Large cardinals and category theory J. Rosick Montseny 2018 A set A - - PowerPoint PPT Presentation

Large cardinals and category theory

• J. Rosický

Montseny 2018

A set A of objects of a category K is called colimit dense if every

• bject of K is a colimit of objects of A.

A set A of objects of a category K is called colimit dense if every

• bject 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

• bject 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

• bject 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

• bject 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

• bject 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 . R2 is dense in Vect.

A set A of objects of a category K is called colimit dense if every

• bject 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 . R2 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

• bject 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 . R2 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) Setop 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) Setop is bounded iff (M) holds. Example (Campion 2018) Since (VP) implies ¬(M), Setop is not bounded under (VP). Let Set2 be the full subcategory of Set consisting of ∅ and sets 2Y . Since Set is the completion of Set2 under retracts, Setop

2 is not bounded. But {∅, 2} is colimit dense in

Setop

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) Setop is bounded iff (M) holds. Example (Campion 2018) Since (VP) implies ¬(M), Setop is not bounded under (VP). Let Set2 be the full subcategory of Set consisting of ∅ and sets 2Y . Since Set is the completion of Set2 under retracts, Setop

2 is not bounded. But {∅, 2} is colimit dense in

Setop

2 .

Question 2. When Setop 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) Setop is bounded iff (M) holds. Example (Campion 2018) Since (VP) implies ¬(M), Setop is not bounded under (VP). Let Set2 be the full subcategory of Set consisting of ∅ and sets 2Y . Since Set is the completion of Set2 under retracts, Setop

2 is not bounded. But {∅, 2} is colimit dense in

Setop

2 .

Question 2. When Setop does have a colimit dense set of objects? Question 3. Is every cocomplete category with a colimit dense set

• f 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) Setop is bounded iff (M) holds. Example (Campion 2018) Since (VP) implies ¬(M), Setop is not bounded under (VP). Let Set2 be the full subcategory of Set consisting of ∅ and sets 2Y . Since Set is the completion of Set2 under retracts, Setop

2 is not bounded. But {∅, 2} is colimit dense in

Setop

2 .

Question 2. When Setop does have a colimit dense set of objects? Question 3. Is every cocomplete category with a colimit dense set

• f 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

• f 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

• f Q containing x but not y.

A coherent choice for Q is a choice of a class AQ ∈ Q where Q ranges through Q such that AQ1 ⊆ AQ2 if Q2 is coarser than Q1.

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

• f Q containing x but not y.

A coherent choice for Q is a choice of a class AQ ∈ Q where Q ranges through Q such that AQ1 ⊆ AQ2 if Q2 is coarser than Q1. 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

• f Q containing x but not y.

A coherent choice for Q is a choice of a class AQ ∈ Q where Q ranges through Q such that AQ1 ⊆ AQ2 if Q2 is coarser than Q1. 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 QX of κ-partitions of it such that every coherent choice for QX has nonempty intersection of the chosen 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

• f Q containing x but not y.

A coherent choice for Q is a choice of a class AQ ∈ Q where Q ranges through Q such that AQ1 ⊆ AQ2 if Q2 is coarser than Q1. 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 QX of κ-partitions of it such that every coherent choice for QX has nonempty intersection of the chosen classes. Clearly (M) ⇒ (WM). Analogously to Theorem 1 we get

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

• f Q containing x but not y.

A coherent choice for Q is a choice of a class AQ ∈ Q where Q ranges through Q such that AQ1 ⊆ AQ2 if Q2 is coarser than Q1. 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 QX of κ-partitions of it such that every coherent choice for QX has nonempty intersection of the chosen classes. Clearly (M) ⇒ (WM). Analogously to Theorem 1 we get Theorem 2. (WM) ⇔ Setop has a small colimit dense subcategory.

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

• f Q containing x but not y.

A coherent choice for Q is a choice of a class AQ ∈ Q where Q ranges through Q such that AQ1 ⊆ AQ2 if Q2 is coarser than Q1. 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 QX of κ-partitions of it such that every coherent choice for QX has nonempty intersection of the chosen classes. Clearly (M) ⇒ (WM). Analogously to Theorem 1 we get Theorem 2. (WM) ⇔ Setop has a small colimit dense subcategory. Problem 1. (WM) ⇔ (M)?

