Les Dérivateurs (The Derivators)

Text of Alexander Grothendieck

Edited par M. Künzer, J. Malgoire, G. Maltsiniotis

This electronic publication of the "Dérivateurs" would have never appeared without the tremendous task of deciphering and typing accomplished by Matthias Künzer and his precious help for the editing. I would like to express my gratitude to him. I would like to thank Jean Malgoire for giving me a copy of the manuscript that Grothendieck gave him in the middle of the nineties and for spending countless hours with me to compare the original with the TeXed version.

The transcription of the manuscript is as faithful as possible. For some minor corrections, or comments of the editors, as well as for the original numbering of the pages, a typewriter font is used within brackets. A question mark in brackets means that we are not sure about the preceding word. The table of contents was established by the editors. Links should point to the (French) main page of derivators, not directly to the ps or pdf files, whose names will change in successive versions. For any remark, comment or correction, please write to

    Georges Maltsiniotis

TABLE OF CONTENTS: ps, pdf.

Chapter I: Généralités sur les (pré)dérivateurs ps, pdf.

Chapter II: Cofinalité (à droite et à gauche) (préliminaire à la cofinalité cohomologique) ps, pdf.

Chapter III: Hom externes dans les dérivateurs ps, pdf.

Chapter IV: Diagrammes substantiels ps, pdf.

Chapter V: Catégories de chemins et localisation ps, pdf.

Chapter VI: HOT ps, pdf.

Chapter VII: Catégories de chemins (2) ps, pdf.

Chapter VIII: 1-types d'homotopie relatifs: leur intégration ... ps, pdf.

Chapter IX: Retour sur les catégories de fractions MW ^-1, comparaison avec Quillen ps, pdf.

Chapter X: Comparaison de Cat avec ^ ps, pdf.

Chapter XI: Hot-fibrations, foncteurs propres et foncteurs lisses etc. (dans Cat) ps, pdf.

Chapter XII: Caractérisation de W_∞ . Foncteurs W-propres, foncteurs W-lisses etc. Sommes amalgamées et carrés W-cocartésiens dans Cat
Sections 1-4: ps, pdf.
Section 5: ps, pdf.
Section 6: ps, pdf.
Appendix: ps, pdf.

Chapter XIII: Catégories de modèles (1) ps, pdf.

Chapter XIV: Carrés h-cartésiens et h-cocartésiens ps, pdf.

Chapter XV: Théorèmes de factorisation. (Modèles (2)) ps, pdf.

Chapitre XVI : Localiseurs fondamentaux dans Cat ps, pdf.

Chapitre XVII : Catégories à fibrations et à cofibrations. (Modèles (3)) ps, pdf.

Chapitre XVIII : Catégories et ensembles accessibles
Sections 1-5 : ps, pdf.
Sections 6-17 and Appendix coming soon.

Chapitre XIX : Modèles (4) ps, pdf.

Some texts related to derivators

Precursory texts:

D. W. Anderson, "Fibrations and Geometric Realizations", Bull. Amer. Math. Soc., Vol. 84, no 5, pp. 765-788 (1978).

D. W. Anderson, "Axiomatic Homotopy Theory", dans Algebraic Topology Waterloo 1978, Lecture Notes in Mathematics 741, pp.520-547 (1979).

J. Franke, "Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence", preprint, (1996).

A. Heller, "Homotopy theories", Memoirs of the American Mathematical Society, Vol. 71, No 383 (1988).

A. Heller, "Stable homotopy theories and stabilization", J. Pure Appl. Algebra, 115, pp. 113-130, (1997).

A. Heller, "Homological algebra and (semi)stable homotopy", J. Pure Appl. Algebra, 115, pp. 131-139, (1997).

A. Heller, "Semistability and infinite loop spaces", J. Pure Appl. Algebra, 154, pp. 213-220, (2000).

B. Keller, "Derived categories and universal problems", Comm. in Alg. 19(3), pp. 699-747 (1991).

Texts written after the electronic publication of Grothendieck's manuscript:

L. Alonso Tarrio, B. Álvarez Diaz, A. Jeremías Lopez, "Unbounded algebraic derivators", Preprint (2021).

K. Arlin, "A higher Whitehead theorem and the embedding of quasicategories in prederivators", Homology Homotopy Appl. 22, no 1, pp. 117-139 (2020).

