Sifted Colimits, Strongly Finitary Monads and Continuous Algebras

TL;DR

This paper characterizes strongly finitary monads on categories like Pos, CPO, and DCPO as those preserving sifted colimits, linking them to varieties of continuous algebras and enriched colimits.

Contribution

It provides a precise characterization of strongly finitary monads via sifted colimits and reflexive coinserters in enriched categories, connecting to continuous algebra varieties.

Findings

Strongly finitary monads preserve sifted colimits.

Varieties of continuous algebras correspond to monadic categories for these monads.

Enriched categories' sifted colimits are studied in detail.

Abstract

We characterize strongly finitary monads on categories $\mathsf{Pos}$, $\mathsf{CPO}$ and $\mathsf{DCPO}$ as precisely those preserving sifted colimits. Or, equivalently, enriched finitary monads preserving reflexive coinserters. We study sifted colimits in general enriched categories. For $\mathsf{CPO}$ and $\mathsf{DCPO}$ we characterize varieties of continuous algebras as precisely the monadic categories for strongly finitary monads.