I know the May snowfalls may make it seem like Minnesota never gets hot enough to sweat, my Southern friend, but mid summer brings very hot and humid commutes, so that even we phlegmatic Northerners have been known to sweat on occasion! Still, my point was not that everyone everywhere could avoid carrying stuff to and fro on their commutes, but that advance planning can minimize that for many, especially for a short commute like the OP.

While I usually purchase lunch most days regardless of whether I'm riding my bike or car, I also periodically bring lunch food on days that I commute by car that I leave in the fridge for days that I eat in. I also take the opportunity on such days to restock my supply of fresh clothes at work that I can change into when needed. That is where the advance planning comes into play.

I have a rack and a backpack that I use when needed, I just found that when I planned ahead a bit more I didn't need them quite as much as I otherwise would have. YMMV.