  We describe a probabilistic method of cross-identifying astrophysical sources
in two catalogs from their positions and positional uncertainties. The
probability that an object is associated with a source from the other catalog,
or that it has no counterpart, is derived under two exclusive assumptions:
first, the classical case of several-to-one associations, and then the more
realistic but more difficult problem of one-to-one associations.
  In either case, the likelihood of observing the objects in the two catalogs
at their effective positions is computed and a maximum likelihood estimator of
the fraction of sources with a counterpart -- a quantity needed to compute the
probabilities of association -- is built. When the positional uncertainty in
one or both catalogs is unknown, this method may be used to estimate its
typical value and even to study its dependence on the size of objects. It may
also be applied when the true centers of a source and of its counterpart at
another wavelength do not coincide.
  To compute the likelihood and association probabilities under the different
assumptions, we developed a Fortran 95 code called "Aspects" ([asp{\epsilon}],
"ASsociation PositionnellE/ProbabilistE de CaTalogues de Sources" in French);
its source files are made freely available. To test Aspects, all-sky mock
catalogs containing up to 10^5 objects were created, forcing either
several-to-one or one-to-one associations. The analysis of these simulations
confirms that, in both cases, the assumption with the highest likelihood is the
right one and that estimators of unknown parameters built for the appropriate
association model are reliable.
