� Background Grothendieck ω -groupoids Iterated injectivity From Lawvere theories to globular theories II ◮ Corresponding diagram for globular theories: Y � Θ 0 � [ G op , Set] G ◮ G is the shape category for globular sets. John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity From Lawvere theories to globular theories II ◮ Corresponding diagram for globular theories: Y � Θ 0 � [ G op , Set] G ◮ G is the shape category for globular sets. ◮ Θ 0 contains the arities for operations in ω -categories, ω -groupoids and so on. These are the globular cardinals (Street) – certain globular sets. John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity From Lawvere theories to globular theories II ◮ Corresponding diagram for globular theories: Y � Θ 0 � [ G op , Set] G ◮ G is the shape category for globular sets. ◮ Θ 0 contains the arities for operations in ω -categories, ω -groupoids and so on. These are the globular cardinals (Street) – certain globular sets. ◮ The globular cardinals are the globular sums of representables. John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity From Lawvere theories to globular theories II ◮ Corresponding diagram for globular theories: Y � Θ 0 � [ G op , Set] G ◮ G is the shape category for globular sets. ◮ Θ 0 contains the arities for operations in ω -categories, ω -groupoids and so on. These are the globular cardinals (Street) – certain globular sets. ◮ The globular cardinals are the globular sums of representables. ◮ A globular theory is an identity on objects functor J : Θ op 0 → T preserving globular products. John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity From Lawvere theories to globular theories II ◮ Corresponding diagram for globular theories: Y � Θ 0 � [ G op , Set] G ◮ G is the shape category for globular sets. ◮ Θ 0 contains the arities for operations in ω -categories, ω -groupoids and so on. These are the globular cardinals (Street) – certain globular sets. ◮ The globular cardinals are the globular sums of representables. ◮ A globular theory is an identity on objects functor J : Θ op 0 → T preserving globular products. John Bourke Grothendieck ω -groupoids as iterated injectives
� �� � � � � � Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals and tables of dimensions ◮ Examples of globular cardinals: • • • • • • John Bourke Grothendieck ω -groupoids as iterated injectives
� �� � � � � � Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals and tables of dimensions ◮ Examples of globular cardinals: • • • • • • ◮ Indexed by (1 , 0 , 1) and (1 , 0 , 2 , 1 , 2) respectively. John Bourke Grothendieck ω -groupoids as iterated injectives
� � �� � � � � Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals and tables of dimensions ◮ Examples of globular cardinals: • • • • • • ◮ Indexed by (1 , 0 , 1) and (1 , 0 , 2 , 1 , 2) respectively. ◮ A table of dimensions (t.o.d.) is a sequence n = ( n 1 , . . . , n k ) of natural numbers such that n 1 > n 2 < n 3 > n 4 < n 5 > . . . < n k and of odd length. John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � �� � � Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals and tables of dimensions ◮ Examples of globular cardinals: • • • • • • ◮ Indexed by (1 , 0 , 1) and (1 , 0 , 2 , 1 , 2) respectively. ◮ A table of dimensions (t.o.d.) is a sequence n = ( n 1 , . . . , n k ) of natural numbers such that n 1 > n 2 < n 3 > n 4 < n 5 > . . . < n k and of odd length. ◮ Index the globular cardinals: Glue the n 2 -target of a n 1 -cell to the n 2 -source of a n 3 -cell. John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � �� � � Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals and tables of dimensions ◮ Examples of globular cardinals: • • • • • • ◮ Indexed by (1 , 0 , 1) and (1 , 0 , 2 , 1 , 