The Mitzvah of building a Sukkah:
It is a command in the Torah to dwell in a Sukkah throughout the seven days of the Sukkos Holiday. As a result of this command, we are required to build a Sukkah prior to Sukkos in order to fulfill the command when the time arrives, and it is thus considered a Mitzvah to do so. Nonetheless, the building of the Sukkah is only considered a preparatory Mitzvah [i.e. Hechsher Mitzvah] as opposed to a Mitzvah with intrinsic value, as it is built simply for the purpose of later fulfilling the Mitzvah of dwelling in it during the holiday of Sukkos. Accordingly, no blessing is recited upon building the Sukkah, in contrast to the ruling by other Mitzvos in which a blessing is said prior to its fulfillment.
The reason for why G-d commanded us to build a Sukkah: The verse states that we are to dwell in a Sukkah for seven days in order so our future generations remember that G-d settled the Jewish people in Sukkos when he took them out of Egypt. The term Sukkos here refers to the clouds of glory, which surrounded the Jewish people for the sake of giving them shade, so the heat and sun do not injure them. Thus, as a replica of this event, G-d commanded us to [build and then] dwell in a Sukkah that is made for the purpose of providing shade in order so we recall the miracles and wonders [done for us in the desert, in which the clouds of glory surrounded us for shade, and served as a protection from the sun]. Therefore, when dwelling in the Sukkah, every person must have intent to do so in order to fulfill G-d’s command to dwell in the Sukkah in commemoration of the Exodus. See Chapter 3 Halacha 1-2!
Is everyone to have their own Sukkah?
From the letter of the law, there is no obligation for one to build his own Sukkah, if he has a neighbors Sukkah or publicly shared Sukkah available, in which he can dwell in throughout the duration of Sukkos. Nonetheless, if feasible, it is proper for each family to build their own Sukkah, just as each person has his own home. [In addition, it is quite difficult to properly fulfill the Mitzvah of dwelling in a Sukkah, in a Sukkah that one does not own, and hence it is proper for each person to have their own Sukkah in order to dwell in it properly.]
Is a Shul required to build a Sukkah?
It is customary for a Shul to have a Sukkah built near the premises of the Shul for the sake of guests to eat in it during Sukkos. [However, individuals who live in the city are not to rely on the Shul’s Sukkah, if they have the premises available to build one near their home.]
Is one obligated to build a Sukkah far from his home if he has no room in or near his property to do so, and there are no neighboring Sukkos available in which he can eat in?
One who does not have room in his property to build a Sukkah [i.e. no porch or backyard with space under the sky] is nevertheless required to build a Sukkah even a distance from his home, and dwell in it throughout Sukkos. [Alternatively, if there are neighboring Sukkos available, one can dwell in another person’s Sukkah, rather than build his own Sukkah.] This comes to negate those who do not build a Sukkah, and do not dwell in any Sukkah over Sukkos, under the claim that they are only required to build a Sukkah near their home, and since they cannot do so therefore they are exempt. [However, some Poskim rule that we do not require one to move to a different city, or to a hotel, for the duration of Sukkos simply for the sake of being able to have a Sukkah nearby, and the matter is to be measured in accordance to the degree of effort that one would pursue to dwell in his own home. Thus, if it is not possible for one to build a Sukkah within his city, such as due to robbers, then he is exempt from the Mitzvah and is not required to move from his city for the duration of Sukkos.]
