Partial Semigroupoid Actions on Sets

We introduce partial semigroupoid actions on sets and demonstrate that each such action admits universal globalization. Our construction extends the universal globalization for partial category actions given by P. Nystedt (Lundström) and the tensor product globalization for strong partial semigroup actions given by G. Kudryavtseva and V. Laan, thereby unifying the theory of partial actions for both categories and semigroups.

💡 Research Summary

This paper, titled “Partial Semigroupoid Actions on Sets,” introduces and develops a unified theory of partial actions for semigroupoids, a broad algebraic structure that encompasses both semigroups and categories as special cases. The central result is that every partial action of a semigroupoid admits a universal globalization, generalizing and unifying prior work on partial category actions and partial semigroup actions.

The authors begin by recalling Exel’s definition of a semigroupoid: a set equipped with a partially defined associative operation. This structure is more general than Tilson’s “graphed semigroupoids” (or semicategories), which correspond precisely to the “categorical” semigroupoids where the sets of right-composable elements for any two morphisms are either disjoint or equal. Important examples include Markov semigroupoids, which may be non-categorical and thus not admit a graph representation.

A partial action α of a semigroupoid S on a set X is defined as a family of subsets {sX} of X and bijections α_s: sX → X_s ⊆ X, satisfying two natural compatibility conditions (P1 and P2) that ensure the actions partially compose according to the product in S. A global action is a partial action where these domains satisfy stronger consistency conditions (G1 and G2), effectively mimicking a well-defined, everywhere-defined action.

The paper then shows how to restrict a global action on a larger set Y to a partial action on a subset X, setting the stage for the inverse problem: globalization. The main constructive theorem demonstrates that for any given partial action α of S on X, one can build a canonical global action β of S on a set Y that contains X, such that restricting β to X recovers α. This constructed β is universal: it is the “smallest” such globalization, meaning any other globalization of α factors uniquely through β via a morphism of global actions.

The construction of Y is explicit. It starts with the set of formal pairs (s, x) for s ∈ S and x ∈ sX, and then imposes an equivalence relation generated by the fundamental identifications forced by the partial action axioms, most notably (s, α_t(x)) ~ (st, x) for x ∈ tX ∩ stX. The action β_s is defined on equivalence classes by β_s(