2) respectively. ◮ A table of dimensions (t.o.d.) is a sequence n = ( n 1 , . . . , n k ) of natural numbers such that n 1 > n 2 < n 3 > n 4 < n 5 > . . . < n k and of odd length. ◮ Index the globular cardinals: Glue the n 2 -target of a n 1 -cell to the n 2 -source of a n 3 -cell.Glue the n 4 -target of that n 3 -cell to the n 4 -source of a n 5 -cell. . . . John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � �� � � Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals and tables of dimensions ◮ Examples of globular cardinals: • • • • • • ◮ Indexed by (1 , 0 , 1) and (1 , 0 , 2 , 1 , 2) respectively. ◮ A table of dimensions (t.o.d.) is a sequence n = ( n 1 , . . . , n k ) of natural numbers such that n 1 > n 2 < n 3 > n 4 < n 5 > . . . < n k and of odd length. ◮ Index the globular cardinals: Glue the n 2 -target of a n 1 -cell to the n 2 -source of a n 3 -cell.Glue the n 4 -target of that n 3 -cell to the n 4 -source of a n 5 -cell. . . . ◮ Colimit formulation of this process is via globular sums. John Bourke Grothendieck ω -groupoids as iterated injectives
� � � Background Grothendieck ω -groupoids Iterated injectivity Globular sums and globular products ◮ The t.o.d. n = ( n 1 , . . . , n k ) determines the diagram n 2 n 4 n k − 1 ❊ ❊ ❊ τ ② ❊ σ τ ② ❊ σ τ ❊ σ . . . � ②②② ❊ ❊ ❊ ② ② ❊ ❊ ❊ ② ② � ② � ② n k − 2 n 1 n 3 n 5 n k in G . John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � � � Background Grothendieck ω -groupoids Iterated injectivity Globular sums and globular products ◮ The t.o.d. n = ( n 1 , . . . , n k ) determines the diagram n 2 n 4 n k − 1 ❊ ❊ ❊ τ ② ❊ σ τ ② ❊ σ τ ❊ σ . . . � ②②② ❊ ❊ ❊ ② ② ❊ ❊ ❊ ② ② � ② � ② n 1 n 3 n 5 n k − 2 n k in G . ◮ So given D : G → C we get a diagram D ( n 2 ) D ( n 4 ) D ( n k − 1 ) D τ � ② D σ D τ � ② D σ D τ � ② D σ ❊ ❊ ❊ ❊ ❊ . . . ❊ ② ② ② D ( n 1 ) D ( n 3 ) D ( n 5 ) D ( n k − 2 ) D ( n k ) whose colimit, denoted D ( n ), is called a D -globular sum. John Bourke Grothendieck ω -groupoids as iterated injectives
� � � Background Grothendieck ω -groupoids Iterated injectivity Globular sums and globular products ◮ The t.o.d. n = ( n 1 , . . . , n k ) determines the diagram n k − 1 n 2 n 4 ❊ ❊ ❊ τ ❊ σ τ ❊ σ τ ❊ σ ② ② � ②②② . . . ② ❊ ② ❊ ❊ ❊ ❊ ❊ ② ② � ② � ② n 1 n 3 n 5 n k − 2 n k in G . ◮ A globular object A : G op → C determines a diagram A ( n 1 ) A ( n 3 ) A ( n 5 ) A ( n k − 2 ) A ( n k ) . . . ❊ ❊ ❊ ❊ ② ❊ ② ❊ ② t � t � t � � ② � ② � ② s s s A ( n 2 ) A ( n 4 ) A ( n k − 1 ) whose limit is called a A - globular product . John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals as globular sums ◮ Consider Y : G → [ G op , Set]. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals as globular sums ◮ Consider Y : G → [ G op , Set]. ◮ Y -globular sums are exactly the globular cardinals. John Bourke Grothendieck ω -groupoids as iterated injectives
� � Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals as globular sums ◮ Consider Y : G → [ G op , Set]. ◮ Y -globular sums are exactly the globular cardinals. ◮ Y (0) Y τ Y σ ❏ ❏ t ❏ t � t Y (1) Y (1) ❏ ❏ ❏ t t � t Y (1 , 0 , 1) So Y (1 , 0 , 1) is ( • → • → • ) as expected. John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � � � � � �� Background Grothendieck ω -groupoids Iterated injectivity Globular cardinals as globular sums ◮ Consider Y : G → [ G op , Set]. ◮ Y -globular sums are exactly the globular cardinals. ◮ Y (0) Y (1) Y τ Y σ Y τ Y σ ❏ ❏ ❏ ❏ t ❏ t ❏ t t � t � t Y (1) Y (2) Y (2) ❚ ❚ � ❥❥❥❥❥❥ ❚ ❚ ❚ ❚ Y (1 , 0 , 2 , 1 , 2) So = • Y (1 , 0 , 2 , 1 , 2) • • John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity The category Θ 0 and globular theories ◮ Reconsider the diagram: Y D � Θ 0 � [ G op , Set] G John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity The category Θ 0 and globular theories ◮ Reconsider the diagram: Y D � Θ 0 � [ G op , Set] G ◮ Θ 0 skeletal full subcategory of [ G op , Set] containing the globular cardinals ( aka Y-globular sums ). John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity The category Θ 0 and globular theories ◮ Reconsider the diagram: Y D � Θ 0 � [ G op , Set] G ◮ Θ 0 skeletal full subcategory of [ G op , Set] containing the globular cardinals ( aka Y-globular sums ). ◮ For canonical representatives, set the objects of Θ 0 to be the t.o.d’s and Θ 0 ( n , m ) = [ G op , Set]( Y ( n ) , Y ( m )). John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity The category Θ 0 and globular theories ◮ Reconsider the diagram: Y D � Θ 0 � [ G op , Set] G ◮ Θ 0 skeletal full subcategory of [ G op , Set] containing the globular cardinals ( aka Y-globular sums ). ◮ For canonical representatives, set the objects of Θ 0 to be the t.o.d’s and Θ 0 ( n , m ) = [ G op , Set]( Y ( n ) , Y ( m )). ◮ D : G → Θ 0 sends n to the t.o.d.( n ) of length 1. John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity The category Θ 0 and globular theories ◮ Reconsider the diagram: Y D � Θ 0 � [ G op , Set] G ◮ Θ 0 skeletal full subcategory of [ G op , Set] containing the globular cardinals ( aka Y-globular sums ). ◮ For canonical representatives, set the objects of Θ 0 to be the t.o.d’s and Θ 0 ( n , m ) = [ G op , Set]( Y ( n ) , Y ( m )). ◮ D : G → Θ 0 sends n to the t.o.d.( n ) of length 1. ◮ So Θ 0 consists of the D -globular sums John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity The category Θ 0 and globular theories ◮ Reconsider the diagram: Y D � Θ 0 � [ G op , Set] G ◮ Θ 0 skeletal full subcategory of [ G op , Set] containing the globular cardinals ( aka Y-globular sums ). ◮ For canonical representatives, set the objects of Θ 0 to be the t.o.d’s and Θ 0 ( n , m ) = [ G op , Set]( Y ( n ) , Y ( m )). ◮ D : G → Θ 0 sends n to the t.o.d.( n ) of length 1. ◮ So Θ 0 consists of the D -globular sums and Θ op consists of 0 the D op -globular products. John Bourke Grothendieck ω -groupoids as iterated injectives
� Background Grothendieck ω -groupoids Iterated injectivity The category Θ 0 and globular theories ◮ Reconsider the diagram: Y D � Θ 0 � [ G op , Set] G ◮ Θ 0 skeletal full subcategory of [ G op , Set] containing the globular cardinals ( aka Y-globular sums ). ◮ For canonical representatives, set the objects of Θ 0 to be the t.o.d’s and Θ 0 ( n , m ) = [ G op , Set]( Y ( n ) , Y ( m )). ◮ D : G → Θ 0 sends n to the t.o.d.( n ) of length 1. ◮ So Θ 0 consists of the D -globular sums and Θ op consists of 0 the D op -globular products. ◮ A globular theory consists of an identity on objects functor J : Θ op 0 → T preserving globular products. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Globular theories ◮ A globular theory consists of an identity on objects functor J : Θ op 0 → T preserving globular products. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Globular theories ◮ A globular theory consists of an identity on objects functor J : Θ op 0 → T preserving globular products. ◮ The category Mod ( T , C ) of T -models in C is the full subcategory of [ T , C ] containing those functors preserving globular products. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Globular theories ◮ A globular theory consists of an identity on objects functor J : Θ op 0 → T preserving globular products. ◮ The category Mod ( T , C ) of T -models in C is the full subcategory of [ T , C ] containing those functors preserving globular products. ◮ Write Mod ( T ) for Mod ( T , Set). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Globular theories ◮ A globular theory consists of an identity on objects functor J : Θ op 0 → T preserving globular products. ◮ The category Mod ( T , C ) of T -models in C is the full subcategory of [ T , C ] containing those functors preserving globular products. ◮ Write Mod ( T ) for Mod ( T , Set). ◮ Forgetful functor Mod ( T ) → [ G op , Set] obtained by restricting along the inclusion G op → Θ op 0 → T . John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Globular theories ◮ A globular theory consists of an identity on objects functor J : Θ op 0 → T preserving globular products. ◮ The category Mod ( T , C ) of T -models in C is the full subcategory of [ T , C ] containing those functors preserving globular products. ◮ Write Mod ( T ) for Mod ( T , Set). ◮ Forgetful functor Mod ( T ) → [ G op , Set] obtained by restricting along the inclusion G op → Θ op 0 → T . ◮ Monadic (Ara). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Contractible theories ◮ Let A : G op → C . A parallel pair of n-cells in A is a pair f , g : X ⇒ A ( n ) such that either n = 0 or s ◦ f = s ◦ g and t ◦ f = t ◦ g . John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � Background Grothendieck ω -groupoids Iterated injectivity Contractible theories ◮ Let A : G op → C . A parallel pair of n-cells in A is a pair f , g : X ⇒ A ( n ) such that either n = 0 or s ◦ f = s ◦ g and t ◦ f = t ◦ g . ◮ A lifting is an arrow h : X → A ( n + 1) such that A ( n + 1) ✇ ✇ h ✇ ✇ s t ✇ ✇ ✇ ✇ f ✇ X � A ( n ) g commutes. John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � Background Grothendieck ω -groupoids Iterated injectivity Contractible theories ◮ Let A : G op → C . A parallel pair of n-cells in A is a pair f , g : X ⇒ A ( n ) such that either n = 0 or s ◦ f = s ◦ g and t ◦ f = t ◦ g . ◮ A lifting is an arrow h : X → A ( n + 1) such that A ( n + 1) ✇ ✇ h ✇ ✇ s t ✇ ✇ ✇ ✇ f ✇ X � A ( n ) g commutes. ◮ A is said to be contractible if each parallel pair has a lifting. John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � Background Grothendieck ω -groupoids Iterated injectivity Contractible theories ◮ Let A : G op → C . A parallel pair of n-cells in A is a pair f , g : X ⇒ A ( n ) such that either n = 0 or s ◦ f = s ◦ g and t ◦ f = t ◦ g . ◮ A lifting is an arrow h : X → A ( n + 1) such that A ( n + 1) ✇ ✇ h ✇ ✇ s t ✇ ✇ ✇ ✇ f ✇ X � A ( n ) g commutes. ◮ A is said to be contractible if each parallel pair has a lifting. ◮ J : Θ op 0 → T is said to be contractible if its underlying globular object J ◦ D op : G op → T is contractible. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck ω -groupoids ◮ A Grothendieck ω -groupoid is an algebra for some contractible globular theory T . John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck ω -groupoids ◮ A Grothendieck ω -groupoid is an algebra for some contractible globular theory T . ◮ Where are the operations for a ω -groupoid in such a T ? John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck ω -groupoids ◮ A Grothendieck ω -groupoid is an algebra for some contractible globular theory T . ◮ Where are the operations for a ω -groupoid in such a T ? ◮ A function composing 1-cells in a globular set A will be a suitable function A ( m ) : A (1 , 0 , 1) → A (1). John Bourke Grothendieck ω -groupoids as iterated injectives
� � � Background Grothendieck ω -groupoids Iterated injectivity Grothendieck ω -groupoids ◮ A Grothendieck ω -groupoid is an algebra for some contractible globular theory T . ◮ Where are the operations for a ω -groupoid in such a T ? ◮ A function composing 1-cells in a globular set A will be a suitable function A ( m ) : A (1 , 0 , 1) → A (1). q (1 , 0 , 1) (1) p s t � (0) (1) John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � � � Background Grothendieck ω -groupoids Iterated injectivity Grothendieck ω -groupoids ◮ A Grothendieck ω -groupoid is an algebra for some contractible globular theory T . ◮ Where are the