 Piskeiy Teshuvos 625:1; 640:7
 See Admur 625:1 who calls it a Mitzvah “It is a Mitzvah to build..as a Mitzvah that comes to one’s hands..”; 641:1 “The Sukkah is not the completion of the Mitzvah” which implies that nonetheless it is the start of the Mitzvah; Rambam Brachos 11:8 “Any Mitzvah that contains another Mitzvah after doing it.”; See Menachos 42b; Rashi Makos 8a; Yerushalmi Brachos 9:3 that even a blessing is recited; Avnei Nezer O.C. 459 that even according to the Bavli there is a Mitzvah to do so; Emek Sheila on Sheilasos Mitzvah 169; Likkutei Sichos 17 p. 188 footnote 56 that even according to the Bavli there is a Mitzvah to do so; See Mikraeiy Kodesh Sukkos 1:25; Kinyan Torah 3:90; Moadim Uzmanim 1:80; Mishneh Halachos 8:237; 10:72
Other opinions: Some Poskim rule that a Hechsher Mitzvah is not considered a Mitzvah at all. [See Chasam Sofer Y.D. 10; Chelkas Yoev 27; Minchas Elazar 4:55]
 Admur 641:1 “The building of the Sukkah is not the completion of the Mitzvah, as the main Mitzvah is only fulfilled when dwells in the Sukkah during the holiday itself”; Chasam Sofer Y.D. 271; Chelkas Yoev 27; Minchas Elazar 4:55; See Menachos 42b; Piskeiy Teshuvos 641:1; So rule regarding Tzitzis: Admur 19:1; Michaber 19:1; Menachos ibid
Other opinions: Some Poskim rule that a Hechsher Mitzvah is also considered a Mitzvah in its own right. [Yerushalmi Brachos 9:3; Avnei Nezer O.C. 459 that so agrees also Bavli; See Biur Halacha 656 “Afilu” and Likkutei Sichos ibid]
 Admur 625:1; Michaber 625:1; Rebbe Eliezer Sukkah 11b; See Piskeiy Teshuvos 625:1
Other opinions: According to Rebbe Akiva [Sukkah ibid] we slept in actual Sukkos, and the reason behind the holiday of Sukkos is to remember all the kindness that Hashem provides us with throughout the year, as in the desert we lived in considerable lack of wealth, not having any land or produce. Thus, in order to remind us of this, and give gratitude to G-d for all that he has provided us with, therefore we sleep outside of our homes in a Sukkah and learn to gain appreciation for all that we have. [Rashbam Sukkah ibid; P”M 625 M”Z 1 agrees with reason of Rashbam, as we rule like Rebbe Akiva versus Rebbe Eliezer]
 Vayikra 23:42
 Admur ibid and 429:4; Michaber ibid; Bach 625; Levush 625; Rashi and Ramban on Vayikra 23:43; Rebbe Eliezer in Sukkah ibid; See Birkeiy Yosef 625 in name of Toras Kohanim for opposite Girsa of opinions; See Bach and Taz 625 for different explanations of their debate.
Other opinions: According to Rebbe Akiva, we slept in actual Sukkahs and not in clouds of glory. [Sukkah ibid]
 So is written in some prints of Admur ibid. In other prints the word says “Vichein”; See Likkutei Sichos 32:140
 Likkutei Sichos 20 p. 267; See Teshuvah Hageonim ibid “However, the people of the city are not to rely on the Shul’s Sukkah to fulfill their obligation, as one only fulfills his obligation with a Sukkah that he makes near his home.”
 Teshuvos Hageonim Tractate Sukkah 51-53 “Regarding that which you asked if today one is obligated to build a Sukkah in synagogues and Batei Medrash or not? The answer is as follows: We know that the Jewish people are only obligated to build a Sukkah near their home where they can eat and drink just as they do in their home, while a synagogue is only designated for prayer. Nonetheless if they build a Sukkah for the sake of the passersby and guests this is a very good thing.”; Machazik Bracha 639:2; Zekan Aaron 165; Kaf Hachaim 639:78; Likkutei Sichos 19 p. 289
 See Teshuvah Hageonim ibid “However, the people of the city are not to rely on the Shul’s Sukkah to fulfill their obligation, as one only fulfills his obligation with a Sukkah that he makes near his home.”; Kaf Hachaim ibid
 Gloss of Chasam Sofer 640; Bikureiy Yaakov 640:13; M”B 640:24; Shaar Hatziyon 640:30
 Chasam Sofer ibid; Shaar Hatziyon ibid
 Mishneh Halachos 6:115; Piskeiy Teshuvos 640:7
 The reason: As we apply here the rule of Teishvu Keiyn Taduru, and one is thus exempt from the Mitzvah.