Skew structures in 2-category theory and homotopy theory John - PowerPoint PPT Presentation

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Skew structures in 2-category theory and homotopy theory John Bourke Department of Mathematics and Statistics Masaryk University CT2015 John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Introduction ◮ A monoidal category C involves invertible maps α : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ), l : I ⊗ A → A and r : A → A ⊗ I . John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Introduction ◮ A monoidal category C involves invertible maps α : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ), l : I ⊗ A → A and r : A → A ⊗ I . ◮ Recently skew monoidal categories have come to attention: non-invertible maps. John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Background on skew structures Homotopical version of the Eilenberg-Kelly theorem Examples and applications of homotopical EK theorem Introduction ◮ A monoidal category C involves invertible maps α : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ), l : I ⊗ A → A and r : A → A ⊗ I . ◮ Recently skew monoidal categories have come to attention: non-invertible maps. ◮ If C is a category with notion of weak equivalence it is natural to consider intermediate case in which the maps are weak equivalences. John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Skew monoidal categories A skew monoidal category (Szlachanyi, 2012) has ◮ A functor ⊗ : C × C → C and unit object I John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Skew monoidal categories A skew monoidal category (Szlachanyi, 2012) has ◮ A functor ⊗ : C × C → C and unit object I and natural maps ◮ α : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) ◮ l : I ⊗ A → A ◮ r : A → A ⊗ I satisfying five axioms. John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Skew monoidal categories A skew monoidal category (Szlachanyi, 2012) has ◮ A functor ⊗ : C × C → C and unit object I and natural maps ◮ α : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) ◮ l : I ⊗ A → A ◮ r : A → A ⊗ I satisfying five axioms. The skew monoidal category C is called monoidal if the transformations α , l and r are invertible. John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Skew monoidal categories A skew monoidal category (Szlachanyi, 2012) has ◮ A functor ⊗ : C × C → C and unit object I and natural maps ◮ α : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) ◮ l : I ⊗ A → A ◮ r : A → A ⊗ I satisfying five axioms. The skew monoidal category C is called monoidal if the transformations α , l and r are invertible. Then three axioms are redundant ( Max Kelly ). John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Skew closed categories A skew closed category (Street, 2013) has ◮ A functor [ − , − ] : C op × C → C and unit object I John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Skew closed categories A skew closed category (Street, 2013) has ◮ A functor [ − , − ] : C op × C → C and unit object I and natural maps ◮ L : [ B , C ] → [[ A , B ] , [ A , C ]] ◮ j : I → [ A , A ] ◮ i : [ I , A ] → A satisfying five axioms. John Bourke Skew structures in 2-category theory and homotopy theory

� Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Skew closed categories A skew closed category (Street, 2013) has ◮ A functor [ − , − ] : C op × C → C and unit object I and natural maps ◮ L : [ B , C ] → [[ A , B ] , [ A , C ]] ◮ j : I → [ A , A ] ◮ i : [ I , A ] → A satisfying five axioms. The skew closed category C is called closed if ◮ i : [ I , A ] → A is invertible, ◮ the function v : C ( A , B ) → C ( I , [ A , B ]) sending f : A → B to [ A , f ] � j [ A , A ] [ A , B ] I is invertible. John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Weak maps and skewness ◮ The iso in a genuine closed category C ( A , B ) ∼ = C ( I , [ A , B ]) says that elements of [ A , B ] are just maps A → B of C . John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Weak maps and skewness ◮ The iso in a genuine closed category C ( A , B ) ∼ = C ( I , [ A , B ]) says that elements of [ A , B ] are just maps A → B of C . ◮ If we fiddle with that, we go skew. John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Weak maps and skewness ◮ The iso in a genuine closed category C ( A , B ) ∼ = C ( I , [ A , B ]) says that elements of [ A , B ] are just maps A → B of C . ◮ If we fiddle with that, we go skew. ◮ For instance, if elements of [ A , B ] are weak maps . John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Weak maps and skewness ◮ The iso in a genuine closed category C ( A , B ) ∼ = C ( I , [ A , B ]) says that elements of [ A , B ] are just maps A → B of C . ◮ If we fiddle with that, we go skew. ◮ For instance, if elements of [ A , B ] are weak maps . ◮ Sometimes forced to look at weak maps to get correct enrichment. John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Example of symmetric monoidal categories ◮ SMCat s category of symmetric monoidal categories and strict maps. John Bourke Skew structures in 2-category theory and homotopy theory

Introduction Skew monoidal categories Background on skew structures Skew closed categories Homotopical version of the Eilenberg-Kelly theorem Skew monoidal versus skew closed Examples and applications of homotopical EK theorem The Eilenberg-Kelly theorem revisited Example of symmetric monoidal categories ◮ SMCat s category of symmetric monoidal categories and strict maps. ◮ If B is symmetric monoidal then ⊗ : B 2 � B is only a pseudomap(!): ( a ⊗ b ) ⊗ ( c ⊗ d ) ∼ = ( a ⊗ c ) ⊗ ( b ⊗ d ). John Bourke Skew structures in 2-category theory and homotopy theory