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) := Ω|QE|, 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) := Ω|QE|, 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) := Ω|QE|, 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) := Ω|QE|, 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) := Ω|QE|, 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) := Ω|S•C|, 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) := Ω|S•C|, 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) := Ω|S•C|, 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) := Ω|S•C|, 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) := Ω|S•C|, 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 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
Ian Coley The K-theory of Derivators 17 March 2018 5 / 11 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? 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 2018 5 / 11 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
Triangulated Derivator K-theory
Les D´erivateurs, [Gro90]
Ian Coley The K-theory of Derivators 17 March 2018 6 / 11 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
Triangulated Derivator K-theory
Les D´erivateurs, [Gro90] Among other things, an enhancement of triangulated categories by encoding higher coherent homotopy data
Ian Coley The K-theory of Derivators 17 March 2018 6 / 11 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
Triangulated Derivator K-theory
Les D´erivateurs, [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 2018 6 / 11 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
Triangulated Derivator K-theory
Les D´erivateurs, [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 2018 6 / 11 Fernando Muro and George Raptis in [MR11] prove that localization and agreement cannot both hold for Maltsiniotis’ definition
Triangulated Derivator K-theory
Les D´erivateurs, [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 2018 6 / 11 Triangulated Derivator K-theory
Les D´erivateurs, [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 2018 6 / 11 Na¨ıveK-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?
K-theory of Derivators Revisited
Two K-theories for a broader class of derivators defined by Fernando Muro and George Raptis in [MR17], na¨ıve and coherent
Ian Coley The K-theory of Derivators 17 March 2018 7 / 11 Open question in 2014: what properties do these K-theories have?
K-theory of Derivators Revisited
Two K-theories for a broader class of derivators defined by Fernando Muro and George Raptis in [MR17], na¨ıve and coherent Na¨ıveK-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 2018 7 / 11 K-theory of Derivators Revisited
Two K-theories for a broader class of derivators defined by Fernando Muro and George Raptis in [MR17], na¨ıve and coherent Na¨ıveK-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 2018 7 / 11 Conjecture (C., almost finished) Na¨ıvederivator K-theory satisfies localization.
Conjecture (C., less confident) These results can be generalized to coherent K-theory.
Results and conjectures
Theorem (C. 2015-7) Na¨ıvederivator K-theory satisfies additivity and takes values in spectra.
Ian Coley The K-theory of Derivators 17 March 2018 8 / 11 Conjecture (C., less confident) These results can be generalized to coherent K-theory.
Results and conjectures
Theorem (C. 2015-7) Na¨ıvederivator K-theory satisfies additivity and takes values in spectra.
Conjecture (C., almost finished) Na¨ıvederivator K-theory satisfies localization.
Ian Coley The K-theory of Derivators 17 March 2018 8 / 11 Results and conjectures
Theorem (C. 2015-7) Na¨ıvederivator K-theory satisfies additivity and takes values in spectra.
Conjecture (C., almost finished) Na¨ıvederivator 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 2018 8 / 11 Thank you!
math.ucla.edu/∼iacoley/derivators
Ian Coley The K-theory of Derivators 17 March 2018 9 / 11 Andrew J. Blumberg, David Gepner, and Gon¸caloTabuada. A universal characterization of higher algebraic K-theory. Geom. Topol., 17(2):733–838, 2013. Denis-Charles Cisinski and Amnon Neeman. Additivity for derivator K-theory. Adv. Math., 217(4):1381–1475, 2008. Alexandre Grothendieck. Les d´erivateurs. https://webusers.imj-prg.fr/ georges.maltsiniotis/groth/Derivateurs.html, 1990. Georges Maltsiniotis. La K-th´eorie d’un d´erivateur triangul´e. In Categories in algebra, geometry and mathematical physics, volume 431 of Contemp. Math., pages 341–368. Amer. Math. Soc., Providence, RI, 2007. Fernando Muro and George Raptis. A note on K-theory and triangulated derivators. Adv. Math., 227(5):1827–1845, 2011. Fernando Muro and George Raptis. K-theory of derivators revisited. Ann. K-Theory, 2(2):303–340, 2017.
Ian Coley The K-theory of Derivators 17 March 2018 10 / 11 Daniel Quillen. Higher algebraic K-theory. I. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 85–147. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973. Marco Schlichting. A note on K-theory and triangulated categories. Invent. Math., 150(1):111–116, 2002. Jean-Louis Verdier. Des cat´egories d´eriv´eesdes cat´egories ab´eliennes. Ast´erisque, (239):xii+253 pp. (1997), 1996. 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 318–419. Springer, Berlin, 1985.
Ian Coley The K-theory of Derivators 17 March 2018 11 / 11