  Spin-splitting induced in a conventional superconductor weakens
superconductivity by destroying spin-singlet and creating spin-triplet Cooper
pairs. We demonstrate theoretically that such an effect is also caused by an
adjacent compensated antiferromagnet, which yields no net spin-splitting. We
find that the antiferromagnet produces N\'eel triplet Cooper pairs, whose
pairing amplitude oscillates rapidly in space similar to the antiferromagnet's
spin. The emergence of these unconventional Cooper pairs reduces the singlet
pairs' amplitude, thereby lowering the superconducting critical temperature. We
develop a quasiclassical Green's functions description of the system employing
a two-sublattice framework. It successfully captures the rapid oscillations in
the Cooper pairs' amplitude at the lattice spacing scale as well as their
smooth variation on the larger coherence length scale. Employing the
theoretical framework thus developed, we investigate this N\'eel proximity
effect in a superconductor/antiferromagnet bilayer as a function of interfacial
exchange, disorder, and chemical potential, finding rich physics. Our findings
also offer insights into experiments which have found a larger than expected
suppression of superconductivity by an adjacent antiferromagnet.