J. Ayoub, "Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique", Astérisque, Vol. 314-315, Thesis (2006) (pdf).

J. Ayoub, "The motivic vanishing cycles and the conservation conjecture", London Math. Soc. Lecture Note Ser., 343, pp. 3-54 (2007) (pdf).

J. Ayoub, "Note sur les opérations de Grothendieck et la réalisation de Betti", J. Inst. Math. Jussieu, 9, pp. 225-263 (2010) (pdf).

J. Ayoub, "Motifs des variétés analytiques rigides", Mém. Soc. Math. Fr. (N.S.), No. 140-141 (2015) (pdf).

J. Ayoub, "La réalisation étale et les opérations de Grothendieck", Ann. Sci. Éc. Norm. Supér. (4) 47, no. 1, pp. 1-145 (2014) (pdf).

J. Ayoub, S. Zucker, "Relative Artin Motives and the reductive Borel-Serre compactification of a locally symmetric variety", Invent. Math. 188, pp. 277-427 (2012) (pdf).

P. Balmer, I. Dell'Ambrogio, "Mackey 2-functors and Mackey 2-motives", EMS Monographs in Mathematics (2020).

P. Balmer, G. Tabuada, "The fundamental isomorphism conjecture via non-commutative motives", Math. Nachr. 286, no. 8-9, pp. 791-798 (2013).

P. Balmer, J. Zhang, "Affine space over triangulated categories: a further invitation to Grothendieck derivators", J. Pure Appl. Algebra 221, no. 7, pp. 1560-1564 (2017).

F. Beckert, M. Groth, "Abstract cubical homotopy theory", Preprint (2018).

F. Beckert, "The bivariant parasimplicial S_.-construction", Preprint (2020).

D.-C. Cisinski, "Images directes cohomologiques dans les catégories de modèles", Annales Mathématiques Blaise Pascal, volume 10, pp. 195-244 (2003).

D.-C. Cisinski, "Propriétés universelles et extensions de Kan dérivées", Theory and Applications of Categories, 20, pp. 605-649 (2008).

D.-C. Cisinski, "Locally constant functors", Math. Proc. Camb. Phil. Soc., 147, pp. 593-614 (2009).

D.-C. Cisinski, "Catégories dérivables", Bull. Soc. Math. France, 138, pp. 317-393 (2010).

D.-C. Cisinski, "Descente par éclatements en K-théorie invariante par homotopie", Ann. of Math. (2) 177, no. 2, pp. 425-448 (2013).

D.-C. Cisinski, F. Déglise, "Triangulated categories of mixed motives", Springer Monographs in Mathematics (2019) pdf.

D.-C. Cisinski, A. Neeman, "Additivity for derivator K-theory", Advances in Mathematics, Volume 217, 4, pp. 1381-1475 (2008).

D.-C. Cisinski, G. Tabuada, "Non-connective K-theory via universal invariants", Compos. Math. 147, no. 4, pp. 1281-1320 (2011).

D.-C. Cisinski, G. Tabuada, "Symmetric monoidal structure on non-commutative motives", J. K-Theory 9, no. 2, pp. 201-268 (2012).

I. Coley, "Stabilization of derivators revisited", J. Homotopy Relat. Struct. 14, no. 2, pp. 525-577 (2019).

I. Coley, "The K-theory of left pointed derivators", Preprint (2020).

I. Coley, "The theory of half derivators", Preprint (2020).

I. Dell'Ambrogio, "Green 2-functors", Preprint (2021).

I. Di Liberti, F. Loregian, "Accessibility and presentability in 2-categories", Preprint (2018).

V. Drinfeld, "DG quotients of DG categories", J. Algebra, 272, pp. 643-691 (2004).

C. Dupont, D. Juteau, "The localization spectral sequence in the motivic setting", Preprint (2020).

D. Fuentes-Keuthan, M. Kedziorek, M. Rovelli, "A model structure on prederivators for (oo,1)-categories", Theory Appl. Categ. 34, Paper No. 39, pp. 1220-1245 (2019).

