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The early Christian teaching Didache states that Christians should not fast like the hypocrites:
Your fasts must not be identical with those of the hypocrites. They fast on Mondays and Thursdays; but you should fast on Wednesdays and Fridays. [Chapter 8]
Who are the hypocrites? A rival Christian group? The Pharisees? Someone else?
It is indeed about the Pharisees. Here's what Shmuel Safrai's paper “Religion in Everyday Life" says:
Mondays and Thursdays, which were synagogue days, when country-folk came to town and the courts sat and the Torah was read, were the favoured days for public and private fasts. People would assemble for prayer, mention the reason for the fast, as follows from a baraita in the Babylonian Talmud. Most texts which mention fasting on Mondays and Thursdays are later than 70 C. E. though some are definitely earlier. Epiphanius says that these were the days of the Pharisees’ fasts in Jesus’ time, and the Didache warns against fasting ‘along with the hypocrites’ (the Pharisees) on these days, urging for Wednesday and Friday instead. The Pharisee in Luke who boasted of his twice-weekly fasting must have meant Mondays and Thursdays. But the custom was confined to certain circles among the Pharisees and their disciples. (page 816)
What translation is the OP using?
Chapter 8. Fasting and Prayer (the Lord's Prayer). But let not your fasts be with the hypocrites, for they fast on the second and fifth day of the week. Rather, fast on the fourth day and the Preparation (Friday). http://www.earlychristianwritings.com/text/didache-roberts.html
IOW, the instruction was not about a fast like the hypocrites, but rather do not fast with the hypocrites on Monday and Thursday. Rather fast on Wednesday and Friday.
There was only one fast day (Atonement) in the OT, but the hypocrites were presumably "pleasing the flesh, while ignoring the spiritual". Don't fast with them, but it's okay (apparently per Didache) to fast like them twice a week.
So, "with them" was wrong, but "like them" was okay on two different days.
PS To answer the OP question, it is the Jews. The Didache is saying don't fast with the Jews. (This, if a Christian did so, amongst other things, was caustically called Judaizing.)
Second Temple Period
During the Second Temple period, daily or biweekly fastings were practiced for reasons of asceticism , especially among women (Judith 8:6; Luke 2:37; TJ, Ḥag 2:2, 77d), but also among men (Luke 18:12; Mark 2:18), or in preparation for an apocalyptic revelation (Dan. 10:3, 12; ii Bar. 12:5; 20:5–21:1; 43:3; iv Ezra 5:13–20; 6:35; Sanh. 65b; TJ, Kil. 9:4, 32b). The Jewish literature of the Second Temple period also advocates fasting as a way of atonement for sins committed either unintentionally (Ps. of Sol. 3:9) or even deliberately (Test. Patr., Sim. 3:4), or to prevent them (ibid., Joseph 3:4; 4:8; 10:1–2). These reasons for fasting were strengthened by the destruction of the Second Temple and even more by the repression of the Bar Kokhba revolt and the subsequent religious persecutions. http://www.jewishvirtuallibrary.org/fasting-and-fast-days
This passage [Mt. 6:16] suggests that Jesus, like the Pharisees and the disciples of John, believes in fasting for piety, but that such behavior should be done secretly and without fanfare. Reading these texts together suggests that Jesus’ gripe with the disciplines of John was that they made their fasting known and public, and Jesus saw this as hypocrisy. Despite this, fasting gained prominence in Christian circles and emerged as an act that reflected personal piety. In addition, many ascetic Christian communities arose that encouraged its members to avoid sex, food, and other physical indulgences that distract from leading a purely spiritual life. http://thetorah.com/why-jews-fast/
So, if you're going to fast, do it privately on whatever days you decide.
