Higher Structures Lisbon July 24-27, 2017 Instituto Superior T - PowerPoint PPT Presentation

� 1b. Methodology We consider functors � � H � � � Topological Algebraic Data Data B such that 1) H is homotopically defined. 2) HB is equivalent to 1. 3) The Topological Data has a notion of connected. 4) For all Algebraic Data A , B A is connected. 5) “Nice” colimits of connected Topological Data are : (a) connected, and (b) preserved by H . Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 4 / 19

� 1b. Methodology We consider functors � � H � � � Topological Algebraic Data Data B such that 1) H is homotopically defined. 2) HB is equivalent to 1. 3) The Topological Data has a notion of connected. 4) For all Algebraic Data A , B A is connected. 5) “Nice” colimits of connected Topological Data are : (a) connected, and (b) preserved by H . The last is a generalised Seifert-van Kampen Theorem. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 4 / 19

� 1b. Methodology We consider functors � � H � � � Topological Algebraic Data Data B such that 1) H is homotopically defined. 2) HB is equivalent to 1. 3) The Topological Data has a notion of connected. 4) For all Algebraic Data A , B A is connected. 5) “Nice” colimits of connected Topological Data are : (a) connected, and (b) preserved by H . The last is a generalised Seifert-van Kampen Theorem. We now recall an Excision Theorem of RB and Philip Higgins, JPAA, (1981) Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 4 / 19

2. Relative Homotopical Excision Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 5 / 19

2. Relative Homotopical Excision Excision deals with Y = X ∪ B , A = X ∩ B , and, e.g., X , B are open in Y and considers the Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 5 / 19

� � 2. Relative Homotopical Excision Excision deals with i � ( B , B ) ( A , A ) Y = X ∪ B , A = X ∩ B , and, e.g., X , B are open in Y and considers the pushout of pairs � ( Y , B ) ( X , A ) j Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 5 / 19

� � 2. Relative Homotopical Excision Excision deals with i � ( B , B ) ( A , A ) Y = X ∪ B , A = X ∩ B , and, e.g., X , B are open in Y and considers the pushout of pairs � ( Y , B ) ( X , A ) j Excision Theorem If ( X , A ) is ( n − 1)-connected, then so also is ( Y , B ) and we get a pushout of modules (crossed if n = 2) Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 5 / 19

� � � � � 2. Relative Homotopical Excision Excision deals with i � ( B , B ) ( A , A ) Y = X ∪ B , A = X ∩ B , and, e.g., X , B are open in Y and considers the pushout of pairs � ( Y , B ) ( X , A ) j Excision Theorem If ( X , A ) is ( n − 1)-connected, then so also is ( Y , B ) and we get a pushout of modules (crossed if n = 2) i ∗ (1 , π 1 ( A )) (1 , π 1 ( B )) j ∗ is “induced” by i ∗ � ( π n ( Y , B ) , π 1 ( B )) ( π n ( X , A ) , π 1 ( A )) j ∗ Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 5 / 19

� � � � � 2. Relative Homotopical Excision Excision deals with i � ( B , B ) ( A , A ) Y = X ∪ B , A = X ∩ B , and, e.g., X , B are open in Y and considers the pushout of pairs � ( Y , B ) ( X , A ) j Excision Theorem If ( X , A ) is ( n − 1)-connected, then so also is ( Y , B ) and we get a pushout of modules (crossed if n = 2) i ∗ (1 , π 1 ( A )) (1 , π 1 ( B )) j ∗ is “induced” by i ∗ � ( π n ( Y , B ) , π 1 ( B )) ( π n ( X , A ) , π 1 ( A )) j ∗ This implies the Relative Hurewicz Theorem by taking B = CA , a cone on A . Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 5 / 19

Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 6 / 19

The case n = 2, X = CA , Y = B ∪ CA and A = ∨ i S 1 i is a wedge of circles gives a 1949 Theorem of Whitehead on free crossed modules. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 6 / 19

