I'm a Ph. D student. I have been fortunate to have an extremely supportive advisor against whom I have only positives and literally no complaints so far. She offers me sound technical guidance, almost always available, refines my problems, never rude, given me several weeks off when I had some personal family emergencies, and of course, helps me get my work published.
My advisor collaborates with a Co-PI from another university (for a couple of projects) who is a few years senior to her. The Co-PI is from a premier high-ranking university, has stellar track record of publications, co-founded manufacturing startups, and is an overall rock star of sorts. Co-PI has several industry connections. I have had few brief interactions with Co-PI, but never worked with him in the projects that my advisor collaborates with.
As I plan to get into industry after I graduate, I would like to further my association with Co-PI in the hopes that : 1. I could get letters of recommendation from him 2. Get contacts in the industry for internships/full-time jobs. 3. Get a peek into how research is done as competitive high-ranking universities.
My concern is, if it would be taken negatively by my advisor if I express my interest to work with Co-PI for a summer/semester. I do not want my advisor to think - "So my student thinks I am not good enough", which is certainly the farthest from the truth. Any advice on how best to discuss the subject with my advisor without complicating things would be helpful. Or if at all its a good idea to work on a different lab for a brief period.