Home | Questions | References | Notations | Dictionary | Library | Links    Questions  Any comments are welcome. Thanks in advance! Please send responses to catgeo@azd.com   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  X = (U + Uc) Ç (V + Vc) = (U Ç V) + (Uc Ç V) + (U Ç Vc) + (Uc Ç Vc).  Thus  U Ç V  and  (U Ç V) + (Uc Ç V) + (U Ç Vc)  are direct subobjects. But  U Ç V = U Ù V, and we have the formula  (U Ç V) + (Uc Ç V) + (U Ç Vc) = U Ú V in B(X) as (U Ç V) + (U Ç Vc) = U  and  (U Ç V) + (Uc Ç V) = V. Thus B(X) is a lattice.  If W is another direct subobject of X, then  X = W Ç Uc + W Ç U + Wc  implies that  (W Ç U)c = W Ç Uc + Wc.  Thus  (W Ç U)c Ç W = [(W Ç Uc) + Wc] Ç W = (W Ç Uc) Ç W. Similarly  (W Ç V)c = W Ç Vc Ç W. We have      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?              Home | Questions | References | Notations | Dictionary | Library | Links