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

• f Q containing x but not y.

A coherent choice for Q is a choice of a class AQ ∈ Q where Q ranges through Q such that AQ1 ⊆ AQ2 if Q2 is coarser than Q1. 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 QX of κ-partitions of it such that every coherent choice for QX has nonempty intersection of the chosen classes. Clearly (M) ⇒ (WM). Analogously to Theorem 1 we get Theorem 2. (WM) ⇔ Setop has a small colimit dense subcategory. Problem 1. (WM) ⇔ (M)? Or, what is the set-theoretical strength of (WM)?

A category K is boundable if it can be fully embedded to some category Str Σ of Σ-structures and homomorphisms.

A category K is boundable if it can be fully embedded to some category Str Σ of Σ-structures and homomorphisms. Any boundable category is concrete.

A category K is boundable if it can be fully embedded to some category Str Σ of Σ-structures and homomorphisms. Any boundable category is concrete. Theorem 3. (Hedrlín, Kučera, Pultr 1973) (M) ⇔ boundable = concrete.

A category K is boundable if it can be fully embedded to some category Str Σ of Σ-structures and homomorphisms. Any boundable category is concrete. Theorem 3. (Hedrlín, Kučera, Pultr 1973) (M) ⇔ boundable = concrete. A is dense iff E : K → SetAop is full and faithful (Isbell 1960).

A category K is boundable if it can be fully embedded to some category Str Σ of Σ-structures and homomorphisms. Any boundable category is concrete. Theorem 3. (Hedrlín, Kučera, Pultr 1973) (M) ⇔ boundable = concrete. A is dense iff E : K → SetAop is full and faithful (Isbell 1960). Since SetAop can be viewed as the category of many-sorted unary algebras, any bounded category is boundable.

A category K is boundable if it can be fully embedded to some category Str Σ of Σ-structures and homomorphisms. Any boundable category is concrete. Theorem 3. (Hedrlín, Kučera, Pultr 1973) (M) ⇔ boundable = concrete. A is dense iff E : K → SetAop is full and faithful (Isbell 1960). Since SetAop can be viewed as the category of many-sorted unary algebras, any bounded category is boundable. The category Ord of ordinals (as an ordered class), its dual and the large discrete category Dis are concrete but not bounded.

A category K is boundable if it can be fully embedded to some category Str Σ of Σ-structures and homomorphisms. Any boundable category is concrete. Theorem 3. (Hedrlín, Kučera, Pultr 1973) (M) ⇔ boundable = concrete. A is dense iff E : K → SetAop is full and faithful (Isbell 1960). Since SetAop can be viewed as the category of many-sorted unary algebras, any bounded category is boundable. The category Ord of ordinals (as an ordered class), its dual and the large discrete category Dis are concrete but not bounded. Vopěnka’s principle (VP) states that Dis (or, equivalently Ord) are not boundable.

A category K is boundable if it can be fully embedded to some category Str Σ of Σ-structures and homomorphisms. Any boundable category is concrete. Theorem 3. (Hedrlín, Kučera, Pultr 1973) (M) ⇔ boundable = concrete. A is dense iff E : K → SetAop is full and faithful (Isbell 1960). Since SetAop can be viewed as the category of many-sorted unary algebras, any bounded category is boundable. The category Ord of ordinals (as an ordered class), its dual and the large discrete category Dis are concrete but not bounded. Vopěnka’s principle (VP) states that Dis (or, equivalently Ord) are not boundable. Theorem 4. (RTA 1990) (VP) ⇔ boundable = bounded.

Theorem 5. The following conditions are equivalent: (1) Setop is bounded, (2) Setop is boundable, (3) Dual of a boundable category is boundable, (4) (M)

Theorem 5. The following conditions are equivalent: (1) Setop is bounded, (2) Setop is boundable, (3) Dual of a boundable category is boundable, (4) (M) (1) ⇔ (4) Theorem 1, (2) ⇔ (4) Theorem 3, (3) ⇒ (2) evident. (4) ⇒ (3) follows from Theorem 1.

