By Pierre Gillibert
This paintings introduces instruments from the sphere of classification idea that give the opportunity to take on a couple of illustration difficulties that experience remained unsolvable thus far (e.g. the selection of the variety of a given functor). the fundamental proposal is: if a functor lifts many items, then it additionally lifts many (poset-indexed) diagrams.
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Example text
3. Denote by Bool???? the category described as follows: Boolean algebras, are the families of • The objects of (Bool ( ???? , called ???? -scaled )) the form ???? = ????, ????(????) ∣ ???? ∈ ???? , where ???? is a Boolean algebra and ????(????) is an ideal of ???? for all ???? ∈ ???? , and the following conditions are satisfied: ) ⋁ ( (????) (i) ???? = ???? ∣????∈???? . ( ) ⋁ (ii) ????(????) ∩ ????(????) = ????(????) ∣ ???? ≥ ????, ???? in ???? , for all ????, ???? ∈ ???? . 2 ???? -Normed Spaces, ???? -Scaled Boolean Algebras 39 • For objects ???? and ???? in Bool???? , a morphism from ???? to ???? is a morphism ???? : ???? → ???? of Boolean algebras such that ???? “(????(????) ) ⊆ ???? (????) , for all ???? ∈ ???? .
Let ????, ???? ∈ ∥????∥. Pick ???? ∈ ???? ∩ ????(????) and ???? ∈ ???? ∩ ????(????) . Then ???? := ???? ∧ ???? (????) (????) belongs ⋁ to ???? ∩ (???? ∩ ???? ), and so there exists a decomposition of the form ???? = (???????? ∣ ???? ∈ ????) in ????, where ???? is a finite subset of ???? ⇑{????, ????} and ???????? ∈ ????(????) for all ???? ∈ ????. As ???? is an ultrafilter, there exists ???? ∈ ???? such that ???????? ∈ ????, and so ???? ∈ ∥????∥, with ???? ≥ ????, ????. This proves that ∥????∥ is directed. Therefore, ∥????∥ is an ideal of ???? . For each ???? ∈ ???? , {???? ∈ Ult ???? ∣ ???? ∈ ∥????∥} = {???? ∈ Ult ???? ∣ ???? ∩ ????(????) ∕= ∅} is obviously an open subset of Ult ????.
1 of ???? , we set ????0 ▽ ⋅ ⋅ ⋅ ▽ ????????−1 := ∪ (▽????
