Sequential Sets

Let ∆∞ = End([∞]) be the monoid of convergent monotonic sequences in N∪{∞}. The category Pre(∆∞) of sequential sets is the category whose objects are sets equipped with an action of this monoid, and whose morphisms are equivariant maps. We call a category C sequentially generated if it is a localization of Pre(∆∞). In this paper we show that several important categories, including the category Cat of categories, the category Topseq of sequentially generated topological spaces, and the category of simplicial sets, are sequentially generated. We hope to convince the reader that sequential sets gives an appropriate and simple setting for several mathematical considerations.