Theorem 5. The following conditions are equivalent: (1) Setop is bounded, (2) Setop is boundable, (3) Dual of a boundable category is boundable, (4) (M) (1) ⇔ (4) Theorem 1, (2) ⇔ (4) Theorem 3, (3) ⇒ (2) evident. (4) ⇒ (3) follows from Theorem 1. Weak Vopěnka’s principle (WVP) states that Ordop is not boundable.

Theorem 5. The following conditions are equivalent: (1) Setop is bounded, (2) Setop is boundable, (3) Dual of a boundable category is boundable, (4) (M) (1) ⇔ (4) Theorem 1, (2) ⇔ (4) Theorem 3, (3) ⇒ (2) evident. (4) ⇒ (3) follows from Theorem 1. Weak Vopěnka’s principle (WVP) states that Ordop is not boundable. We have (VP) ⇒ (WVP) ⇒ ¬(M)

Theorem 5. The following conditions are equivalent: (1) Setop is bounded, (2) Setop is boundable, (3) Dual of a boundable category is boundable, (4) (M) (1) ⇔ (4) Theorem 1, (2) ⇔ (4) Theorem 3, (3) ⇒ (2) evident. (4) ⇒ (3) follows from Theorem 1. Weak Vopěnka’s principle (WVP) states that Ordop is not boundable. We have (VP) ⇒ (WVP) ⇒ ¬(M) (VP) is stronger than ¬(M) because it implies the existence of a proper class of extendible cardinals.

Theorem 5. The following conditions are equivalent: (1) Setop is bounded, (2) Setop is boundable, (3) Dual of a boundable category is boundable, (4) (M) (1) ⇔ (4) Theorem 1, (2) ⇔ (4) Theorem 3, (3) ⇒ (2) evident. (4) ⇒ (3) follows from Theorem 1. Weak Vopěnka’s principle (WVP) states that Ordop is not boundable. We have (VP) ⇒ (WVP) ⇒ ¬(M) (VP) is stronger than ¬(M) because it implies the existence of a proper class of extendible cardinals. The position of (WVP) between (VP) and ¬(M) is open.

Theorem 6. (RTA 1990) (WVP) is equivalent to the fact that every complete bounded category is cocomplete.

Theorem 6. (RTA 1990) (WVP) is equivalent to the fact that every complete bounded category is cocomplete. Any cocomplete bounded category is complete, without any set theory (Gabriel, Ulmer 1971).

Theorem 6. (RTA 1990) (WVP) is equivalent to the fact that every complete bounded category is cocomplete. Any cocomplete bounded category is complete, without any set theory (Gabriel, Ulmer 1971). The statement of Theorem 6 can be reformulated to: every full subcategory of Str Σ closed under limits is reflective.

Theorem 6. (RTA 1990) (WVP) is equivalent to the fact that every complete bounded category is cocomplete. Any cocomplete bounded category is complete, without any set theory (Gabriel, Ulmer 1971). The statement of Theorem 6 can be reformulated to: every full subcategory of Str Σ closed under limits is reflective. A “thick” category Ord of ordinals is any category whose objects are ordinals and a morphism i → j exists iff i ≤ j.

Theorem 6. (RTA 1990) (WVP) is equivalent to the fact that every complete bounded category is cocomplete. Any cocomplete bounded category is complete, without any set theory (Gabriel, Ulmer 1971). The statement of Theorem 6 can be reformulated to: every full subcategory of Str Σ closed under limits is reflective. A “thick” category Ord of ordinals is any category whose objects are ordinals and a morphism i → j exists iff i ≤ j. Any Ord is concrete but not bounded.