The case n = 2, X = CA , Y = B ∪ CA and A = ∨ i S 1 i is a wedge of circles gives a 1949 Theorem of Whitehead on free crossed modules. This Excision Theorem is a special case, or application, of a Seifert-van Kampen Theorem for filtered spaces; see the book Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 6 / 19

The case n = 2, X = CA , Y = B ∪ CA and A = ∨ i S 1 i is a wedge of circles gives a 1949 Theorem of Whitehead on free crossed modules. This Excision Theorem is a special case, or application, of a Seifert-van Kampen Theorem for filtered spaces; see the book Nonabelian Algebraic Topology Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 6 / 19

The case n = 2, X = CA , Y = B ∪ CA and A = ∨ i S 1 i is a wedge of circles gives a 1949 Theorem of Whitehead on free crossed modules. This Excision Theorem is a special case, or application, of a Seifert-van Kampen Theorem for filtered spaces; see the book Nonabelian Algebraic Topology (EMS, 2011, 703 pp) which realises the methodology of the second slide to give an exposition of the border between homology and homotopy, using Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 6 / 19

The case n = 2, X = CA , Y = B ∪ CA and A = ∨ i S 1 i is a wedge of circles gives a 1949 Theorem of Whitehead on free crossed modules. This Excision Theorem is a special case, or application, of a Seifert-van Kampen Theorem for filtered spaces; see the book Nonabelian Algebraic Topology (EMS, 2011, 703 pp) which realises the methodology of the second slide to give an exposition of the border between homology and homotopy, using Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 6 / 19

The case n = 2, X = CA , Y = B ∪ CA and A = ∨ i S 1 i is a wedge of circles gives a 1949 Theorem of Whitehead on free crossed modules. This Excision Theorem is a special case, or application, of a Seifert-van Kampen Theorem for filtered spaces; see the book Nonabelian Algebraic Topology (EMS, 2011, 703 pp) which realises the methodology of the second slide to give an exposition of the border and not requiring the usual singular between homology and homotopy, homology. using Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 6 / 19

The case n = 2, X = CA , Y = B ∪ CA and A = ∨ i S 1 i is a wedge of circles gives a 1949 Theorem of Whitehead on free crossed modules. This Excision Theorem is a special case, or application, of a Seifert-van Kampen Theorem for filtered spaces; see the book Nonabelian Algebraic Topology (EMS, 2011, 703 pp) which realises the methodology of the second slide to give an exposition of the border and not requiring the usual singular between homology and homotopy, homology. using Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 6 / 19

The case n = 2, X = CA , Y = B ∪ CA and A = ∨ i S 1 i is a wedge of circles gives a 1949 Theorem of Whitehead on free crossed modules. This Excision Theorem is a special case, or application, of a Seifert-van Kampen Theorem for filtered spaces; see the book Nonabelian Algebraic Topology (EMS, 2011, 703 pp) which realises the methodology of the second slide to give an exposition of the border and not requiring the usual singular between homology and homotopy, homology. using Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 6 / 19

3. Origin of my work with Loday In November 1981 as part of a visit to France, and at the suggestion of Michel Zisman, I visited Jean-Louis Loday in Strasbourg to give a seminar on the work with Philip Higgins. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 7 / 19

3. Origin of my work with Loday In November 1981 as part of a visit to France, and at the suggestion of Michel Zisman, I visited Jean-Louis Loday in Strasbourg to give a seminar on the work with Philip Higgins. Jean-Louis got the point of the talk, and expected there could be a van Kampen Theorem for his cat n -groups functor on n -cubes of pointed spaces. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 7 / 19

3. Origin of my work with Loday In November 1981 as part of a visit to France, and at the suggestion of Michel Zisman, I visited Jean-Louis Loday in Strasbourg to give a seminar on the work with Philip Higgins. Jean-Louis got the point of the talk, and expected there could be a van Kampen Theorem for his cat n -groups functor on n -cubes of pointed spaces. He also told me of a conjecture he had. I interpreted this conjecture as a Triadic Hurewicz Theorem, Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 7 / 19

