The K-theory of Derivators Ian Coley University of California, Los Angeles 17 March 2018 Ian Coley The K-theory of Derivators 17 March 2018 1 / 11 Functorial assignment of vector bundles on X (an abelian category) to a `Klasse' Specifically, K0(X ) is the quotient of the free abelian group on isomorphism classes of vector bundles V on X by the relation 0 ! V 0 ! V ! V 00 ! 0 =) [V ] = [V 0] + [V 00] Quickly generalized to: same thing but for finitely-generated projective modules P on a ring R Alexander Grothendieck, the initial mathematician 1957, Grothendieck-Riemann-Roch Theorem classifying smooth algebraic varieties X Ian Coley The K-theory of Derivators 17 March 2018 2 / 11 Specifically, K0(X ) is the quotient of the free abelian group on isomorphism classes of vector bundles V on X by the relation 0 ! V 0 ! V ! V 00 ! 0 =) [V ] = [V 0] + [V 00] Quickly generalized to: same thing but for finitely-generated projective modules P on a ring R Alexander Grothendieck, the initial mathematician 1957, Grothendieck-Riemann-Roch Theorem classifying smooth algebraic varieties X Functorial assignment of vector bundles on X (an abelian category) to a `Klasse' Ian Coley The K-theory of Derivators 17 March 2018 2 / 11 Quickly generalized to: same thing but for finitely-generated projective modules P on a ring R Alexander Grothendieck, the initial mathematician 1957, Grothendieck-Riemann-Roch Theorem classifying smooth algebraic varieties X Functorial assignment of vector bundles on X (an abelian category) to a `Klasse' Specifically, K0(X ) is the quotient of the free abelian group on isomorphism classes of vector bundles V on X by the relation 0 ! V 0 ! V ! V 00 ! 0 =) [V ] = [V 0] + [V 00] Ian Coley The K-theory of Derivators 17 March 2018 2 / 11 Alexander Grothendieck, the initial mathematician 1957, Grothendieck-Riemann-Roch Theorem classifying smooth algebraic varieties X Functorial assignment of vector bundles on X (an abelian category) to a `Klasse' Specifically, K0(X ) is the quotient of the free abelian group on isomorphism classes of vector bundles V on X by the relation 0 ! V 0 ! V ! V 00 ! 0 =) [V ] = [V 0] + [V 00] Quickly generalized to: same thing but for finitely-generated projective modules P on a ring R Ian Coley The K-theory of Derivators 17 March 2018 2 / 11 Developed exact categories as a weakening of abelian categories but still suitable for K-theory `All at once' approach unifying existing ideas for K0, K1, K2 The Q construction, Q : ExCat ! SSet, and K(E) := ΩjQEj; KnE := πnK(E) New and persistent notion of encoding K-groups as homotopy groups of a space constructed combinatorially from E Daniel Quillen and higher Kn Higher Algebraic K-theory I, [Qui73] Ian Coley The K-theory of Derivators 17 March 2018 3 / 11 `All at once' approach unifying existing ideas for K0, K1, K2 The Q construction, Q : ExCat ! SSet, and K(E) := ΩjQEj; KnE := πnK(E) New and persistent notion of encoding K-groups as homotopy groups of a space constructed combinatorially from E Daniel Quillen and higher Kn Higher Algebraic K-theory I, [Qui73] Developed exact categories as a weakening of abelian categories but still suitable for K-theory Ian Coley The K-theory of Derivators 17 March 2018 3 / 11 The Q construction, Q : ExCat ! SSet, and K(E) := ΩjQEj; KnE := πnK(E) New and persistent notion of encoding K-groups as homotopy groups of a space constructed combinatorially from E Daniel Quillen and higher Kn Higher Algebraic K-theory I, [Qui73] Developed exact categories as a weakening of abelian categories but still suitable for K-theory `All at once' approach unifying existing ideas for K0, K1, K2 Ian Coley The K-theory of Derivators 17 March 2018 3 / 11 New and persistent notion of encoding K-groups as homotopy groups of a space constructed combinatorially from E Daniel Quillen and higher Kn Higher Algebraic K-theory I, [Qui73] Developed exact categories as a weakening of abelian categories but still suitable for K-theory `All at once' approach unifying existing ideas for K0, K1, K2 The Q construction, Q : ExCat ! SSet, and K(E) := ΩjQEj; KnE := πnK(E) Ian Coley The K-theory of Derivators 17 March 2018 3 / 11 Daniel Quillen and higher Kn Higher Algebraic K-theory I, [Qui73] Developed exact categories as a weakening of abelian categories but still suitable for K-theory `All at once' approach unifying existing ideas for K0, K1, K2 The Q construction, Q : ExCat ! SSet, and K(E) := ΩjQEj; KnE := πnK(E) New and persistent notion of encoding K-groups as homotopy groups of a space constructed combinatorially from E Ian Coley The K-theory of Derivators 17 March 2018 3 / 11 Developed categories with cofibrations and weak equivalences as a further weakening of exact categories The S construction, S : WaldCat ! SSet, and K(C) := ΩjS•Cj; KnC := πnK(C) Four