Theorem 6. (RTA 1990) (WVP) is equivalent to the fact that every complete bounded category is cocomplete. Any cocomplete bounded category is complete, without any set theory (Gabriel, Ulmer 1971). The statement of Theorem 6 can be reformulated to: every full subcategory of Str Σ closed under limits is reflective. A “thick” category Ord of ordinals is any category whose objects are ordinals and a morphism i → j exists iff i ≤ j. Any Ord is concrete but not bounded. Semiweak Vopěnka’s principle (sWVP) states that no Ordop is boundable.

Theorem 6. (RTA 1990) (WVP) is equivalent to the fact that every complete bounded category is cocomplete. Any cocomplete bounded category is complete, without any set theory (Gabriel, Ulmer 1971). The statement of Theorem 6 can be reformulated to: every full subcategory of Str Σ closed under limits is reflective. A “thick” category Ord of ordinals is any category whose objects are ordinals and a morphism i → j exists iff i ≤ j. Any Ord is concrete but not bounded. Semiweak Vopěnka’s principle (sWVP) states that no Ordop is boundable. (sWVP) is between (VP) and (WVP). Its position is not known.

Theorem 6. (RTA 1990) (WVP) is equivalent to the fact that every complete bounded category is cocomplete. Any cocomplete bounded category is complete, without any set theory (Gabriel, Ulmer 1971). The statement of Theorem 6 can be reformulated to: every full subcategory of Str Σ closed under limits is reflective. A “thick” category Ord of ordinals is any category whose objects are ordinals and a morphism i → j exists iff i ≤ j. Any Ord is concrete but not bounded. Semiweak Vopěnka’s principle (sWVP) states that no Ordop is boundable. (sWVP) is between (VP) and (WVP). Its position is not known. Are there other “boundability created” axioms?

Theorem 6. (RTA 1990) (WVP) is equivalent to the fact that every complete bounded category is cocomplete. Any cocomplete bounded category is complete, without any set theory (Gabriel, Ulmer 1971). The statement of Theorem 6 can be reformulated to: every full subcategory of Str Σ closed under limits is reflective. A “thick” category Ord of ordinals is any category whose objects are ordinals and a morphism i → j exists iff i ≤ j. Any Ord is concrete but not bounded. Semiweak Vopěnka’s principle (sWVP) states that no Ordop is boundable. (sWVP) is between (VP) and (WVP). Its position is not known. Are there other “boundability created” axioms? Theorem 7. (AR 1993) (sWVP) is equivalent to the fact that every full subcategory of Str Σ closed under products and retracts is weakly reflective.

An object A in K is presentable if there is a regular cardinal κ such that K(A, −) : K → Set preserves κ-directed colimits.

An object A in K is presentable if there is a regular cardinal κ such that K(A, −) : K → Set preserves κ-directed colimits. Theorem 8. (RTA 1990) (VP) is equivalent to the fact that every

• bject of a bounded category is presentable.

An object A in K is presentable if there is a regular cardinal κ such that K(A, −) : K → Set preserves κ-directed colimits. Theorem 8. (RTA 1990) (VP) is equivalent to the fact that every

• bject of a bounded category is presentable.

Any cocomplete category containing a colimit dense set of presentable objects is bounded.

An object A in K is presentable if there is a regular cardinal κ such that K(A, −) : K → Set preserves κ-directed colimits. Theorem 8. (RTA 1990) (VP) is equivalent to the fact that every

• bject of a bounded category is presentable.

Any cocomplete category containing a colimit dense set of presentable objects is bounded. In fact, these categories are locally presentable.

An object A in K is presentable if there is a regular cardinal κ such that K(A, −) : K → Set preserves κ-directed colimits. Theorem 8. (RTA 1990) (VP) is equivalent to the fact that every

• bject of a bounded category is presentable.

Any cocomplete category containing a colimit dense set of presentable objects is bounded. In fact, these categories are locally presentable. A category K is accessible if there is a regular cardinal κ such that K has κ-directed colimits and each object of K is a κ-directed colimit of κ-presentable objects. Locally presentable categories are precisely cocomplete accessible ones.

An object A in K is presentable if there is a regular cardinal κ such that K(A, −) : K → Set preserves κ-directed colimits. Theorem 8. (RTA 1990) (VP) is equivalent to the fact that every

• bject of a bounded category is presentable.