3. Origin of my work with Loday In November 1981 as part of a visit to France, and at the suggestion of Michel Zisman, I visited Jean-Louis Loday in Strasbourg to give a seminar on the work with Philip Higgins. Jean-Louis got the point of the talk, and expected there could be a van Kampen Theorem for his cat n -groups functor on n -cubes of pointed spaces. He also told me of a conjecture he had. I interpreted this conjecture as a Triadic Hurewicz Theorem, which from the RB/PJH point of view should be deduced from a Triadic Excision Theorem, Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 7 / 19

3. Origin of my work with Loday In November 1981 as part of a visit to France, and at the suggestion of Michel Zisman, I visited Jean-Louis Loday in Strasbourg to give a seminar on the work with Philip Higgins. Jean-Louis got the point of the talk, and expected there could be a van Kampen Theorem for his cat n -groups functor on n -cubes of pointed spaces. He also told me of a conjecture he had. I interpreted this conjecture as a Triadic Hurewicz Theorem, which from the RB/PJH point of view should be deduced from a Triadic Excision Theorem, which itself should be deduced from a Triadic van Kampen Theorem. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 7 / 19

3. Origin of my work with Loday In November 1981 as part of a visit to France, and at the suggestion of Michel Zisman, I visited Jean-Louis Loday in Strasbourg to give a seminar on the work with Philip Higgins. Jean-Louis got the point of the talk, and expected there could be a van Kampen Theorem for his cat n -groups functor on n -cubes of pointed spaces. He also told me of a conjecture he had. I interpreted this conjecture as a Triadic Hurewicz Theorem, which from the RB/PJH point of view should be deduced from a Triadic Excision Theorem, which itself should be deduced from a Triadic van Kampen Theorem. Jean-Louis suggested a more general theorem could be easier to prove! Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 7 / 19

� � 4. Squares of pointed spaces J-L’s methods involves in the first dimension a square of pointed spaces b � B A Z := g x � Y X f Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 8 / 19

� � 4. Squares of pointed spaces J-L’s methods involves in the first dimension a square of pointed spaces b � B A Z := g x � Y X f Now we expand this to a diagram of fibrations, Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 8 / 19

� � � � � � � � � � � 4. Squares of pointed spaces J-L’s methods involves in the first dimension a square of pointed spaces b � B F (Z) F ( x ) F ( g ) A Z := g x � Y b � B X F ( b ) A f x g Now we expand this to a diagram of fibrations, � X � Y F ( f ) f Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 8 / 19

� � � � � � � � � � � 4. Squares of pointed spaces J-L’s methods involves in the first dimension a square of pointed spaces b � B F (Z) F ( x ) F ( g ) A Z := g x � Y b � B X F ( b ) A f x g Now we expand this to a diagram of fibrations, � X � Y F ( f ) f and say Z is connected if all these spaces are connected. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 8 / 19

� � � � � � � � � � � 4. Squares of pointed spaces J-L’s methods involves in the first dimension a square of pointed spaces b � B F (Z) F ( x ) F ( g ) A Z := g x � Y b � B X F ( b ) A f x g Now we expand this to a diagram of fibrations, � X � Y F ( f ) f and say Z is connected if all these spaces are connected. Then we use π 1 to form the square of groups Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 8 / 19

� � � � � � � � � � � � � � 4. Squares of pointed spaces J-L’s methods involves in the first dimension a square of pointed spaces b � B F (Z) F ( x ) F ( g ) A Z := g x � Y b � B X F ( b ) A f x g Now we expand this to a diagram of fibrations, � X � Y F ( f ) f and say Z is connected if all these spaces are connected. Then we use π 1 to form the square of groups π 1 ( F (Z)) π 1 ( F ( x )) Π(Z) := � π 1 ( A ) π 1 ( F ( b )) Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 8 / 19

In the case X , B ⊆ Y , A = X ∩ B this is equivalent to the classical diagram Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 9 / 19

