How Did the Alexandrian Expatriate Church in Jerusalem Function?

No one before me (at least that I can see) has delved very deeply into the Jerusalem Church as an Alexandrian expatriate community. I have spent many years thinking about the so-called Catechetical School of Alexandria supposing, as most people do, that it was a kind of yeshiva. Now I am puzzled by my recognition that a similar Christian body seems to have existed in Palestine. Origen certainly had something associated with his person at Caesarea. Clement must have also established a similar body in Jerusalem or wherever he was staying. The question in my mind of course is not only how this yeshiva 'fit' within not only the Alexandrian Church or - more importantly - the Alexandrian expatriate Church that seemed to have grown up around a massive exodus from Egypt to Palestine.

This couldn't have been the same ecclesiastical structure as we see associated with the fourth century Church. But what was it? How did Christians assemble? How large was this religious body really?

It is very difficult to make heads or tails of this important phenomenon.