"To engage the culture" means to actively participate, interact, and contribute to the customs, beliefs, values, practices, and ideas of a particular society or group of people. Full definition
It is not a distraction, but an opportunity to have other means of honest employment and also engage the culture in a way he wouldn't be able to from a church office. (christianitytoday.com)
We've been working on a developing a strong research - engaged culture for several years and increasingly we've been talking about methodology and evidence. (teacherhead.com)
For to engage a culture there must first be a culture to engage. (firstthings.com)