� � � In the case X , B ⊆ Y , A = X ∩ B this is equivalent to the classical diagram π 3 ( Y ; B , X ) π 2 ( X , A ) � π 1 ( A ) π 2 ( B , A ) Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 9 / 19

� � � In the case X , B ⊆ Y , A = X ∩ B this is equivalent to the classical diagram π 3 ( Y ; B , X ) π 2 ( X , A ) � π 1 ( A ) π 2 ( B , A ) With the operations of π 1 ( A ) on the other groups Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 9 / 19

� � � In the case X , B ⊆ Y , A = X ∩ B this is equivalent to the classical diagram π 3 ( Y ; B , X ) π 2 ( X , A ) � π 1 ( A ) π 2 ( B , A ) With the operations of π 1 ( A ) on the other groups and the generalised Whitehead product h : π 2 ( B , A ) × π 2 ( X , A ) → π 3 ( Y ; B , X ) Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 9 / 19

� � � In the case X , B ⊆ Y , A = X ∩ B this is equivalent to the classical diagram π 3 ( Y ; B , X ) π 2 ( X , A ) � π 1 ( A ) π 2 ( B , A ) With the operations of π 1 ( A ) on the other groups and the generalised Whitehead product h : π 2 ( B , A ) × π 2 ( X , A ) → π 3 ( Y ; B , X ) this gives a structure called a Crossed Square. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 9 / 19

� � � In the case X , B ⊆ Y , A = X ∩ B this is equivalent to the classical diagram π 3 ( Y ; B , X ) π 2 ( X , A ) � π 1 ( A ) π 2 ( B , A ) With the operations of π 1 ( A ) on the other groups and the generalised Whitehead product h : π 2 ( B , A ) × π 2 ( X , A ) → π 3 ( Y ; B , X ) this gives a structure called a Crossed Square. So we have a functor H : (Squares of pointed spaces) → (Crossed Squares) . We also say ( Y ; B , X ) is connected if the square Z of spaces on the previous slide is connected. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 9 / 19

� � � 5. Crossed Squares λ ′ L N h : M × N → L and P acts on L , M , N ν λ � P so M , N act on L , M , N M µ Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 10 / 19

� � � 5. Crossed Squares λ ′ L N h : M × N → L and P acts on L , M , N ν λ � P so M , N act on L , M , N M µ A crossed square should be thought of as a crossed module of crossed modules. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 10 / 19

� � � 5. Crossed Squares λ ′ L N h : M × N → L and P acts on L , M , N ν λ � P so M , N act on L , M , N M µ A crossed square should be thought of as a crossed module of crossed modules. Two basic rules are: h ( mm ′ , n ) = h ( m m ′ , m n ) h ( m , n ) , Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 10 / 19

� � � 5. Crossed Squares λ ′ L N h : M × N → L and P acts on L , M , N ν λ � P so M , N act on L , M , N M µ A crossed square should be thought of as a crossed module of crossed modules. Two basic rules are: h ( mm ′ , n ) = h ( m m ′ , m n ) h ( m , n ) , h ( m , nn ′ ) = h ( m , n ) h ( n m , n n ′ ) , Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 10 / 19

� � � 5. Crossed Squares λ ′ L N h : M × N → L and P acts on L , M , N ν λ � P so M , N act on L , M , N M µ A crossed square should be thought of as a crossed module of crossed modules. Two basic rules are: h ( mm ′ , n ) = h ( m m ′ , m n ) h ( m , n ) , h ( m , nn ′ ) = h ( m , n ) h ( n m , n n ′ ) , Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 10 / 19

� � � 5. Crossed Squares λ ′ L N h : M × N → L and P acts on L , M , N ν λ � P so M , N act on L , M , N M µ A crossed square should be thought of as a crossed module of crossed modules. Two basic rules are: h ( mm ′ , n ) = h ( m m ′ , m n ) h ( m , n ) , h ( m , nn ′ ) = h ( m , n ) h ( n m , n n ′ ) , h is a biderivation, cf a rule for commutators Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 10 / 19

