The church in Babylon, chosen together with you, greets you, as does Mark, my son (1Peter 5:13, HBFV).
Peter's letter warns the church at Babylon that it does them no spiritual good to suffer the consequences for any sins committed. It is, however, commendable to God to endure hardships and trials for his name's sake.
In this you yourselves greatly rejoice; though for the present, if it is necessary, you are in distress for a little while by various trials . . .
If you are reviled for the name of Christ, you are blessed because the Spirit of glory and the Spirit of God is resting upon you; on their part He is blasphemed, but on your part He is glorified. Assuredly, let none of you suffer as a murderer, or a thief, or an evil doer, or as an overlording busybody in other people’s lives.
Yet if anyone is suffering as a Christian, he should not be ashamed; but let him glorify God because of this . . . (1Peter 1:6, 4:14 - 16).
The apostle in Babylon
In spite of what some may believe, there is no reason to think that Peter was symbolically referring to Rome when he greets those in Asia Minor (1Peter 1:1) from the church in Babylon.
The church in Babylon, chosen together with you, greets you, as does Mark, my son. Greet one another with a kiss of love. Peace be with all of you who are in Christ Jesus. Amen (1Peter 5:13 - 14).
According to the Illustrated Bible Dictionary, Peter was referencing the former capital city of the Babylonian Empire. This makes a great deal of sense, as it was inhabited by many Jews and Jewish converts when he wrote his epistle around 64 A.D. The view the he was referring to the church in the literal city of Babylon is also supported by other Bible commentaries such as Matthew Henry's, Jamieson Fausset and Brown's, Adam Clarke's and others.