Any cocomplete category containing a colimit dense set of presentable objects is bounded. In fact, these categories are locally presentable. A category K is accessible if there is a regular cardinal κ such that K has κ-directed colimits and each object of K is a κ-directed colimit of κ-presentable objects. Locally presentable categories are precisely cocomplete accessible ones. Any accessible category is bounded and has all objects presentable.

An object A in K is presentable if there is a regular cardinal κ such that K(A, −) : K → Set preserves κ-directed colimits. Theorem 8. (RTA 1990) (VP) is equivalent to the fact that every

• bject of a bounded category is presentable.

Any cocomplete category containing a colimit dense set of presentable objects is bounded. In fact, these categories are locally presentable. A category K is accessible if there is a regular cardinal κ such that K has κ-directed colimits and each object of K is a κ-directed colimit of κ-presentable objects. Locally presentable categories are precisely cocomplete accessible ones. Any accessible category is bounded and has all objects presentable. A is a generator of K if E : K → SetAop is faithful.

An object A in K is presentable if there is a regular cardinal κ such that K(A, −) : K → Set preserves κ-directed colimits. Theorem 8. (RTA 1990) (VP) is equivalent to the fact that every

• bject of a bounded category is presentable.

Any cocomplete category containing a colimit dense set of presentable objects is bounded. In fact, these categories are locally presentable. A category K is accessible if there is a regular cardinal κ such that K has κ-directed colimits and each object of K is a κ-directed colimit of κ-presentable objects. Locally presentable categories are precisely cocomplete accessible ones. Any accessible category is bounded and has all objects presentable. A is a generator of K if E : K → SetAop is faithful. Theorem 9. Let K be a cocomplete category having a small colimit dense subcategory whose presentable members form a

• generator. Then, assuming (VP), K is bounded.

A full subcategory of Str Σ is reductive if it consists of reducts of a theory T in a signature Σ′ ⊇ Σ. Here, Σ, Σ′ and T could be infinitary.

A full subcategory of Str Σ is reductive if it consists of reducts of a theory T in a signature Σ′ ⊇ Σ. Here, Σ, Σ′ and T could be infinitary. Recall that a cardinal κ ≥ λ is called Lλ,ω-compact if any κ-complete filter extends to a λ-complete ultrafilter.

A full subcategory of Str Σ is reductive if it consists of reducts of a theory T in a signature Σ′ ⊇ Σ. Here, Σ, Σ′ and T could be infinitary. Recall that a cardinal κ ≥ λ is called Lλ,ω-compact if any κ-complete filter extends to a λ-complete ultrafilter. Theorem 10. (JR 1994) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every reductive category is bounded and has all objects presentable.

A full subcategory of Str Σ is reductive if it consists of reducts of a theory T in a signature Σ′ ⊇ Σ. Here, Σ, Σ′ and T could be infinitary. Recall that a cardinal κ ≥ λ is called Lλ,ω-compact if any κ-complete filter extends to a λ-complete ultrafilter. Theorem 10. (JR 1994) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every reductive category is bounded and has all objects presentable. Quaetion 4. What is the set-theoretical strength of this claim?

A full subcategory of Str Σ is reductive if it consists of reducts of a theory T in a signature Σ′ ⊇ Σ. Here, Σ, Σ′ and T could be infinitary. Recall that a cardinal κ ≥ λ is called Lλ,ω-compact if any κ-complete filter extends to a λ-complete ultrafilter. Theorem 10. (JR 1994) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every reductive category is bounded and has all objects presentable. Quaetion 4. What is the set-theoretical strength of this claim? We even do not know whether it depends on set theory at all.

A full subcategory of Str Σ is reductive if it consists of reducts of a theory T in a signature Σ′ ⊇ Σ. Here, Σ, Σ′ and T could be infinitary. Recall that a cardinal κ ≥ λ is called Lλ,ω-compact if any κ-complete filter extends to a λ-complete ultrafilter. Theorem 10. (JR 1994) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every reductive category is bounded and has all objects presentable. Quaetion 4. What is the set-theoretical strength of this claim? We even do not know whether it depends on set theory at all. Theorem 11. (WVP) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every complete reductive category is cocomplete.

