From First Corinthians by George T. Montague, SM, commenting on I Corinthians 1:1-3:
The addressee is not an individual but a community, the church of God… in Corinth. Though the letter was apparently delivered by Timothy (1 Cor 4:17), who probably read it to the community, offering his own comments to explain when necessary, the absence of a leader-addressee indicates that Paul still feels so close to the community he founded that he can address them directly, though we may assume there were local leaders (16:15–16; 1 Thess 5:12–13).He is himself the authority without equal but also one who has an unparalleled affection for his spiritual children (1 Cor 4:15).
By this time, if the number of converts has grown beyond the capacity of one house, which is likely, there would be more than one house church in Corinth (1:11), each no doubt having some kind of coordinator. Yet it is significant that he does not speak of churches in the plural but only of a singular church in Corinth—a precursor of dioceses with multiple parishes.
If “church” here refers to the sum of house churches, it is easy to see how it could be applied to the one Church universal, as it so clearly is in the letter to the Ephesians. Thus in the desert narrative of the Old Testament, the “assembly of the Lord” includes all the tribes gathered as one. There the Israelites were the people the Lord called out of Egypt to be his own. So God has called the Christians out of the world to a saving union with Christ (1 Cor 1:9). Calling the community the “church of God” links the Corinthian church with the Jewish Christian churches of God in Judea (1 Thess 2:14), as well as those of Galatia, Asia, and Macedonia. Already in this title “church,” Paul is inviting them to think beyond their personal interests.