M. Gallauer Alves de Souza, "Traces in monoidal derivators, and homotopy colimits", Adv. Math. 261, pp. 26-84 (2014).

M. Gallauer Alves de Souza, "tt-geometry of Tate motives over algebraically closed fields", Compos. Math. 155, no. 10, pp. 1888-1923 (2019).

G. Garkusha, "Systems of diagram categories and K-theory". I, Algebra i Analiz 18 no. 6, pp. 136-181 (2006), St. Petersburg Math. J. 18, no. 6, pp. 957-996 (2007).

G. Garkusha, "Systems of diagram categories and K-theory". II, Math. Z. 249, no. 3, pp. 641-682 (2005).

M. Groth, "Derivators, pointed derivators, and stable derivators", Algebr. Geom. Topol. 13, no. 1, pp. 313-374 (2013).

M. Groth, "Monoidal derivators and additive derivators", Preprint (2012).

M. Groth, "Revisiting the canonicity of canonical triangulations", Theory Appl. Categ. 33, Paper No. 14, pp. 350-389 (2018).

M. Groth, "Characterizations of abstract stable homotopy theories", Preprint (2016).

M. Groth, K. Ponto, M. Shulman, "The additivity of traces in monoidal derivators", J. K-Theory 14, no. 3, pp. 422-494 (2014).

M. Groth, K. Ponto, M. Shulman, " Mayer-Vietoris sequences in stable derivators", Homology Homotopy Appl. 16, no. 1, pp. 265-294 (2014).

M. Groth, M. Rahn, "Higher symmetries in abstract stable homotopy theories", Contemp. Math. 758, pp. 91-193 (2020).

M. Groth, M. Shulman, "Generalized stability for abstract homotopy theories", Ann. K-Theory 6, no. 1, pp. 1-28 (2021).

M. Groth, J. Stovicek, "Tilting theory via stable homotopy theory", J. Reine Angew. Math. 743, pp. 29-90 (2018).

M. Groth, J. Stovicek, "Tilting theory for trees via stable homotopy theory", J. Pure Appl. Algebra 220, no. 6, pp. 2324-2363 (2016).

M. Groth, J. Stovicek, " Abstract representation theory of Dynkin quivers of type A", Adv. Math. 293, pp. 856-941 (2016).

M. Groth, J. Stovicek, "Abstract tilting theory for quivers and related categories", Ann. K-Theory 3, no. 1, pp. 71-124, (2018).

L. Guetta, "Homology of strict ω-categories", Thesis (2021).

F. Hörmann, "Fibered Multiderivators and (co)homological descent", Theory Appl. Categ. 32, Paper No. 38, pp. 1258-1362 (2017).

F. Hörmann, "Six functor formalisms and fibered multiderivators", Selecta Math. (N.S.) 24, no. 4, pp. 2841-2925 (2018).

F. Hörmann, "Derivator six functor formalisms -- definition and construction I", Preprint (2017).

F. Hörmann, "Enlargement of (fibered) derivators", J. Pure Appl. Algebra 224, no. 3, pp. 1023-1063 (2020).

F. Hörmann, "Derivator six functor formalisms -- construction II", Preprint (2019).

F. Hörmann, "Model category structures on simplicial objects", Preprint (2021).

F. Hörmann, "Higher stacks as diagrams", Preprint (2021).

M. Hrbek, T. Nakamura, "Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings", J. Pure Appl. Algebra 225, no. 4 (2021).

M. Hrbek, S. Pavon, "Singular equivalences to locally coherent hearts of commutative noetherian rings", Preprint (2021).

F. Ivorra, S. Morel, "The four operations on perverse motives", Preprint (2019).

F. Jin, E. Yang, "Künneth formulas for motives and additivity of traces", Adv. Math. 376 (2021).

B. Keller, P. Nicolás, "Weight structures and simple dg modules for positive dg algebras", Int. Math. Res. Not. IMRN 2013, no. 5, pp. 1028-1078 (2013).

M. Künzer, " Heller triangulated categories", Homology, Homotopy and Applications, vol. 9 (2), pp. 233-320 (2007).

I. Lagkas-Nicolos, "Levelwise modules over separable monads on stable derivators", J. Pure Appl. Algebra 222, no. 7, 1704-1726 (2018).