A full subcategory of Str Σ is reductive if it consists of reducts of a theory T in a signature Σ′ ⊇ Σ. Here, Σ, Σ′ and T could be infinitary. Recall that a cardinal κ ≥ λ is called Lλ,ω-compact if any κ-complete filter extends to a λ-complete ultrafilter. Theorem 10. (JR 1994) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every reductive category is bounded and has all objects presentable. Quaetion 4. What is the set-theoretical strength of this claim? We even do not know whether it depends on set theory at all. Theorem 11. (WVP) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every complete reductive category is cocomplete. This follows from Theorems 6 and 10.

A full subcategory of Str Σ is reductive if it consists of reducts of a theory T in a signature Σ′ ⊇ Σ. Here, Σ, Σ′ and T could be infinitary. Recall that a cardinal κ ≥ λ is called Lλ,ω-compact if any κ-complete filter extends to a λ-complete ultrafilter. Theorem 10. (JR 1994) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every reductive category is bounded and has all objects presentable. Quaetion 4. What is the set-theoretical strength of this claim? We even do not know whether it depends on set theory at all. Theorem 11. (WVP) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every complete reductive category is cocomplete. This follows from Theorems 6 and 10. Question 5. What is the set-theoretical strength of this claim?

A full subcategory of Str Σ is reductive if it consists of reducts of a theory T in a signature Σ′ ⊇ Σ. Here, Σ, Σ′ and T could be infinitary. Recall that a cardinal κ ≥ λ is called Lλ,ω-compact if any κ-complete filter extends to a λ-complete ultrafilter. Theorem 10. (JR 1994) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every reductive category is bounded and has all objects presentable. Quaetion 4. What is the set-theoretical strength of this claim? We even do not know whether it depends on set theory at all. Theorem 11. (WVP) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then every complete reductive category is cocomplete. This follows from Theorems 6 and 10. Question 5. What is the set-theoretical strength of this claim? [R] wrongly claims that it follows from the assumption of Theorem 10.

Reductive classes of Σ-structures are also called pseudoelementary.

Reductive classes of Σ-structures are also called pseudoelementary. A class of Σ-structures is called universal if it is closed under submodels.

Reductive classes of Σ-structures are also called pseudoelementary. A class of Σ-structures is called universal if it is closed under submodels. Theorem 12. Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then any reductive and universal class K ⊆ Str Σ is closed under κ-directed colimits for some κ.

Reductive classes of Σ-structures are also called pseudoelementary. A class of Σ-structures is called universal if it is closed under submodels. Theorem 12. Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then any reductive and universal class K ⊆ Str Σ is closed under κ-directed colimits for some κ. This result was proved by Makkai and Paré (1989) under the stronger assumption of the existence of a proper class of strongly compact cardinals. The formulation above is due to [Brooke-Taylor, JR 2017] and, independently, to [Booney, Unger 2017].

Reductive classes of Σ-structures are also called pseudoelementary. A class of Σ-structures is called universal if it is closed under submodels. Theorem 12. Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then any reductive and universal class K ⊆ Str Σ is closed under κ-directed colimits for some κ. This result was proved by Makkai and Paré (1989) under the stronger assumption of the existence of a proper class of strongly compact cardinals. The formulation above is due to [Brooke-Taylor, JR 2017] and, independently, to [Booney, Unger 2017]. I am not sure what is the set-theoretic strength of Theorem 12. [BU] proved the equivalence of (1) for every cardinal λ there is an Lλ,ω-compact cardinal, (2) every reductive and universal class of Σ-structures is bounded, has all objects presentable and is closed under κ-directed colimits for some κ.

Reductive categories coincide with full images of accessible functors.

Reductive categories coincide with full images of accessible functors. Reductive and universal classes of Σ-structures coincide with powerful images of accessible functors.

Reductive categories coincide with full images of accessible functors. Reductive and universal classes of Σ-structures coincide with powerful images of accessible functors. Condition (2) above says that every powerful image of an accessible functor is accessible.

