Background information on Bashō's career

Teimon school

From Encyclopedia of Japan (Kodansha) accessed via JapanKnowledge:

Matsunaga Teitoku 松永貞徳 1571－1653. Classical scholar and poet. Founder of the Teimon school of haikai (the prototype of haiku). Born in Kyōto. He studied with Hosokawa Yūsai, Satomura Jōha, and scholars close to Toyotomi Hideyoshi, whom he served as secretary. The principal encyclopedist of his time, Teitoku compiled lexicons and commentaries on classics such as the Tsurezuregusa of Yoshida Kenkō. He strove to liberate classical learning from the tradition of secret oral transmission. His most important haikai work is Gosan (1651), in which he elaborated the rules of haikai composition, thus establishing haikai as a genre of poetry. Among Teitoku's disciples were Kitamura Kigin and Yasuhara Teishitsu.

Danrin school (談林派)

From Encyclopedia of Japan (Kodansha) accessed via JapanKnowledge:

A school of haikai ... that arose as a radical revolt against the conservatism of the predominant Teimon school led by Matsunaga Teitoku and that held the stage between 1673 and 1683. Its leader was Nishiyama Sōin (1605－82), a renowned renga master and haikai poet, but it was his disciples who earned it an avant-garde reputation. This group's style was demonstrated in the Ikutama manku, a sequence of 10,000 verses composed by Ihara Saikaku and others in 1673. It was not long before the Bashō style of haikai took the reins from Danrin, but Bashō readily acknowledged the liberating influence of Sōin and the Danrin school.