CatDat

Implication Details

Assumptions: finitely accessible, left cancellative

Conclusions: generalized variety

Proof: Let $\C$ be a finitely accessible left cancellative category. The proof of this result shows that every sifted diagram $\I \to \C$ factors through the preorder reflection of $\I$, and hence reduces to a filtered diagram. Since $\C$ has filtered colimits, it therefore also has sifted colimits. It follows that every functor on $\C$ preserving filtered colimits automatically preserves sifted colimits. In particular, for representable functors this means that every finitely presentable object is automatically strongly finitely presentable. Now let $G$ be a set of finitely presentable objects in $\C$ generating all objects via filtered colimits. The claim follows because every filtered colimit is sifted and the objects in $G$ are strongly finitely presentable.