properties worth emphasizing: additivity, localization, approximation, agreement Localization implies that K(C) is actually an infinite loop space, a.k.a. a spectrum Friedhelm Waldhausen and the last good idea Algebraic K-theory of spaces, [Wal85] Ian Coley The K-theory of Derivators 17 March 2018 4 / 11 The S construction, S : WaldCat ! SSet, and K(C) := ΩjS•Cj; KnC := πnK(C) Four properties worth emphasizing: additivity, localization, approximation, agreement Localization implies that K(C) is actually an infinite loop space, a.k.a. a spectrum Friedhelm Waldhausen and the last good idea Algebraic K-theory of spaces, [Wal85] Developed categories with cofibrations and weak equivalences as a further weakening of exact categories Ian Coley The K-theory of Derivators 17 March 2018 4 / 11 Four properties worth emphasizing: additivity, localization, approximation, agreement Localization implies that K(C) is actually an infinite loop space, a.k.a. a spectrum Friedhelm Waldhausen and the last good idea Algebraic K-theory of spaces, [Wal85] Developed categories with cofibrations and weak equivalences as a further weakening of exact categories The S construction, S : WaldCat ! SSet, and K(C) := ΩjS•Cj; KnC := πnK(C) Ian Coley The K-theory of Derivators 17 March 2018 4 / 11 Localization implies that K(C) is actually an infinite loop space, a.k.a. a spectrum Friedhelm Waldhausen and the last good idea Algebraic K-theory of spaces, [Wal85] Developed categories with cofibrations and weak equivalences as a further weakening of exact categories The S construction, S : WaldCat ! SSet, and K(C) := ΩjS•Cj; KnC := πnK(C) Four properties worth emphasizing: additivity, localization, approximation, agreement Ian Coley The K-theory of Derivators 17 March 2018 4 / 11 Friedhelm Waldhausen and the last good idea Algebraic K-theory of spaces, [Wal85] Developed categories with cofibrations and weak equivalences as a further weakening of exact categories The S construction, S : WaldCat ! SSet, and K(C) := ΩjS•Cj; KnC := πnK(C) Four properties worth emphasizing: additivity, localization, approximation, agreement Localization implies that K(C) is actually an infinite loop space, a.k.a. a spectrum Ian Coley The K-theory of Derivators 17 March 2018 4 / 11 Every exact category E gives rise to a triangulated category DbE Unfortunately, the canonical map K(E) ! K(DbE) on Waldhausen K-theory not a (weak) homotopy equivalence So can we find a good K-theory for triangulated categories that satisfies agreement? Marco Schlichting [Sch02] proves that any K-theory on triangulated categories can't satisfy both agreement and localization We need another solution! Triangulated K-theory Jean-Louis Verdier [Ver96] invents triangulated categories in his thesis (1963, published 1996) Ian Coley The K-theory of Derivators 17 March 2018 5 / 11 Unfortunately, the canonical map K(E) ! K(DbE) on Waldhausen K-theory not a (weak) homotopy equivalence So can we find a good K-theory for triangulated categories that satisfies agreement? Marco Schlichting [Sch02] proves that any K-theory on triangulated categories can't satisfy both agreement and localization We need another solution! Triangulated K-theory Jean-Louis Verdier [Ver96] invents triangulated categories in his thesis (1963, published 1996) Every exact category E gives rise to a triangulated category DbE Ian Coley The K-theory of Derivators 17 March 2018 5 / 11 So can we find a good K-theory for triangulated categories that satisfies agreement? Marco Schlichting [Sch02] proves that any K-theory on triangulated categories can't satisfy both agreement and localization We need another solution! Triangulated K-theory Jean-Louis Verdier [Ver96] invents triangulated categories in his thesis (1963, published 1996) Every exact category E gives rise to a triangulated category DbE Unfortunately, the canonical map K(E) ! K(DbE) on Waldhausen K-theory not a (weak) homotopy equivalence Ian Coley The K-theory of Derivators 17 March 2018 5 / 11 Marco Schlichting [Sch02] proves that any K-theory on triangulated categories can't satisfy both agreement and localization We need another solution! Triangulated K-theory Jean-Louis Verdier [Ver96] invents triangulated categories in his thesis (1963, published 1996) Every exact category E gives rise to a triangulated category DbE Unfortunately, the canonical map K(E) ! K(DbE) on Waldhausen K-theory not a (weak) homotopy equivalence So can we find a good K-theory for triangulated categories that satisfies agreement? Ian Coley The K-theory of Derivators 17 March 2018 5 / 11 We need another solution! Triangulated K-theory Jean-Louis Verdier [Ver96] invents triangulated categories in his thesis (1963, published 1996) Every exact category E gives rise to a triangulated category DbE Unfortunately, the canonical