By J. Adamek, J. Rosicky
The options of a in the community presentable classification and an available class are super necessary in formulating connections among common algebra, version idea, common sense, and computing device technology. the purpose of this booklet is to supply an exposition of either the speculation and the functions of those different types at a degree obtainable to graduate scholars. The techniques of lambda-presentable gadgets, in the community lambda-presentable different types, and lambda-accessible different types are mentioned intimately. The authors turn out that Freyd's primarily algebraic different types are exactly the in the community presentable different types. within the ultimate bankruptcy, they deal with a few complicated themes in version idea.
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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.