On the 1-type of Waldhausen K -theory . Muro 1 A. Tonks 2 F 1 - PowerPoint PPT Presentation

On the 1-type of Waldhausen K -theory . Muro 1 A. Tonks 2 F 1 Max-Planck-Institut für Mathematik, Bonn, Germany 2 London Metropolitan University, London, UK ICM Satellite Conference on K -theory and Noncommutative Geometry, Valladolid 2006 university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

Goal Understanding K 1 in the same clear way we understand K 0 . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

Waldhausen categories We use the following notation for the basic structure of a Waldhausen category W : Zero object ∗ . Weak equivalences A ∼ → A ′ . Cofiber sequences A ֌ B ։ B / A . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

Waldhausen categories We use the following notation for the basic structure of a Waldhausen category W : Zero object ∗ . Weak equivalences A ∼ → A ′ . Cofiber sequences A ֌ B ։ B / A . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

Waldhausen categories We use the following notation for the basic structure of a Waldhausen category W : Zero object ∗ . Weak equivalences A ∼ → A ′ . Cofiber sequences A ֌ B ։ B / A . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

Waldhausen categories We use the following notation for the basic structure of a Waldhausen category W : Zero object ∗ . Weak equivalences A ∼ → A ′ . Cofiber sequences A ֌ B ։ B / A . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

K 0 of a Waldhausen category The abelian group K 0 W is generated by the symbols [ A ] for any object A in W . These symbols satisfy the following relations: [ ∗ ] = 0, [ A ] = [ A ′ ] for any weak equivalence A ∼ → A ′ , [ B / A ] + [ A ] = [ B ] for any cofiber sequence A ֌ B ։ B / A . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

K 0 of a Waldhausen category The abelian group K 0 W is generated by the symbols [ A ] for any object A in W . These symbols satisfy the following relations: [ ∗ ] = 0, [ A ] = [ A ′ ] for any weak equivalence A ∼ → A ′ , [ B / A ] + [ A ] = [ B ] for any cofiber sequence A ֌ B ։ B / A . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

K 0 of a Waldhausen category The abelian group K 0 W is generated by the symbols [ A ] for any object A in W . These symbols satisfy the following relations: [ ∗ ] = 0, [ A ] = [ A ′ ] for any weak equivalence A ∼ → A ′ , [ B / A ] + [ A ] = [ B ] for any cofiber sequence A ֌ B ։ B / A . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

K 0 of a Waldhausen category The abelian group K 0 W is generated by the symbols [ A ] for any object A in W . These symbols satisfy the following relations: [ ∗ ] = 0, [ A ] = [ A ′ ] for any weak equivalence A ∼ → A ′ , [ B / A ] + [ A ] = [ B ] for any cofiber sequence A ֌ B ։ B / A . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

K -theory of a Waldhausen category The K -theory of a Waldhausen category W is a spectrum K W . The spectrum K W was defined by Waldhausen by using the S . -construction which associates a simplicial category wS . W to any Waldhausen category. A simplicial category is regarded as a bisimplicial set by taking levelwise the nerve of a category. university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

K -theory of a Waldhausen category The K -theory of a Waldhausen category W is a spectrum K W . The spectrum K W was defined by Waldhausen by using the S . -construction which associates a simplicial category wS . W to any Waldhausen category. A simplicial category is regarded as a bisimplicial set by taking levelwise the nerve of a category. university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

The nature of our algebraic model A stable quadratic module C consists of a diagram of groups �− , −� ∂ C ab 0 ⊗ C ab − → C 1 − → C 0 0 such that � a , b � = −� b , a � , ∂ � a , b � = − b − a + b + a , � ∂ c , ∂ d � = − d − c + d + c . The homotopy groups of C are π 0 C = Coker ∂ , π 1 C = Ker ∂ . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

The nature of our algebraic model A stable quadratic module C consists of a diagram of groups �− , −� ∂ C ab 0 ⊗ C ab − → C 1 − → C 0 0 such that � a , b � = −� b , a � , ∂ � a , b � = − b − a + b + a , � ∂ c , ∂ d � = − d − c + d + c . The homotopy groups of C are π 0 C = Coker ∂ , π 1 C = Ker ∂ . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