operations for a ω -groupoid in such a T ? ◮ A function composing 1-cells in a globular set A will be a suitable function A ( m ) : A (1 , 0 , 1) → A (1). q (1 , 0 , 1) (1) (1) p s s t s ◦ p t � (0) (1) (1 , 0 , 1) � (0) t ◦ q John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � � � � � Background Grothendieck ω -groupoids Iterated injectivity Grothendieck ω -groupoids ◮ A Grothendieck ω -groupoid is an algebra for some contractible globular theory T . ◮ Where are the operations for a ω -groupoid in such a T ? ◮ A function composing 1-cells in a globular set A will be a suitable function A ( m ) : A (1 , 0 , 1) → A (1). q (1 , 0 , 1) (1) (1) ✇ ✇ m ✇ ✇ p s s ✇ t ✇ ✇ ✇ s ◦ p ✇ t � (0) (1) (1 , 0 , 1) (0) t ◦ q John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity ◮ What about the associator? John Bourke Grothendieck ω -groupoids as iterated injectives
� � � Background Grothendieck ω -groupoids Iterated injectivity ◮ What about the associator? (2) s t m ◦ ( m , 1) (1 , 0 , 1 , 0 , 1) � (1) m ◦ (1 , m ) John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � Background Grothendieck ω -groupoids Iterated injectivity ◮ What about the associator? (2) ♠ ♠ ♠ ♠ ♠ a ♠ ♠ ♠ s ♠ t ♠ ♠ ♠ ♠ ♠ m ◦ ( m , 1) ♠ (1 , 0 , 1 , 0 , 1) � (1) m ◦ (1 , m ) John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � Background Grothendieck ω -groupoids Iterated injectivity ◮ What about the associator? (2) ♠ ♠ ♠ ♠ ♠ a ♠ ♠ ♠ s t ♠ ♠ ♠ ♠ ♠ ♠ m ◦ ( m , 1) ♠ (1 , 0 , 1 , 0 , 1) � (1) m ◦ (1 , m ) ◮ Equivalence inverses for 1-cells? John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � � � � Background Grothendieck ω -groupoids Iterated injectivity ◮ What about the associator? (2) ♠ ♠ ♠ ♠ ♠ a ♠ ♠ ♠ s ♠ t ♠ ♠ ♠ ♠ ♠ m ◦ ( m , 1) ♠ (1 , 0 , 1 , 0 , 1) � (1) m ◦ (1 , m ) ◮ Equivalence inverses for 1-cells? (1) s t t (1) � (0) s John Bourke Grothendieck ω -groupoids as iterated injectives
� � � � � � � � Background Grothendieck ω -groupoids Iterated injectivity ◮ What about the associator? (2) ♠ ♠ ♠ ♠ ♠ a ♠ ♠ ♠ s ♠ t ♠ ♠ ♠ ♠ ♠ m ◦ ( m , 1) ♠ (1 , 0 , 1 , 0 , 1) � (1) m ◦ (1 , m ) ◮ Equivalence inverses for 1-cells? (1) q q q q inv q q s t q q q q q q t q (1) � (0) s John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck weak ω -groupoids ◮ I gave slight oversimplification: Grothendieck defined specifically theories for weak ω -groupoids , but there is a contractible theory ˜ Θ for strict ω -groupoids (Ara). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck weak ω -groupoids ◮ I gave slight oversimplification: Grothendieck defined specifically theories for weak ω -groupoids , but there is a contractible theory ˜ Θ for strict ω -groupoids (Ara). ◮ Grothendieck: weak ω -groupoid = model of cellular contractible theory (or coherator ). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck weak ω -groupoids ◮ I gave slight oversimplification: Grothendieck defined specifically theories for weak ω -groupoids , but there is a contractible theory ˜ Θ for strict ω -groupoids (Ara). ◮ Grothendieck: weak ω -groupoid = model of cellular contractible theory (or coherator ). ◮ Note: if T coherator and S contractible there exists T → S and so Mod ( S ) → Mod ( T ). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck weak ω -groupoids ◮ I gave slight oversimplification: Grothendieck defined specifically theories for weak ω -groupoids , but there is a contractible theory ˜ Θ for strict ω -groupoids (Ara). ◮ Grothendieck: weak ω -groupoid = model of cellular contractible theory (or coherator ). ◮ Note: if T coherator and S contractible there exists T → S and so Mod ( S ) → Mod ( T ). So Grothendieck ω -groupoids are weak ω -groupoids – no contradiction with earlier definition. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck weak ω -groupoids ◮ I gave slight oversimplification: Grothendieck defined specifically theories for weak ω -groupoids , but there is a contractible theory ˜ Θ for strict ω -groupoids (Ara). ◮ Grothendieck: weak ω -groupoid = model of cellular contractible theory (or coherator ). ◮ Note: if T coherator and S contractible there exists T → S and so Mod ( S ) → Mod ( T ). So Grothendieck ω -groupoids are weak ω -groupoids – no contradiction with earlier definition. ◮ Before definition need to understand the category GlobTh of globular theories – this is just full subcategory of Θ 0 / Cat. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck weak ω -groupoids ◮ I gave slight oversimplification: Grothendieck defined specifically theories for weak ω -groupoids , but there is a contractible theory ˜ Θ for strict ω -groupoids (Ara). ◮ Grothendieck: weak ω -groupoid = model of cellular contractible theory (or coherator ). ◮ Note: if T coherator and S contractible there exists T → S and so Mod ( S ) → Mod ( T ). So Grothendieck ω -groupoids are weak ω -groupoids – no contradiction with earlier definition. ◮ Before definition need to understand the category GlobTh of globular theories – this is just full subcategory of Θ 0 / Cat. ◮ Note: GlobTh is locally finitely presentable. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Grothendieck weak ω -groupoids ◮ I gave slight oversimplification: Grothendieck defined specifically theories for weak ω -groupoids , but there is a contractible theory ˜ Θ for strict ω -groupoids (Ara). ◮ Grothendieck: weak ω -groupoid = model of cellular contractible theory (or coherator ). ◮ Note: if T coherator and S contractible there exists T → S and so Mod ( S ) → Mod ( T ). So Grothendieck ω -groupoids are weak ω -groupoids – no contradiction with earlier definition. ◮ Before definition need to understand the category GlobTh of globular theories – this is just full subcategory of Θ 0 / Cat. ◮ Note: GlobTh is locally finitely presentable. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , together with sets J n of parallel pairs in T n , John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , together with sets J n of parallel pairs in T n , such that T n +1 is obtained from T n by freely adjoining a lifting for each parallel pair in J n . John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , together with sets J n of parallel pairs in T n , such that T n +1 is obtained from T n by freely adjoining a lifting for each parallel pair in J n . ◮ Call T cellular if it admits a cellular presentation. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , together with sets J n of parallel pairs in T n , such that T n +1 is obtained from T n by freely adjoining a lifting for each parallel pair in J n . ◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω -groupoid is a model for a cellular contractible theory (a coherator ). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , together with sets J n of parallel pairs in T n , such that T n +1 is obtained from T n by freely adjoining a lifting for each parallel pair in J n . ◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω -groupoid is a model for a cellular contractible theory (a coherator ). ◮ Canonical coherator : J n = all parallel pairs in T n . John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , together with sets J n of parallel pairs in T n , such that T n +1 is obtained from T n by freely adjoining a lifting for each parallel pair in J n . ◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω -groupoid is a model for a cellular contractible theory (a coherator ). ◮ Canonical coherator : J n = all parallel pairs in T n . In T 0 we have the parallel pair s ◦ p , t ◦ q : (1 , 0 , 1) ⇒ (0). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , together with sets J n of parallel pairs in T n , such that T n +1 is obtained from T n by freely adjoining a lifting for each parallel pair in J n . ◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω -groupoid is a model for a cellular contractible theory (a coherator ). ◮ Canonical coherator : J n = all parallel pairs in T n . In T 0 we have the parallel pair s ◦ p , t ◦ q : (1 , 0 , 1) ⇒ (0). So composition m : (1 , 0 , 1) → (1) appears in T 1 . John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , together with sets J n of parallel pairs in T n , such that T n +1 is obtained from T n by freely adjoining a lifting for each parallel pair in J n . ◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω -groupoid is a model for a cellular contractible theory (a coherator ). ◮ Canonical coherator : J n = all parallel pairs in T n . In T 0 we have the parallel pair s ◦ p , t ◦ q : (1 , 0 , 1) ⇒ (0). So composition m : (1 , 0 , 1) → (1) appears in T 1 . Associator appears in T 2 . . . John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity Cellularity and weak ω -groupoids ◮ A cellular presentation of a globular theory T is a chain of globular theories Θ 0 = T 0 → T 1 → . . . → T n . . . → T with colimit T , together with sets J n of parallel pairs in T n , such that T n +1 is obtained from T n by freely adjoining a lifting for each parallel pair in J n . ◮ Call T cellular if it admits a cellular presentation. ◮ A Grothendieck weak ω -groupoid is a model for a cellular contractible theory (a coherator ). ◮ Canonical coherator : J n = all parallel pairs in T n . In T 0 we have the parallel pair s ◦ p , t ◦ q : (1 , 0 , 1) ⇒ (0). So composition m : (1 , 0 , 1) → (1) appears in T 1 . Associator appears in T 2 . . . ◮ Only add liftings – no equalities. Cellularity = weakness. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity What has been done? ◮ Consider D : G → Top sending n to n -disk. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity What has been done? ◮ Consider D : G → Top sending n to n -disk. Induces globular theory T D with T D ( n , m ) = Top ( D ( m ) , D ( n )). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity What has been done? ◮ Consider D : G → Top sending n to n -disk. Induces globular theory T D with T D ( n , m ) = Top ( D ( m ) , D ( n )). T D is contractible since each globular sum D ( n ) is a contractible space. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity What has been done? ◮ Consider D : G → Top sending n to n -disk. Induces globular theory T D with T D ( n , m ) = Top ( D ( m ) , D ( n )). T D is contractible since each globular sum D ( n ) is a contractible space. Kan construction gives fundamental ω -groupoid functor ˜ D : Top → Mod ( T D ) where ˜ DX ( n ) = Top ( D ( n ) , X ). (Grothendieck 1983, Maltsiniotis 2010). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity What has been done? ◮ Consider D : G → Top sending n to n -disk. Induces globular theory T D with T D ( n , m ) = Top ( D ( m ) , D ( n )). T D is contractible since each globular sum D ( n ) is a contractible space. Kan construction gives fundamental ω -groupoid functor ˜ D : Top → Mod ( T D ) where ˜ DX ( n ) = Top ( D ( n ) , X ). (Grothendieck 1983, Maltsiniotis 2010).Very simple! John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity What has been done? ◮ Consider D : G → Top sending n to n -disk. Induces globular theory T D with T D ( n , m ) = Top ( D ( m ) , D ( n )). T D is contractible since each globular sum D ( n ) is a contractible space. Kan construction gives fundamental ω -groupoid functor ˜ D : Top → Mod ( T D ) where ˜ DX ( n ) = Top ( D ( n ) , X ). (Grothendieck 1983, Maltsiniotis 2010).Very simple! ◮ Each type in dependent type theory with identity types