Semigroupoids (also called semicategories) provide a natural language that unifies the theories of groupoids and inverse semigroups, and appear naturally associated to fibred spaces. In this talk we will define algebraic bundles over topological semigroupoids and the associated (graded) sectional algebras, in a manner similar to that of a sectional algebra of a Fell bundle. Several well-known constructions may be regarded as particular cases of this construction. We then prove generalizations of recent results which have been obtained e.g. in the study of Steinberg algebras. Namely, we relate semidirect products of semigroupoids and crossed products algebras ; skew products of graded (semi)groupoids and smash products of algebras ; direct products of semigroupoids and tensor products ; and quotient semigroupoids and quotient algebras. I will finish the presentation with a few natural open questions.

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