Large cardinals and category theory J. Rosický Montseny 2018
A set A of objects of a category K is called colimit dense if every object of K is a colimit of objects of A .
A set A of objects of a category K is called colimit dense if every object of K is a colimit of objects of A . R is colimit dense in the category Vect of vector spaces over R because every vector space is a coproduct of copies of R .
A set A of objects of a category K is called colimit dense if every object of K is a colimit of objects of A . R is colimit dense in the category Vect of vector spaces over R because every vector space is a coproduct of copies of R . A set A of objects of a category K is called dense if every object K ∈ K is a colimit of its canonical diagram consisting of morphisms A → K , A ∈ A .
A set A of objects of a category K is called colimit dense if every object of K is a colimit of objects of A . R is colimit dense in the category Vect of vector spaces over R because every vector space is a coproduct of copies of R . A set A of objects of a category K is called dense if every object K ∈ K is a colimit of its canonical diagram consisting of morphisms A → K , A ∈ A . A category having a dense set of objects is called bounded .
A set A of objects of a category K is called colimit dense if every object of K is a colimit of objects of A . R is colimit dense in the category Vect of vector spaces over R because every vector space is a coproduct of copies of R . A set A of objects of a category K is called dense if every object K ∈ K is a colimit of its canonical diagram consisting of morphisms A → K , A ∈ A . A category having a dense set of objects is called bounded . R is not dense in Vect . The reason is that any homogeneous map F : V → W is compatible with R → V .
A set A of objects of a category K is called colimit dense if every object of K is a colimit of objects of A . R is colimit dense in the category Vect of vector spaces over R because every vector space is a coproduct of copies of R . A set A of objects of a category K is called dense if every object K ∈ K is a colimit of its canonical diagram consisting of morphisms A → K , A ∈ A . A category having a dense set of objects is called bounded . R is not dense in Vect . The reason is that any homogeneous map F : V → W is compatible with R → V . R 2 is dense in Vect.
A set A of objects of a category K is called colimit dense if every object of K is a colimit of objects of A . R is colimit dense in the category Vect of vector spaces over R because every vector space is a coproduct of copies of R . A set A of objects of a category K is called dense if every object K ∈ K is a colimit of its canonical diagram consisting of morphisms A → K , A ∈ A . A category having a dense set of objects is called bounded . R is not dense in Vect . The reason is that any homogeneous map F : V → W is compatible with R → V . R 2 is dense in Vect. Question 1. Is every category with a colimit dense set of objects bounded?
A set A of objects of a category K is called colimit dense if every object of K is a colimit of objects of A . R is colimit dense in the category Vect of vector spaces over R because every vector space is a coproduct of copies of R . A set A of objects of a category K is called dense if every object K ∈ K is a colimit of its canonical diagram consisting of morphisms A → K , A ∈ A . A category having a dense set of objects is called bounded . R is not dense in Vect . The reason is that any homogeneous map F : V → W is compatible with R → V . R 2 is dense in Vect. Question 1. Is every category with a colimit dense set of objects bounded? [Adámek, Herrlich, Reiterman 1989] showed that the positive answer implies Vopěnka’s principle.
In [JR, Trnková, Adámek 1990] we also claimed that the answer is positive under VP and copied it in our book. Recently, Tim Campion found a gap in our proof and provided a counter-example.
In [JR, Trnková, Adámek 1990] we also claimed that the answer is positive under VP and copied it in our book. Recently, Tim Campion found a gap in our proof and provided a counter-example. Let (M) denote the non-existence of a proper class of measurable cardinals.
In [JR, Trnková, Adámek 1990] we also claimed that the answer is positive under VP and copied it in our book. Recently, Tim Campion found a gap in our proof and provided a counter-example. Let (M) denote the non-existence of a proper class of measurable cardinals. Theorem 1. (Isbell 1960) Set op is bounded iff (M) holds.
In [JR, Trnková, Adámek 1990] we also claimed that the answer is positive under VP and copied it in our book. Recently, Tim Campion found a gap in our proof and provided a counter-example. Let (M) denote the non-existence of a proper class of measurable cardinals. Theorem 1. (Isbell 1960) Set op is bounded iff (M) holds. Example (Campion 2018) Since (VP) implies ¬ (M), Set op is not bounded under (VP). Let Set 2 be the full subcategory of Set consisting of ∅ and sets 2 Y . Since Set is the completion of Set 2 under retracts, Set op 2 is not bounded. But {∅ , 2 } is colimit dense in Set op 2 .
