What are the contributions of Beck and Chevalley respectively associated with the Beck-Chevalley condition ?
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?
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:
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?