I. Lagkas-Nicolos, "Levelwise Modules and Localization in Derivators", Thesis (2018).

R. Laking, "Purity in compactly generated derivators and t-structures with Grothendieck hearts", Math. Z. 295, no. 3-4, pp. 1615-1641 (2020).

T. Lenz, "Homotopy (Pre-)Derivators of Cofibration Categories and Quasi-Categories", Algebr. Geom. Topol. 18, no 6, pp. 3601-3646 (2018).

F. Loregian, "Localization theory for derivators", Preprint (2018).

F. Loregian, S. Virili, "Factorization systems on (stable) derivators", Preprint (2017).

G. Maltsiniotis, "Introduction à la théorie des dérivateurs, d'après Grothendieck", Preprint (2001) (ps) (pdf).

G. Maltsiniotis, "Exposés sur les dérivateurs et les dérivateurs triangulés", scan of handwritten notes taken by D.-C. Cisinski (2001) (pdf).

G. Maltsiniotis, "La K-théorie d'un dérivateur triangulé", followed by an appendix by B. Keller, in "Categories in Algebra, Geometry and Mathematical Physics", Contemp. Math. 431, pp. 341-373 (2007) (ps) (pdf).

G. Maltsiniotis, Lecture notes on derivators in Seville (2009)

G. Maltsiniotis, Carrés exacts homotopiques, et dérivateurs, Cah. Topol. Géom. Différ. Catég. 53, no. 1, pp. 3-63 (2012).

L. Martini, C. E. Parra, "Local coherence of hearts associated with Thomason filtrations", Preprint (2021).

G. C. Modoi, "Reasonable triangulated categories have filtered enhancements", Proc. Amer. Math. Soc. 147, no. 7, pp. 2761-2773 (2019).

G. C. Modoi, "Weight structures cogenerated by weak cocompact objects", Preprint (2019).

F. Muro, "Maltsiniotis's first conjecture for K₁", Int. Math. Res. Not. IMRN 2008, no 4, pp. 1-31.

F. Muro, "Homotopy units in A-infinity algebras", Trans. Amer. Math. Soc. 368, no. 3, pp. 2145-2184 (2016).

F. Muro, A. Tonks, M. Witte, "On determinant functors and K-theory", Publ. Mat. 59, no. 1, pp. 137-233 (2015).

F. Muro, G. Raptis, "A note on K-theory and triangulated derivators", Adv. Math. 227, no. 5, pp. 1827-1845 (2011).

F. Muro, G. Raptis, "K-theory of derivators revisited", Ann. K-Theory 2, no. 2, pp. 303-340 (2017).

H. Nakaoka, "Biset functors as module Mackey functors, and its relation to derivators", Comm. Algebra 44, no. 12, pp. 5105-5148 (2016).

P. Nicolás, "The bar derived category of a curved dg algebra", J. Pure Appl. Algebra, 212, pp. 2633-2659 (2008).

D. Park, "Triangulated categories of motives over fs log schemes", Thesis (2016).

D. Park, "Equivariant cd-structures and descent theory", J. Pure Appl. Algebra 223, no. 10, pp. 4123-4152 (2019).

C. E. Parra, M. Saorin, "The HRS tilting process and Grothendieck hearts of t-structures", Contemp. Math. 769, pp. 209-241 (2021).

I. Patchkoria, "The derived category of complex periodic K-theory localized at an odd prime", Adv. Math. 309, pp. 392-435 (2017).

K. Ponto, M. Shulman, "The linearity of traces in monoidal categories and bicategories", Theory Appl. Categ. 31, No. 23, pp. 594-689 (2016).

K. Ponto, M. Shulman, "The linearity of fixed point invariants", Contemp. Math., 707, pp. 89-120 (2018).

M. Porta, "Universal property of triangulated derivators via Keller's towers", Preprint (2015).

A. Radulescu-Banu, "Cofibrations in Homotopy Theory ", Preprint (2006).

G. Raptis, "Higher homotopy categories, higher derivators, and K-theory", Preprint (2019).

O. Renaudin, "Théories homotopiques de Quillen combinatoires et dérivateurs de Grothendieck", Preprint (2005).

O. Renaudin, "Plongement de certaines théories homotopiques de Quillen dans les dérivateurs", J. Pure Appl. Algebra, 213, pp. 1916-1935 (2009).