� � � 6. Squares of squares A standard trick is that a (pushout) square of pointed spaces A B Z := � Y X Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 11 / 19

� � � 6. Squares of squares A standard trick is that a (pushout) square of pointed spaces A B Z := � Y X can be turned into Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 11 / 19

� � � � � 6. Squares of squares A standard trick is that a A A � A B (pushout) square of pointed A A A B spaces Z := � A B A A A B X X X Y Z := � Y X can be turned into Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 11 / 19

� � � � � 6. Squares of squares A standard trick is that a A A � A B (pushout) square of pointed A A A B spaces Z := � A B A A A B X X X Y Z := a (pushout) square of squares. � Y X can be turned into Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 11 / 19

� � � � � 6. Squares of squares A standard trick is that a A A � A B (pushout) square of pointed A A A B spaces Z := � A B A A A B X X X Y Z := a (pushout) square of squares. � Y If Z is a connected square so X also are all the other vertices can be turned into of this square of squares. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 11 / 19

� � � � � 6. Squares of squares A standard trick is that a A A � A B (pushout) square of pointed A A A B spaces Z := � A B A A A B X X X Y Z := a (pushout) square of squares. � Y If Z is a connected square so X also are all the other vertices can be turned into of this square of squares. Part of the van Kampen Theorem for squares gives the converse, concluding that the square Z is connected. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 11 / 19

� � � � � 6. Squares of squares A standard trick is that a A A � A B (pushout) square of pointed A A A B spaces Z := � A B A A A B X X X Y Z := a (pushout) square of squares. � Y If Z is a connected square so X also are all the other vertices can be turned into of this square of squares. Part of the van Kampen Theorem for squares gives the converse, concluding that the square Z is connected. Considering “squares of squares”, Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 11 / 19

� � � � � 6. Squares of squares A standard trick is that a A A � A B (pushout) square of pointed A A A B spaces Z := � A B A A A B X X X Y Z := a (pushout) square of squares. � Y If Z is a connected square so X also are all the other vertices can be turned into of this square of squares. Part of the van Kampen Theorem for squares gives the converse, concluding that the square Z is connected. Considering “squares of squares”, or “cubes of cubes”, is analogous to using skeleta of CW-complexes, but allows also n -cubes of r -cubes! Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 11 / 19

7. A nonabelian tensor product Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 12 / 19

7. A nonabelian tensor product By the algebraic part of the van Kampen Theorem for squares, applying Π to the square of squares Z gives a pushout of crossed squares of the form Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 12 / 19

� � � � 7. A nonabelian tensor product � � � � By the algebraic part of the 1 1 1 N van Kampen Theorem for 1 P 1 P squares, applying Π to the square of squares Z gives a pushout of crossed squares of the form � � � � 1 1 L N M P M P Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 12 / 19

� � � � 7. A nonabelian tensor product � � � � By the algebraic part of the 1 1 1 N van Kampen Theorem for 1 P 1 P squares, applying Π to the square of squares Z gives a pushout of crossed squares of the form � � � � 1 1 L N M P M P What should be L ? Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 12 / 19

� � � � 7. A nonabelian tensor product � � � � By the algebraic part of the 1 1 1 N van Kampen Theorem for 1 P 1 P squares, applying Π to the square of squares Z gives a pushout of crossed squares of the form � � � � 1 1 L N M P M P What should be L ? It has to be home for a new h : M × N → L . Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 12 / 19

� � � � 7. A nonabelian tensor product � � � � By the algebraic part of the 1 1 1 N van Kampen Theorem for 1 P 1 P squares, applying Π to the square of squares Z gives a pushout of crossed squares of the form � � � � 1 1 L N M P M P What should be L ? It has to be home for a new h : M × N → L . This turns out to be the universal biderivation, so we write it h : M × N → M ⊗ N , a nonabelian tensor product. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 12 / 19