In [JR, Trnková, Adámek 1990] we also claimed that the answer is positive under VP and copied it in our book. Recently, Tim Campion found a gap in our proof and provided a counter-example. Let (M) denote the non-existence of a proper class of measurable cardinals. Theorem 1. (Isbell 1960) Set op is bounded iff (M) holds. Example (Campion 2018) Since (VP) implies ¬ (M), Set op is not bounded under (VP). Let Set 2 be the full subcategory of Set consisting of ∅ and sets 2 Y . Since Set is the completion of Set 2 under retracts, Set op 2 is not bounded. But {∅ , 2 } is colimit dense in Set op 2 . Question 2. When Set op does have a colimit dense set of objects?
In [JR, Trnková, Adámek 1990] we also claimed that the answer is positive under VP and copied it in our book. Recently, Tim Campion found a gap in our proof and provided a counter-example. Let (M) denote the non-existence of a proper class of measurable cardinals. Theorem 1. (Isbell 1960) Set op is bounded iff (M) holds. Example (Campion 2018) Since (VP) implies ¬ (M), Set op is not bounded under (VP). Let Set 2 be the full subcategory of Set consisting of ∅ and sets 2 Y . Since Set is the completion of Set 2 under retracts, Set op 2 is not bounded. But {∅ , 2 } is colimit dense in Set op 2 . Question 2. When Set op does have a colimit dense set of objects? Question 3. Is every cocomplete category with a colimit dense set of objects bounded?
In [JR, Trnková, Adámek 1990] we also claimed that the answer is positive under VP and copied it in our book. Recently, Tim Campion found a gap in our proof and provided a counter-example. Let (M) denote the non-existence of a proper class of measurable cardinals. Theorem 1. (Isbell 1960) Set op is bounded iff (M) holds. Example (Campion 2018) Since (VP) implies ¬ (M), Set op is not bounded under (VP). Let Set 2 be the full subcategory of Set consisting of ∅ and sets 2 Y . Since Set is the completion of Set 2 under retracts, Set op 2 is not bounded. But {∅ , 2 } is colimit dense in Set op 2 . Question 2. When Set op does have a colimit dense set of objects? Question 3. Is every cocomplete category with a colimit dense set of objects bounded? Following [AHR], the positive answer implies (VP).
Let κ be a cardinal. A partitions Q on a set X is called a κ -partition if it has less than κ classes.
Let κ be a cardinal. A partitions Q on a set X is called a κ -partition if it has less than κ classes. Let Q be a set of κ -partitions on X . We say that Q is separating if for any distinct elements x , y ∈ X there is a class in some partition of Q containing x but not y .
Let κ be a cardinal. A partitions Q on a set X is called a κ -partition if it has less than κ classes. Let Q be a set of κ -partitions on X . We say that Q is separating if for any distinct elements x , y ∈ X there is a class in some partition of Q containing x but not y . A coherent choice for Q is a choice of a class A Q ∈ Q where Q ranges through Q such that A Q 1 ⊆ A Q 2 if Q 2 is coarser than Q 1 .
Let κ be a cardinal. A partitions Q on a set X is called a κ -partition if it has less than κ classes. Let Q be a set of κ -partitions on X . We say that Q is separating if for any distinct elements x , y ∈ X there is a class in some partition of Q containing x but not y . A coherent choice for Q is a choice of a class A Q ∈ Q where Q ranges through Q such that A Q 1 ⊆ A Q 2 if Q 2 is coarser than Q 1 . If Q consists of all κ -partitions on X then coherent choices for Q coincide with κ -complete ultrafilters on X .
Let κ be a cardinal. A partitions Q on a set X is called a κ -partition if it has less than κ classes. Let Q be a set of κ -partitions on X . We say that Q is separating if for any distinct elements x , y ∈ X there is a class in some partition of Q containing x but not y . A coherent choice for Q is a choice of a class A Q ∈ Q where Q ranges through Q such that A Q 1 ⊆ A Q 2 if Q 2 is coarser than Q 1 . If Q consists of all κ -partitions on X then coherent choices for Q coincide with κ -complete ultrafilters on X . (WM) There is a cardinal κ such that for every set X there exists a separating set Q X of κ -partitions of it such that every coherent choice for Q X has nonempty intersection of the chosen classes.
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