Questions
What are the contributions of Beck and Chevalley respectively associated
with the
Consider a locally finitely presentable category What conditions on Is there any paper dealing with this kind of questions?
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:
X = (U + U)^{c} Ç
(V + V)^{c} = (U Ç
V) + (UÇ^{c}
V) + (U Ç V)^{c}
+ (UÇ^{c} V). ^{c}U Ç V U Ç V) + (UÇ^{c}
V) + (U Ç
V) ^{c}U Ç V = U Ù
V,U Ç V) + (UÇ^{c}
V) + (U Ç
V) = ^{c}U Ú VB(X) as (U Ç
V) + (U Ç V)^{c}
= U and (U Ç
V) + (UÇ^{c}
V) = V. Thus B(X) is a lattice.
If W is another direct subobject of X, then
X = W Ç U
+ ^{c}W Ç U + W^{c} W Ç U)^{c}
= W Ç U + ^{c}W. ^{c}W Ç U)
Ç ^{c}W = [(W Ç
U) + ^{c}W] Ç
^{c}W = (W Ç U)
Ç ^{c}W.W Ç V)
= ^{c}W Ç V Ç
^{c}W. W Ç
(U Ú V)
= W Ç [(U Ç
V) + (UÇ^{c}
V) + (U Ç V)]^{c}
= W Ç U Ç
V + W Ç UÇ^{c}
V + W Ç U Ç
V
^{c} = (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)
Ç (W Ç
V) + (W Ç U)Ç^{c}
(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
U is the complement of ^{c}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? |