  We present the first-principles investigation of the structural, electronic,
and magnetic properties of SrCoO$_{3-\delta}$ ($\delta=0, 0.25, 0.5$) to
understand the multivalent nature of Co ions in SrCoO$_{3-\delta}$ along the
line of topotactic transition between perovskite SrCoO$_{3}$ and brownmillerite
SrCoO$_{2.5}$. From the on-site Coulomb interaction $U$-dependent ground state
of stoichiometric SrCoO$_{3}$, we show the proximity of its metallic
ferromagnetic ground state to other antiferromagnetic states. The structural
and magnetic properties of SrCoO$_{3-\delta}$ depending on their oxygen-content
provide an interesting insight into the relationship between the Co-Co
distances and the magnetic couplings so that the spin-state transition of Co
spins can understood by the change of $pd$-hybridization depending on the Co-Co
distances. The \emph{strong} suppression of the $dp\sigma$-hybridization
between Co $d$ and O $p$ orbitals in brownmillerite SrCoO$_{2.5}$ brings on the
high-spin state of Co$^{3+}$ $d^{6}$ and is responsible for the
antiferromagnetically ordered insulating ground state. The increase of
effective Co-Co distances driven by the presence of oxygen vacancies in
SrCoO$_{3-\delta}$ is consistent with the reduction of the effective
$pd$-hybridization between Co $d$ and O $p$ orbitals. We conclude that the
configuration of neighboring Co spins is shown to be crucial to their local
electronic structure near the metal-to-insulator transition along the line of
the topotactic transition in SrCoO$_{3-\delta}$. Incidentally, we also find
that the \textit{I2mb} symmetry of SrCoO$_{2.5}$ is energetically stable and
exhibits ferroelectricity via the ordering of CoO$_{4}$ tetrahedra, where this
polar lattice can be stabilized by the presence of a large activation barrier.
