Why should Hatmanah of something cold be forbidden?

Rashi: This is a decree due to Hatmanah in order to warm food.

Why would R. Yosi have sat submissively in front of Rebbi?

Rashi: It is due to the honor of the Nasi (Rebbi); he would not have ruled in front of him, even though he was greater than Rebbi.

Tosfos: We find that R. Yishmael b'Rebbi Yosi asked Rav Chama bar Bisa 'should I tell you my father's opinion, or Rebbi's?' Rav Chama asked to hear Rebbi's. R. Yishmael said, would a great Chacham abandon the teacher's (R. Yosi's) opinion, and ask to hear the Talmid's (Rebbi's) opinion?! (Nidah 14b) This shows that R. Yosi was greater. R. Tam said, initially Rebbi was R. Yosi's Talmid. After, Rebbi became a greater Chacham, like it says there that Rav Chama held that because Rebbi was Rosh Yeshivah, he was always around Chachamim, so his teachings were [more] sharp.

Rav Elyashiv #1: R. Yosi was greater, like it says in Nidah. However, we find that R. Akiva argued with his Rebbi (R. Eliezer), even though his Rebbi was greater than himself! If something is truly forbidden, one may not retract due to honor of his Rebbi, who disagrees! Also, since R. Yosi was greater, what is the Chidush that Rebbi showed honor to him? Even if R. Yosi would have been submissive to him, this does not change the Halachah! I answer that here, surely Hatmanah does not apply. The only question was whether to decree lest one be Matmin something hot. Rebbi thought to forbid. When he heard that R. Yosi permitted, he retracted. Alternatively, even if the question was whether or not to say 'Lo Pelug' (and forbid every Hatmanah on Shabbos, even something cold), one may not retract due to honor of his Rebbi. Rather, Rebbi weighed the matter, and concluded not to say 'Lo Pelug', but he attributed his retraction to 'the elder already ruled', and all should follow him. He acted as if his own opinion did not matter. (NOTE: Perhaps Rebbi was unsure, and initially thought to be stringent, but when he heard that R. Yosi permitted, Rebbi's Safek did not override what was clear to R. Yosi! And even without this, what is difficult if Rebbi gave more weight to R. Yosi's opinion than to his own? Once, Rebbi commanded to check whether a Yevamah is an adult, and may do Chalitzah. When he heard that R. Yosi said 'the Torah says "Ish", so the Yavam must be an adult, but she may be a minor', he retracted and said 'the elder already ruled' (Yevamos 105b). Initially, Rebbi held that a minor Yevamah's Chalitzah is invalid! - PF)

Rav Elyashiv #2: Perhaps Rebbi already was greater than R. Yosi (like Rav Chama held - Nidah 14b), but amidst his great humility, he acted as if R. Yosi was greater, and he deferred to him. Nowadays we do not have a Nasi or great Chacham like Rebbi, that one may not rule unlike him.

What is the source that Rebbi showed honor to R. Yosi?

Maharsha: It is not that Rebbi retracted. If a Chacham permitted, another Chacham may not forbid! Rather, it is because he said 'the elder has already ruled!'(NOTE: Surely, the Klal that one may not forbid what another Chacham permitted refers to ruling in a particular case. Here they argued in general about Hatmanah of something cold! - PF)

When did Rav Nachman tell his slave to be Matmin water, and to bring hot water from a Nochri chef?

Rashi: The Hatmanah was on Shabbos, and the hot water was on a weekday. He did not forbid due to Bishul Akum.

Rav Elyashiv: Just like water is permitted, also tea, which is essentially water. Poskim discussed whether coffee is permitted. Why did Rashi need to say that the hot water was on a weekday? Even if it was on Shabbos, the Nochri heated it for the majority of the city, who are Nochrim! Some Poskim are stringent if a Nochri cooked food for himself, either due to Muktzah and Nolad, or lest a Yisrael come to request the Nochri to cook for him. These do not apply to water! It seems that Rashi said a weekday for a bigger Chidush. It is not common for Nochrim to cook for Yisraelim on Shabbos, so we did not decree Bishul Akum. It is common on weekdays. Even so, Bishul Akum does not apply to water. There are two opinions about whether a Yisrael may eat bread that a Goy baked on Shabbos (OC 325:4). Acharonim ask, it should be forbidden due to Pas Akum! I say that there was no decree about Pas Akum baked on Shabbos, for it is not common for Yisrael to eat it. (NOTE: Is it not common to eat after Shabbos bread that a Nochri baked on Shabbos? Perhaps it was not common on Sunday, for most people baked generously for Shabbos, and there was extra left over. Also, that is the Nochrim's holiday! Before preservatives and refrigeration, bread did not last well, so it was not common to eat bread baked two or more days ago. - PF) Rishonim argued about this, e.g. if a Nochri cooked for a Choleh, may healthy people eat from it immediately after Shabbos (YD Sof 113). The Ra'ah holds that Bishul Akum does not apply, for it is rare that a Yisrael will eat from it, so concern for intermarriage does not apply.

Why should an important person be stringent?

Rashi: One who sees that he is lenient, he will be more lenient.

Iyun Yakov: I wrote (Minchas Yakov 75:30) that nowadays, we are not concerned for this (an important person need not be stringent).

Rav Elyashiv: Rashi does not mean that an important person must be stringent in every case. It is only in cases like this, that people do not distinguish between foods eaten raw and other foods, so they will come to permit all foods.