  We study a one-dimensional array of bosons with infinite-range interactions
mediated by a laser-driven dissipative optical cavity. The cavity-mediated
infinite-range interactions open up a new pathway to fermionization, hitherto
only known for dipolar bosons due to their long-range interactions. In
parameter ranges attainable in state-of-the-art experiments, we systematically
compare observables for bosons and fermions with infinite-range interactions.
At large enough laser pump power, many observables, including density
distributions in real and momentum space, correlation functions, eigenvalues of
the one-body density matrix, and superradiance order parameter, become
identical for bosons and fermions. We map out the emergence of this
cavity-induced fermionization as a function of pump power and contact
interactions. We discover that cavity-mediated interactions can compensate a
reduction by several orders of magnitude in the strength of the contact
interactions needed to trigger fermionization.
