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6/3/98 What are the contributions of Beck and Chevalley respectively associated with the Beck-Chevalley condition ? 6/2/98 Consider a locally finitely presentable category C. Denote by Fin(C) the full subcategory of finitely presentable objects. Since C is uniquely determined by the subcategory Fin(C), the following question makes sense: What conditions on Fin(C) will ensure that C is regular? Is there any paper dealing with this kind of questions? 6/1/98 In Categorical Geometry, Chapter 4, Section 4.3 I proved the following:
If X is any object we denote by B(X) the poset of direct subobjects of X. The following Proposition 4.3.5 holds for any extensive category: Proposition 4.3.5. B(X) is a Boolean algebra. Proof. Suppose U and V are two direct subobjects of an object X. Then If W is another direct subobject of X, then W Ç (U Ú V) = W Ç [(U Ç V) + (Uc Ç V) + (U Ç Vc)] = W Ç U Ç V + W Ç Uc Ç V + W Ç U Ç Vc = (W Ç U) Ç (W Ç V) + (W Ç Uc) Ç (W Ç V) + (W Ç U) Ç (W Ç Vc) = (W Ç U) Ç (W Ç V) + (W Ç U)c Ç (W Ç V) + (W Ç U) Ç (W Ç V)c = (W Ç U) Ú (W Ç V). This shows that B(X) is a distributive lattice. Clearly Uc is the complement of U in B(X). Thus B(X) is a Boolean algebra. n This has been proved by Diers in [Diers 1986, p.24, Proposition 1.3.3] in the dual situation for objects in a locally indecomposable category (= the dual of a coherent analytic category). Since this is a very fundamental fact, I would like to know whether it has already been covered in literature? |