gives rise to a Batanin ω -groupoid (van den Berg–Garner, LeFanu-Lumsdaine around 2009/2010). More complex than above. Easily done using Grothendieck ω -groupoids (Bourke 2016). John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity What has been done? ◮ Consider D : G → Top sending n to n -disk. Induces globular theory T D with T D ( n , m ) = Top ( D ( m ) , D ( n )). T D is contractible since each globular sum D ( n ) is a contractible space. Kan construction gives fundamental ω -groupoid functor ˜ D : Top → Mod ( T D ) where ˜ DX ( n ) = Top ( D ( n ) , X ). (Grothendieck 1983, Maltsiniotis 2010).Very simple! ◮ Each type in dependent type theory with identity types gives rise to a Batanin ω -groupoid (van den Berg–Garner, LeFanu-Lumsdaine around 2009/2010). More complex than above. Easily done using Grothendieck ω -groupoids (Bourke 2016). ◮ Dimitri Ara’s thesis (2010) contains many things. It showed that Batanin’s globular operads are the same as augmented homogenous globular theories. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity What has been done II? ◮ Guillaume Brunerie (2013) constructed a dependent type theory with identity types capturing ω -groupoids. Natural counterpart of Grothendieck ω -groupoids. Precise relationship expected to be written up soon. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck ω -groupoids Iterated injectivity What has been done II? ◮ Guillaume Brunerie (2013) constructed a dependent type theory with identity types capturing ω -groupoids. Natural counterpart of Grothendieck ω -groupoids. Precise relationship expected to be written up soon. ◮ Recent thesis of Remy Tuyeras relating to model structures and homotopy hypothesis. Last talk today! John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck weak ω -groupoids and iterated injectives Grothendieck ω -groupoids Properties of categories of iterated injectives Iterated injectivity Algebraic injectivity ◮ Goal now: reformulate Grothendieck weak ω -groupoids as iterated algebraic injectives . John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck weak ω -groupoids and iterated injectives Grothendieck ω -groupoids Properties of categories of iterated injectives Iterated injectivity Algebraic injectivity ◮ Goal now: reformulate Grothendieck weak ω -groupoids as iterated algebraic injectives . ◮ Algebraic injectivity first arose from Garner’s study of algebraic weak factorisation systems. John Bourke Grothendieck ω -groupoids as iterated injectives
Background Grothendieck weak ω -groupoids and iterated injectives Grothendieck ω -groupoids Properties of categories of iterated injectives Iterated injectivity Algebraic injectivity ◮ Goal now: reformulate Grothendieck weak ω -groupoids as iterated algebraic injectives . ◮ Algebraic injectivity first arose from Garner’s study of algebraic weak factorisation systems. ◮ We work in context of locally finitely presentable categories: C l.f.p. and J a set of morphisms between f.p. objects. John Bourke Grothendieck ω -groupoids as iterated injectives
� � Background Grothendieck weak ω -groupoids and iterated injectives Grothendieck ω -groupoids Properties of categories of iterated injectives Iterated injectivity Algebraic injectivity ◮ Goal now: reformulate Grothendieck weak ω -groupoids as iterated algebraic injectives . ◮ Algebraic injectivity first arose from Garner’s study of algebraic weak factorisation systems. ◮ We work in context of locally finitely presentable categories: C l.f.p. and J a set of morphisms between f.p. objects. ◮ Objects of Inj ( J ) are pairs ( X , x ) where X ∈ C and x is a lifting function f � X i ✁ ✁ ✁ α ∈ J ✁ ✁ ✁ x ( α, f ) ✁ ✁ j John Bourke Grothendieck ω -groupoids as iterated injectives
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