The nature of our algebraic model A stable quadratic module C consists of a diagram of groups �− , −� ∂ C ab 0 ⊗ C ab − → C 1 − → C 0 0 such that � a , b � = −� b , a � , ∂ � a , b � = − b − a + b + a , � ∂ c , ∂ d � = − d − c + d + c . The homotopy groups of C are π 0 C = Coker ∂ , π 1 C = Ker ∂ . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

The nature of our algebraic model A stable quadratic module C consists of a diagram of groups �− , −� ∂ C ab 0 ⊗ C ab − → C 1 − → C 0 0 such that � a , b � = −� b , a � , ∂ � a , b � = − b − a + b + a , � ∂ c , ∂ d � = − d − c + d + c . The homotopy groups of C are π 0 C = Coker ∂ , π 1 C = Ker ∂ . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

The nature of our algebraic model A stable quadratic module C consists of a diagram of groups �− , −� ∂ C ab 0 ⊗ C ab − → C 1 − → C 0 0 such that � a , b � = −� b , a � , ∂ � a , b � = − b − a + b + a , � ∂ c , ∂ d � = − d − c + d + c . The homotopy groups of C are π 0 C = Coker ∂ , π 1 C = Ker ∂ . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

The nature of our algebraic model A stable quadratic module C consists of a diagram of groups �− , −� ∂ C ab 0 ⊗ C ab − → C 1 − → C 0 0 such that � a , b � = −� b , a � , ∂ � a , b � = − b − a + b + a , � ∂ c , ∂ d � = − d − c + d + c . The homotopy groups of C are π 0 C = Coker ∂ , π 1 C = Ker ∂ . university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

A symmetric monoidal category A stable quadratic module C gives rise to a symmetric monoidal category smc C with object set C 0 , morphisms ( c 0 , c 1 ): c 0 → c 0 + ∂ c 1 for c 0 ∈ C 0 and c 1 ∈ C 1 . The symmetry isomorphism is defined by the bracket ∼ ( c 0 + c ′ 0 , � c 0 , c ′ 0 � ): c 0 + c ′ → c ′ − = 0 + c 0 . 0 university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

A symmetric monoidal category A stable quadratic module C gives rise to a symmetric monoidal category smc C with object set C 0 , morphisms ( c 0 , c 1 ): c 0 → c 0 + ∂ c 1 for c 0 ∈ C 0 and c 1 ∈ C 1 . The symmetry isomorphism is defined by the bracket ∼ ( c 0 + c ′ 0 , � c 0 , c ′ 0 � ): c 0 + c ′ → c ′ − = 0 + c 0 . 0 university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

A symmetric monoidal category A stable quadratic module C gives rise to a symmetric monoidal category smc C with object set C 0 , morphisms ( c 0 , c 1 ): c 0 → c 0 + ∂ c 1 for c 0 ∈ C 0 and c 1 ∈ C 1 . The symmetry isomorphism is defined by the bracket ∼ ( c 0 + c ′ 0 , � c 0 , c ′ 0 � ): c 0 + c ′ → c ′ − = 0 + c 0 . 0 university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

A symmetric monoidal category A stable quadratic module C gives rise to a symmetric monoidal category smc C with object set C 0 , morphisms ( c 0 , c 1 ): c 0 → c 0 + ∂ c 1 for c 0 ∈ C 0 and c 1 ∈ C 1 . The symmetry isomorphism is defined by the bracket ∼ ( c 0 + c ′ 0 , � c 0 , c ′ 0 � ): c 0 + c ′ → c ′ − = 0 + c 0 . 0 university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

The classifying spectrum Segal’s construction associates a classifying spectrum B M to any symmetric monoidal category M . The spectrum B smc C has homotopy groups concentrated in dimensions 0 and 1. Moreover, ∼ π 0 B smc C π 0 C , = ∼ π 1 B smc C π 1 C . = university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

The classifying spectrum Segal’s construction associates a classifying spectrum B M to any symmetric monoidal category M . The spectrum B smc C has homotopy groups concentrated in dimensions 0 and 1. Moreover, ∼ π 0 B smc C π 0 C , = ∼ π 1 B smc C π 1 C . = university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory

The classifying spectrum Segal’s construction associates a classifying spectrum B M to any symmetric monoidal category M . The spectrum B smc C has homotopy groups concentrated in dimensions 0 and 1. Moreover, ∼ π 0 B smc C π 0 C , = ∼ π 1 B smc C π 1 C . = university-logo F. Muro, A. Tonks On the 1-type of Waldhausen K -theory