� � � � 7. A nonabelian tensor product � � � � By the algebraic part of the 1 1 1 N van Kampen Theorem for 1 P 1 P squares, applying Π to the square of squares Z gives a pushout of crossed squares of the form � � � � 1 1 L N M P M P What should be L ? It has to be home for a new h : M × N → L . This turns out to be the universal biderivation, so we write it h : M × N → M ⊗ N , a nonabelian tensor product. This is an algebraic example Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 12 / 19

� � � � 7. A nonabelian tensor product � � � � By the algebraic part of the 1 1 1 N van Kampen Theorem for 1 P 1 P squares, applying Π to the square of squares Z gives a pushout of crossed squares of the form � � � � 1 1 L N M P M P What should be L ? It has to be home for a new h : M × N → L . This turns out to be the universal biderivation, so we write it h : M × N → M ⊗ N , a nonabelian tensor product. This is an algebraic example of identifications in dimensions < 3 producing structure in dimension 3. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 12 / 19

Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 13 / 19

Standard example: M , N ✂ P are normal subgroups of P and κ : M ⊗ N → P , m ⊗ n �→ [ m , n ] . is well defined. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 13 / 19

Standard example: M , N ✂ P are normal subgroups of P and κ : M ⊗ N → P , m ⊗ n �→ [ m , n ] . is well defined. Special case: M = N = P : the crossed square Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 13 / 19

� � � P ⊗ P P Standard example: M , N ✂ P are normal subgroups of P and 1 � P P κ : M ⊗ N → P , 1 m ⊗ n �→ [ m , n ] . is well defined. Special case: M = N = P : the crossed square Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 13 / 19

� � � P ⊗ P P Standard example: M , N ✂ P are normal subgroups of P and 1 � P P κ : M ⊗ N → P , 1 m ⊗ n �→ [ m , n ] . gives the homotopy 3-type of SK ( P , 1), is well defined. Special case: M = N = P : the crossed square Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 13 / 19

� � � P ⊗ P P Standard example: M , N ✂ P are normal subgroups of P and 1 � P P κ : M ⊗ N → P , 1 m ⊗ n �→ [ m , n ] . gives the homotopy 3-type of SK ( P , 1), allowing descriptions of is well defined. Special case: π 2 , π 3 , and Whitehead product M = N = P : π 2 × π 2 → π 3 as the crossed square ([ x ] , [ y ]) �→ ( x ⊗ y )( y ⊗ x ). Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 13 / 19

� � � P ⊗ P P Standard example: M , N ✂ P are normal subgroups of P and 1 � P P κ : M ⊗ N → P , 1 m ⊗ n �→ [ m , n ] . gives the homotopy 3-type of SK ( P , 1), allowing descriptions of is well defined. Special case: π 2 , π 3 , and Whitehead product M = N = P : π 2 × π 2 → π 3 as the crossed square ([ x ] , [ y ]) �→ ( x ⊗ y )( y ⊗ x ). Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 13 / 19

� � � P ⊗ P P Standard example: M , N ✂ P are normal subgroups of P and 1 � P P κ : M ⊗ N → P , 1 m ⊗ n �→ [ m , n ] . gives the homotopy 3-type of SK ( P , 1), allowing descriptions of is well defined. Special case: π 2 , π 3 , and Whitehead product M = N = P : π 2 × π 2 → π 3 as the crossed square ([ x ] , [ y ]) �→ ( x ⊗ y )( y ⊗ x ). So this brings the nonabelian tensor product into homotopy theory. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 13 / 19

� � � P ⊗ P P Standard example: M , N ✂ P are normal subgroups of P and 1 � P P κ : M ⊗ N → P , 1 m ⊗ n �→ [ m , n ] . gives the homotopy 3-type of SK ( P , 1), allowing descriptions of is well defined. Special case: π 2 , π 3 , and Whitehead product M = N = P : π 2 × π 2 → π 3 as the crossed square ([ x ] , [ y ]) �→ ( x ⊗ y )( y ⊗ x ). So this brings the nonabelian tensor product into homotopy theory. My web bibliography on the nonabelian tensor product www.groupoids.org.uk/nonabtens.html now has 160 entries, dating from 1952, with most interest from group theorists, because of the commutator connection, and the fun of calculating examples. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 13 / 19

