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We have seen above that locally presentable categories can be described as categories of set-valued functors preserving A-small limits. We will now show that, more generally, set-valued functors preserving specified limits (or, still more generally, turning specified cones to limits) always form a locally presentable category. Such categories are known as categories of models of limit sketches. Let us elaborate on this concept by specifying a collection of cones and working with set-valued functors turning the specified cones into limits.

C. REPRESENTATION THEOREM /I m3=id i 29 (transitivity) (5) An example of an orthogonality class which is not a small-orthogonality class: complete join-semilattices in the category Pos* of all posets and all functions preserving (all existing) joins. , downwards closed sets I C P closed under all existing joins) ordered by inclusion. p = Ix EPI x

3) CPO and Top are not locally finitely presentable. (4) The category of finite sets is not locally finitely presentable since it is not cocomplete. , each element is a directed join of finite elements). 11 Theorem. A category is locally finitely presentable if it is cocomplete, and has a strong generator formed by finitely presentable objects. PROOF. The necessity is clear. To prove the sufficiency, let K be a cocomplete category with a strong generator A formed by finitely presentable objects.

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