  An ind-variety is an inductive limit of closed embeddings of algebraic
varieties and an ind-group is a group object in the category of ind-varieties.
These notions were first introduced by Shafarevich in the study of the
automorphism group of affine spaces and have been studied by many authors
afterwards. An ind-torus is an ind-group obtained as an inductive limit of
closed embeddings of algebraic tori that are also algebraic group
homomorphisms. In this paper, we introduce the natural definition of toric
ind-varieties as ind-varieties having an ind-torus as an open set and such that
the action of the ind-torus on itself by translations extends to a regular
action on the whole ind-variety. We are brought to introduce and study
pro-affine semigroup that turn out to be unital semigroups isomorphic to closed
subsemigroups of the group of arbitrary integer sequences with the product
topology such that their projection to the first i-th coordinates is finitely
generated for all positive integers i. Our main result is a duality between the
categories of affine toric ind-varieties and the the category of pro-affine
semigroups.