8. Categorical background The forgetful functor � L � N Φ : �→ ( M → P , N → P ) M P has a left adjoint Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 14 / 19

8. Categorical background The forgetful functor � L � N Φ : �→ ( M → P , N → P ) M P has a left adjoint � M ⊗ N � N D : ( M → P , N → P ) �→ M P Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 14 / 19

8. Categorical background The forgetful functor � L � N Φ : �→ ( M → P , N → P ) M P has a left adjoint � M ⊗ N � N D : ( M → P , N → P ) �→ M P and Φ is a fibration and cofibration of categories. These aspects are relevant to homotopical excision Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 14 / 19

8. Categorical background The forgetful functor � L � N Φ : �→ ( M → P , N → P ) M P has a left adjoint � M ⊗ N � N D : ( M → P , N → P ) �→ M P and Φ is a fibration and cofibration of categories. These aspects are relevant to homotopical excision and to calculate colimits of crossed squares. Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 14 / 19

8. Categorical background The forgetful functor � L � N Φ : �→ ( M → P , N → P ) M P has a left adjoint � M ⊗ N � N D : ( M → P , N → P ) �→ M P and Φ is a fibration and cofibration of categories. These aspects are relevant to homotopical excision and to calculate colimits of crossed squares. Φ also has a right adjoint of the form Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 14 / 19

8. Categorical background The forgetful functor � L � N Φ : �→ ( M → P , N → P ) M P has a left adjoint � M ⊗ N � N D : ( M → P , N → P ) �→ M P and Φ is a fibration and cofibration of categories. These aspects are relevant to homotopical excision and to calculate colimits of crossed squares. Φ also has a right adjoint of the form � M × P N � N R : ( M → P , N → P ) �→ M P Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 14 / 19

8. Categorical background The forgetful functor � L � N Φ : �→ ( M → P , N → P ) M P has a left adjoint � M ⊗ N � N D : ( M → P , N → P ) �→ M P and Φ is a fibration and cofibration of categories. These aspects are relevant to homotopical excision and to calculate colimits of crossed squares. Φ also has a right adjoint of the form � M × P N � N R : ( M → P , N → P ) �→ M P Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 14 / 19

9. Excision for unions of 3 sets Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 15 / 19

9. Excision for unions of 3 sets Now extend Homotopical Excision from Y = X ∪ B Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 15 / 19

9. Excision for unions of 3 sets Now extend Homotopical Excision from Y = X ∪ B to the case Y = X ∪ B 1 ∪ B 2 , all open in Y . Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 15 / 19

9. Excision for unions of 3 sets Now extend Homotopical Excision from Y = X ∪ B to the case Y = X ∪ B 1 ∪ B 2 , all open in Y . We set B 0 = B 1 ∩ B 2 , A i = X ∩ B i , i = 0 , 1 , 2, Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 15 / 19

9. Excision for unions of 3 sets Now extend Homotopical Excision from Y = X ∪ B to the case Y = X ∪ B 1 ∪ B 2 , all open in Y . We set B 0 = B 1 ∩ B 2 , A i = X ∩ B i , i = 0 , 1 , 2, giving a 3-cube: Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 15 / 19

� � � � � � � � � � � � � � 9. Excision for unions of 3 sets Now extend Homotopical Excision from Y = X ∪ B to the case Y = X ∪ B 1 ∪ B 2 , all open in Y . We set B 0 = B 1 ∩ B 2 , A i = X ∩ B i , i = 0 , 1 , 2, giving a 3-cube: A 0 A 2 2 A 1 X 1 Z := 3 B 0 B 2 � Y B 1 Ronnie Brown () Higher Structures Lisbon July 24-27, 2017 Instituto Superior T´ ecnico, Lisbon Homotopical 15 / 19