The K-theory of Derivators
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1 The K-theory of Derivators Ian Coley University of California, Los Angeles 17 March 2018 Ian Coley The K-theory of Derivators 17 March / 11
2 Alexander Grothendieck, the initial mathematician 1957, Grothendieck-Riemann-Roch Theorem classifying smooth algebraic varieties X Ian Coley The K-theory of Derivators 17 March / 11
3 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 / 11
4 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, K 0 (X ) is the quotient of the free abelian group on isomorphism classes of vector bundles V on X by the relation 0 V V V 0 = [V ] = [V ] + [V ] Ian Coley The K-theory of Derivators 17 March / 11
5 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, K 0 (X ) is the quotient of the free abelian group on isomorphism classes of vector bundles V on X by the relation 0 V V V 0 = [V ] = [V ] + [V ] 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 / 11
6 Daniel Quillen and higher K n Higher Algebraic K-theory I, [Qui73] Ian Coley The K-theory of Derivators 17 March / 11
7 Daniel Quillen and higher K n 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 / 11
8 Daniel Quillen and higher K n 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 K 0, K 1, K 2 Ian Coley The K-theory of Derivators 17 March / 11
9 Daniel Quillen and higher K n 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 K 0, K 1, K 2 The Q construction, Q : ExCat SSet, and K(E) := Ω QE, K n E := π n K(E) Ian Coley The K-theory of Derivators 17 March / 11
10 Daniel Quillen and higher K n 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 K 0, K 1, K 2 The Q construction, Q : ExCat SSet, and K(E) := Ω QE, K n E := π n K(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 / 11
11 Friedhelm Waldhausen and the last good idea Algebraic K-theory of spaces, [Wal85] Ian Coley The K-theory of Derivators 17 March / 11
12 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 / 11
13 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) := Ω S C, K n C := π n K(C) Ian Coley The K-theory of Derivators 17 March / 11
14 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) := Ω S C, K n C := π n K(C) Four properties worth emphasizing: additivity, localization, approximation, agreement Ian Coley The K-theory of Derivators 17 March / 11
15 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) := Ω S C, K n C := π n K(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 / 11
16 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 / 11
17 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 D b E Ian Coley The K-theory of Derivators 17 March / 11
18 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 D b E Unfortunately, the canonical map K(E) K(D b E) on Waldhausen K-theory not a (weak) homotopy equivalence Ian Coley The K-theory of Derivators 17 March / 11
19 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 D b E Unfortunately, the canonical map K(E) K(D b E) 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 / 11
20 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 D b E Unfortunately, the canonical map K(E) K(D b E) 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 Ian Coley The K-theory of Derivators 17 March / 11
21 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 D b E Unfortunately, the canonical map K(E) K(D b E) 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! Ian Coley The K-theory of Derivators 17 March / 11
22 Triangulated Derivator K-theory Les Dérivateurs, [Gro90] Ian Coley The K-theory of Derivators 17 March / 11
23 Triangulated Derivator K-theory Les Dérivateurs, [Gro90] Among other things, an enhancement of triangulated categories by encoding higher coherent homotopy data Ian Coley The K-theory of Derivators 17 March / 11
24 Triangulated Derivator K-theory Les Dérivateurs, [Gro90] Among other things, an enhancement of triangulated categories by encoding higher coherent homotopy data Other choices: dg-categories, stable -categories (very different approach, see [BGT13]) Ian Coley The K-theory of Derivators 17 March / 11
25 Triangulated Derivator K-theory Les Dérivateurs, [Gro90] Among other things, an enhancement of triangulated categories by encoding higher coherent homotopy data Other choices: dg-categories, stable -categories (very different approach, see [BGT13]) K-theory of a triangulated derivator defined by Georges Maltsiniotis in [Mal07] Ian Coley The K-theory of Derivators 17 March / 11
