  A quark+diquark Faddeev equation treatment of the baryon bound state problem
in Poincar\'e-invariant quantum field theory is used to deliver parameter-free
predictions for all six $\Delta$-baryon elastic weak form factors. Amongst the
results, it is worth highlighting that there are two distinct classes of such
$\Delta$-baryon form factors, $(g_1, g_3, G_{\pi\Delta\Delta})$, $(h_1, h_3,
H_{\pi\Delta\Delta})$, the functions within each of which are separately linked
via partial conservation of axial current (PCAC) and Goldberger-Treiman (GT)
relations. Respectively within each class, the listed form factors possess
qualitatively the same structural features as the nucleon axial, induced
pseudoscalar, and pion-nucleon coupling form factors. For instance, the
$\Delta$-baryon $g_1$ axial form factor can reliably be approximated by a
dipole function, characterised by an axial charge $g_A^{\Delta^+}=0.71(9)$ and
mass-scale $m_A^\Delta=0.95(2)m_\Delta$. Moreover, the two distinct
$\Delta$-baryon PCAC form factor relations are satisfied to a high degree of
accuracy on a large range of $Q^2$; the associated GT relations present good
approximations only on $Q^2/m_\Delta^2 \simeq 0$; and pion pole dominance
approximations are reliable within both classes. There are two
$\pi\Delta\Delta$ couplings: $g_{\pi\Delta\Delta} = 10.46(1.88)$;
$h_{\pi\Delta\Delta}= 35.73(3.75)$; and the associated form factors are soft.
Such couplings commonly arise in phenomenology, which may therefore benefit
from our analyses. A flavour decomposition of the axial charges reveals that
quarks carry $71$\% of the $\Delta$-baryon spin. The analogous result for the
proton is $\approx 65$\%.
