PL/4 is the moment. 4000*10*12/4 = 120,000 lb-in.
M/fb = S so 120,000/36000 = S = 3.33
Using a load factor of 5 gives Srequired = 16.67
Smallest/lightest beam that will work is a S6x12.5 with an S = 22.1, or a load factor of 6.6 times. That margin will account for the weight of the beam and hoist.
I see gearhead did the same calculation, and concluded it was unsafe, and recommends a 10x35 with an S = 147. Take your choice, his is a lot more conservative recommendation.
I did some quick googling, and find that a factor of safety of 8 or 9 is recommended for lifting apparatus.
So, using that, S required = 9 x 3.33 = 29.97
An S8x18.4 has S = 57.6, so meets that criteria. It would have a factor of safety of 57.6/3.33 = 17.3.
The harrington chart referenced above (
https://www.harringtonhoists.com/tech_support/edocs/EDOC 0367 rev02.pdf) recommends a S10x25.4 beam with an S of 124 for a 10 foot span and 2 ton load. The factor of safety would be roughly double the S8x18.4, or about 35.
So, a few different choices of recommendations available. You can never go wrong going bigger, except harder to handle the weight of the beam.
If I was going to install the beam permanently in place, and it is rigidly attached, level and not twist, I'd be pretty comfortable at the S6. If it was going to be on a portable stand, level, hard floor, I'd be wanting to go with the S8. Twisted, not level, soft floor, I'd be looking at the S10.