As I can't comment on the accepted answer, I would like to suggest tweaking that solution to use border-width to prevent global color styling of the caret, less code and includes automatic styling of the dropup caret.
.caret {
border-width: 8px;
}
In case you need a different shaped caret, you can use left, right, top, bottom:
border-left-width: 8px;
border-right-width: 8px;
border-top-width: 10px;
border-bottom-width: 10px;
or
border-width: 10px 8px 10px 8px;
etc.