  We propose a simple bose-fermi model in two dimensions, with a coupling that
converts pairs of opposite spin fermions into localized bosons and vice versa.
We show that tracing out one of the degrees, either the bosons or fermions,
generates temperature-dependent long range effective interactions between
bosons as well as effective attractive interactions between fermions. Using
Monte Carlo techniques we obtain the thermodynamic properties and phase
stiffness as a function of temperature, dominated by vortex-antivortex
unbinding of the bosons. Remarkably in the fermion sector we observe a
temperature-induced BCS-BEC crossover signaled by a distinct change of their
spectral properties: the minimum gap locus moves from the Fermi wave vector to
the $\Gamma$ point. Such a model is relevant for describing aspects of high
$T_c$ superconductivity in cuprates and pnictides, superconducting islands on
graphene, and bose-fermi mixtures in cold atomic systems.
