  In 1970s Segal outlined proofs of two theorems relating spaces of Fredholm
and self-adjoint Fredholm operators with Quillen's constructions used to define
higher algebraic K-theory. In the present paper we provide detailed proofs of
these theorems of Segal (different from the ones outlined by Segal) and then
transplant some further ideas of Quillen to the real of Fredholm and
self-adjoint Fredholm operators. The main results may be considered as partial
analogues of the equivalence of Quillen's two definition of higher algebraic
K-theory. Along the way we relate spaces of Fredholm and self-adjoint Fredholm
operators with more concrete classifying spaces. These results are partially
motivated by applications to the index theory in a related paper.