Reductive categories coincide with full images of accessible functors. Reductive and universal classes of Σ-structures coincide with powerful images of accessible functors. Condition (2) above says that every powerful image of an accessible functor is accessible. An example of a powerful image of an accessible functors is the category of free abelian groups.

Reductive categories coincide with full images of accessible functors. Reductive and universal classes of Σ-structures coincide with powerful images of accessible functors. Condition (2) above says that every powerful image of an accessible functor is accessible. An example of a powerful image of an accessible functors is the category of free abelian groups. Theorem 13. (Bagaria, Casacuberta, Mathias, JR 2015) Assume the existence of a proper class of supercompact cardinals. Then every Σ2 full subcategory K of Str Σ is bounded and has all

• bjects presentable.

Reductive categories coincide with full images of accessible functors. Reductive and universal classes of Σ-structures coincide with powerful images of accessible functors. Condition (2) above says that every powerful image of an accessible functor is accessible. An example of a powerful image of an accessible functors is the category of free abelian groups. Theorem 13. (Bagaria, Casacuberta, Mathias, JR 2015) Assume the existence of a proper class of supercompact cardinals. Then every Σ2 full subcategory K of Str Σ is bounded and has all

• bjects presentable.

Theorem 14. (BCMR) Assume the existence of a proper class of C(n − 2) cardinals, n ≥ 3. Then every Σn full subcategory K of Str Σ is bounded and has all objects presentable.

Let Set<ω be the category of eevntually constant sequences (Xn)n∈ω of sets where morphisms are eventually constant sequences (fn) of mappings.

Let Set<ω be the category of eevntually constant sequences (Xn)n∈ω of sets where morphisms are eventually constant sequences (fn) of mappings. Set<ω is a non-full subcategory of Setω.

Let Set<ω be the category of eevntually constant sequences (Xn)n∈ω of sets where morphisms are eventually constant sequences (fn) of mappings. Set<ω is a non-full subcategory of Setω. Theorem 14. (Paré, JR 2013) Assuming the existence of an Lω1,ω-compact cardinal, Set<ω is accessible.

Let Set<ω be the category of eevntually constant sequences (Xn)n∈ω of sets where morphisms are eventually constant sequences (fn) of mappings. Set<ω is a non-full subcategory of Setω. Theorem 14. (Paré, JR 2013) Assuming the existence of an Lω1,ω-compact cardinal, Set<ω is accessible. Question 6. Do we need set theory for this?

Let Set<ω be the category of eevntually constant sequences (Xn)n∈ω of sets where morphisms are eventually constant sequences (fn) of mappings. Set<ω is a non-full subcategory of Setω. Theorem 14. (Paré, JR 2013) Assuming the existence of an Lω1,ω-compact cardinal, Set<ω is accessible. Question 6. Do we need set theory for this? Set<ω is a colimit of locally presentable categories and functors preserving directed colimits Set

F01

− − − − − → Set2

F12

− − − − − → . . . Setn

Fnn+1

− − − − − − − → . . . Here Fnn+1(X1, . . . , Xn) = (X1, . . . , Xn, Xn) and similarly for morphisms.

Let Set<ω be the category of eevntually constant sequences (Xn)n∈ω of sets where morphisms are eventually constant sequences (fn) of mappings. Set<ω is a non-full subcategory of Setω. Theorem 14. (Paré, JR 2013) Assuming the existence of an Lω1,ω-compact cardinal, Set<ω is accessible. Question 6. Do we need set theory for this? Set<ω is a colimit of locally presentable categories and functors preserving directed colimits Set

F01

− − − − − → Set2

F12

− − − − − → . . . Setn

Fnn+1

− − − − − − − → . . . Here Fnn+1(X1, . . . , Xn) = (X1, . . . , Xn, Xn) and similarly for morphisms. Theorem 15. (PR) Assume that for every cardinal λ there is an Lλ,ω-compact cardinal. Then directed colimits of accessible categories and embeddings preserving κ-directed colimits for some κ are accessible.