26 Triangulated Derivator K-theory Les Dérivateurs, [Gro90] Among other things, an enhancement of triangulated categories by encoding higher coherent homotopy data Other choices: dg-categories, stable -categories (very different approach, see [BGT13]) K-theory of a triangulated derivator defined by Georges Maltsiniotis in [Mal07] Denis-Charles Cisinski and Amnon Neeman [CN08] prove additivity for triangulated derivator K-theory Ian Coley The K-theory of Derivators 17 March / 11
27 Triangulated Derivator K-theory Les Dérivateurs, [Gro90] Among other things, an enhancement of triangulated categories by encoding higher coherent homotopy data Other choices: dg-categories, stable -categories (very different approach, see [BGT13]) K-theory of a triangulated derivator defined by Georges Maltsiniotis in [Mal07] Denis-Charles Cisinski and Amnon Neeman [CN08] prove additivity for triangulated derivator K-theory Fernando Muro and George Raptis in [MR11] prove that localization and agreement cannot both hold for Maltsiniotis definition Ian Coley The K-theory of Derivators 17 March / 11
28 K-theory of Derivators Revisited Two K-theories for a broader class of derivators defined by Fernando Muro and George Raptis in [MR17], naïve and coherent Ian Coley The K-theory of Derivators 17 March / 11
29 K-theory of Derivators Revisited Two K-theories for a broader class of derivators defined by Fernando Muro and George Raptis in [MR17], naïve and coherent Naïve K-theory is essentially the same as Maltsiniotis, but coherent K-theory satisfies agreement with Waldhausen K-theory! Ian Coley The K-theory of Derivators 17 March / 11
30 K-theory of Derivators Revisited Two K-theories for a broader class of derivators defined by Fernando Muro and George Raptis in [MR17], naïve and coherent Naïve K-theory is essentially the same as Maltsiniotis, but coherent K-theory satisfies agreement with Waldhausen K-theory! Open question in 2014: what properties do these K-theories have? Ian Coley The K-theory of Derivators 17 March / 11
31 Results and conjectures Theorem (C ) Naïve derivator K-theory satisfies additivity and takes values in spectra. Ian Coley The K-theory of Derivators 17 March / 11
32 Results and conjectures Theorem (C ) Naïve derivator K-theory satisfies additivity and takes values in spectra. Conjecture (C., almost finished) Naïve derivator K-theory satisfies localization. Ian Coley The K-theory of Derivators 17 March / 11
33 Results and conjectures Theorem (C ) Naïve derivator K-theory satisfies additivity and takes values in spectra. Conjecture (C., almost finished) Naïve derivator K-theory satisfies localization. Conjecture (C., less confident) These results can be generalized to coherent K-theory. Ian Coley The K-theory of Derivators 17 March / 11
34 Thank you! math.ucla.edu/ iacoley/derivators Ian Coley The K-theory of Derivators 17 March / 11
35 Andrew J. Blumberg, David Gepner, and Gonçalo Tabuada. A universal characterization of higher algebraic K-theory. Geom. Topol., 17(2): , Denis-Charles Cisinski and Amnon Neeman. Additivity for derivator K-theory. Adv. Math., 217(4): , Alexandre Grothendieck. Les dérivateurs. georges.maltsiniotis/groth/derivateurs.html, Georges Maltsiniotis. La K-théorie d un dérivateur triangulé. In Categories in algebra, geometry and mathematical physics, volume 431 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, Fernando Muro and George Raptis. A note on K-theory and triangulated derivators. Adv. Math., 227(5): , Fernando Muro and George Raptis. K-theory of derivators revisited. Ann. K-Theory, 2(2): , Ian Coley The K-theory of Derivators 17 March / 11
36 Daniel Quillen. Higher algebraic K-theory. I. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages Lecture Notes in Math., Vol Springer, Berlin, Marco Schlichting. A note on K-theory and triangulated categories. Invent. Math., 150(1): , Jean-Louis Verdier. Des catégories dérivées des catégories abéliennes. Astérisque, (239):xii+253 pp. (1997), With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis. Friedhelm Waldhausen. Algebraic K-theory of spaces. In Algebraic and geometric topology (New Brunswick, N.J., 1983), volume 1126 of Lecture Notes in Math., pages Springer, Berlin, Ian Coley The K-theory of Derivators 17 March / 11
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