J. Richardson, "Enriched Derivators", Preprint (2020).

B. Rodríguez González, "Realizable homotopy colimits", Theory Appl. Categ. 29, No. 22, pp. 609-634 (2014).

B. Rodríguez González, "A derivability criterion based on the existence of adjunctions", Preprint (2014).

M. Saorín, J. Stovicek, S. Virili, "t-Structures on stable derivators and Grothendieck hearts", Preprint (2017).

M. Saorín, J. Stovicek, "t-Structures with Grothendieck hearts via functor categories", Preprint (2020).

S. Schwede, "Topological triangulated categories", J. Topol. 6, no. 4, pp. 857-914 (2013).

M. Shulman, "The derivator of setoids", Preprint (2021).

W. Soergel, R. Virk, M. Wendt (with an appendix by F. Hörmann and M. Wendt), "Equivariant motives and geometric representation theory", Preprint (2018).

J. Stovicek, " Derived equivalences induced by big cotilting modules", Adv. Math. 263, pp. 45-87 (2014).

J. Stovicek, D. Pospisil, "On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings", Trans. Amer. Math. Soc. 368, no. 9, pp. 6325-6361 (2016).

N. P. Strickland, N. Bellumat, "Iterated chromatic localisation", Preprint (2019).

G. Tabuada, "Théorie homotopique des DG-catégories", Thesis (2007).

G. Tabuada, "Higher K-theory via universal invariants", Duke Math. J., 145, pp. 121-206 (2008).

G. Tabuada, "Topological Hochschild and cyclic homology for Differential graded categories", Preprint (2008).

G. Tabuada, "Generalized spectral categories, topological Hochschild homology and trace maps", Algebr. Geom. Topol., 10, no 1, pp. 137--213 (2010).

G. Tabuada, "Universal suspension via Non-commutative motives", J. Noncommut. Geom. 5, no. 4, pp. 573-589 (2011).

G. Tabuada, "Bivariant cyclic cohomology and Connes' bilinear pairings in non-commutative motives", J. Noncommut. Geom. 9, no. 2, pp. 265-285 (2015).

G. Tabuada, "Products, multiplicative Chern characters, and finite coefficients via non-commutative motives", J. Pure Appl. Algebra 217, no. 7, pp. 1279-1293 (2013).

G. Tabuada, "Weight structure on Kontsevich's noncommutative mixed motives", Homology Homotopy Appl. 14, no. 2, pp. 129-142 (2012).

G. Tabuada, "A guided tour through the garden of noncommutative motives", Topics in noncommutative geometry, pp. 259-276, Clay Math. Proc., 16, Amer. Math. Soc. (2012).

G. Tabuada, "A simple criterion for extending natural transformations to higher K-theory", Doc. Math. 16, pp. 657-668 (2011).

G. Tabuada, "A1-homotopy theory of noncommutative motives", J. Noncommut. Geom. 9, no. 3, pp. 851-875 (2015).

G. Tabuada, "Voevodsky's mixed motives versus Kontsevich's noncommutative mixed motives", Adv. Math. 264, pp. 506-545 (2014).

B. Toën, Homotopical and Higher Categorical Structures in Algebraic Geometry, Habilitation thesis (2003).

B. Toën, G. Vezzosi, A remark on K-theory and S-categories, Topology, 43, pp. 765-791 (2004).

B. Toën, G. Vezzosi, Algèbres simpliciales S^1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs, Compos. Math. 147, no. 6, pp. 1979-2000 (2011).

S. Virili, "Morita theory for stable derivators", Preprint (2018).

T. Walde, "Hall monoidal categories and categorical modules", Preprint (2016).

J. Zhang, "Affine lines over derivators: Properties", Preprint (2017).

Some other pages dedicated to derivators

In Wikipedia

In the n-category-Lab

The 2001 Seminar on Derivators

The program

The texts of some lectures

Pages dedicated to the mathematical texts of Alexander Grothendieck

"Pursuing Stacks" and correspondence.

Archives Grothendieck at Montpellier

LaTeX version of SGA3