמות בר״ה דהאי נמי עקירת חפץ ממקומו הוא.

These are three apparently similar answers to your question, but are in fact disparate. Ramban simply states that anything that has moved 4 Amot has changed domain. Baal HaMaor connects this to the 4 Amah area that acquires objects for a person, a connection that bears much thought. Ran does not even mention 4 Amot; he says simply that anything that has been forcibly removed from its previous location can be denoted as having changed domain.

In terms of the other cases you mentioned, such as receiving and accepting in the same domain, there is no prohibition at all. The only prohibitions are moving from one domain to the other (מרה״י לרה״ר או להיפך), and moving an object 4 Amot in a public domain (ד׳ אמות ברה״ר).

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