Final answer:
Alexios I Komnenos, the Byzantine emperor, invited Pope Urban II to declare the first crusade, which was intended as military support but evolved into a broader holy mission.The correct option is 2.
Step-by-step explanation:
The Byzantine emperor who invited Pope Urban II to declare the first crusade was Alexios I Komnenos. In 1095, with his empire threatened by external forces, Alexios I reached out to the pope to request military support. However, Pope Urban II responded in a manner that was unexpected by the emperor. He framed the military assistance as a holy mission to liberate Jerusalem, hence initiating the first crusade, which was broader in scope than what Alexios had anticipated.