## OPERADS, ALGEBRAS, MODULES, AND MOTIVES

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Citations: | 6 - 0 self |

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@MISC{Kriz_operads,algebras,,

author = {Igor Kriz and J. P. May},

title = {OPERADS, ALGEBRAS, MODULES, AND MOTIVES},

year = {}

}

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### Abstract

Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and conversion theorems III Derived categories from a topological point of view IV Rational derived categories and mixed Tate motives V Derived categories of modules over E ∞ algebras In differential algebra, operads are systems of parameter chain complexes for multiplication on various types of differential graded algebras “up to homotopy”, for example commutative algebras, n-Lie algebras, n-braid algebras, etc. Our primary focus is the development of the concomitant theory of modules up to homotopy and the study of both classical derived categories of modules over DGA’s and derived categories of modules up to homotopy over DGA’s up to homotopy. Examples of such derived categories provide the appropriate setting for one approach to mixed Tate motives in algebraic geometry, both rational and integral.

### Citations

553 | Homological Algebra - Cartan, Eilenberg - 1956 |

514 |
Categories for the working mathematician
- MacLane
- 1971
(Show Context)
Citation Context ... standard coherence isomorphisms (= associativity and commutativity constraints) on the tensor product pass to the derived category, which is thus a symmetric monoidal closed category in the sense of =-=[43, 36]-=-. There are general accounts of duality theory in such a context in the literature of both algebraic geometry [21, §1], [19] and algebraic topology [23]; we follow [40, III §§1–2]. Observe first that,... |

244 |
Koszul duality for operads
- Ginzburg, Kapranov
- 1993
(Show Context)
Citation Context ...appear in print until the 1987 paper of Hinich and Schechtman [31]. Applications of such algebraic operads and their actions have appeared in a variety of contexts in other recent papers, for example =-=[27, 28, 29, 32, 34, 33, 56]-=-. In the algebraic setting, an operad C consists of suitably related chain complexes C (j) with actions by the symmetric groups Σj. An action of C on a chain complex A is specified by suitably related... |

226 |
Closed categories
- Eilenberg, Kelly
- 1966
(Show Context)
Citation Context ... standard coherence isomorphisms (= associativity and commutativity constraints) on the tensor product pass to the derived category, which is thus a symmetric monoidal closed category in the sense of =-=[43, 36]-=-. There are general accounts of duality theory in such a context in the literature of both algebraic geometry [21, §1], [19] and algebraic topology [23]; we follow [40, III §§1–2]. Observe first that,... |

156 | Equivariant sheaves and functors - Bernstein, Lunts - 1994 |

143 |
Faisceaux pervers
- Bĕılinson, Bernstein, et al.
- 1982
(Show Context)
Citation Context ...this theory to study the derived category DA of cohomologically bounded below A-modules, where A is a cohomologically connected commutative DGA over a field of characteristic zero. In the language of =-=[3]-=-, we shall give the triangulated category DA a t-structure. Its heart HA will be the Abelian subcategory of modules whose indecomposable elements have homology concentrated in degree zero. In the lang... |

126 | Batalin-Vilkovisky algebras and two-dimensional topological field theories
- Getzler
- 1994
(Show Context)
Citation Context ...appear in print until the 1987 paper of Hinich and Schechtman [31]. Applications of such algebraic operads and their actions have appeared in a variety of contexts in other recent papers, for example =-=[27, 28, 29, 32, 34, 33, 56]-=-. In the algebraic setting, an operad C consists of suitably related chain complexes C (j) with actions by the symmetric groups Σj. An action of C on a chain complex A is specified by suitably related... |

116 | The Homology of Iterated Loop Spaces - Cohen, Lada, et al. - 1976 |

73 | Algebraic cycles and higher K-theory - Bloch - 1986 |

62 | Conditionally convergent spectral sequences, Contemp
- Boardman
- 1999
(Show Context)
Citation Context ..., AND MOTIVES 55 each fixed (p, q), only finitely many of the differentials (4.10) are non-zero. (The best study of the convergence of spectral sequences, unfortunately still unpublished, is given in =-=[10]-=-.) Our construction of the spectral sequences follows [30], which is a precursor of the present approach to derived categories. Let ɛ : P → N be a quasi-isomorphism of left A-modules, where P is a cel... |

58 |
The moving lemma for higher Chow groups
- Bloch
- 1994
(Show Context)
Citation Context ...roups by taking mod p homology, then these groups admit homology operations just like those familiar in algebraic topology. A harder moving lemma of Bloch, first claimed in [7] and recently proven in =-=[9]-=-, implies that CH r (X, q) ⊗ Q ∼ = (Kq(X) ⊗ Q) (r) . Here the right side is the n r -eigenspace of the Adams operation ψ n for any n > 1 (which is independent of n); Kq(X) ⊗ Q is the direct sum of the... |

56 |
Infinitesimal computations in topology
- Sullivan
- 1978
(Show Context)
Citation Context ...generalized nilpotent representations of the co-Lie algebra γA. Part IV is really a chapter in rational homotopy theory, and it may well have applications to that subject. As was observed by Sullivan =-=[58]-=-, a co-Lie algebra γ determines a structure of DGA on the exterior algebra ∧(γ[−1]), where γ[−1] is a copy of γ concentrated in degree one. For a cohomologically connected DGA A,6 IGOR KRIZ AND J. P.... |

55 |
gories Tannakiennes
- Deligne, Cate
- 1991
(Show Context)
Citation Context ...ory, which is thus a symmetric monoidal closed category in the sense of [43, 36]. There are general accounts of duality theory in such a context in the literature of both algebraic geometry [21, §1], =-=[19]-=- and algebraic topology [23]; we follow [40, III §§1–2]. Observe first that, by an easy direct inspection of definitions, the functor HomA(M, N) preserves cofiber sequences in both variables. (Actuall... |

48 |
Categories and cohomology theories
- Segal
- 1974
(Show Context)
Citation Context ...f further algebraic analogs [47, 49, 26, 52]. In particular, the proofs in Part II areOPERADS, ALGEBRAS, MODULES, AND MOTIVES 5 exactly analogous to a topological comparison between Segal’s Γ-spaces =-=[55]-=- and spaces with operad actions that is given in [26]. While these results can be expected to have other applications, the motivation came from algebraic geometry. For a variety X, Bloch [7] defined t... |

45 |
with contributions by
- May, Quinn, et al.
- 1977
(Show Context)
Citation Context ...ycles is (t). Since s and t run through Z, the analogy is with stable homotopy theory: that is where negative dimensional spheres live. In fact, the modern description of the stable homotopy category =-=[40]-=- translates directly into our new description of the derived category. (The preamble of [40] explains the relationship with earlier treatments of the stable homotopy category, which do not have the sa... |

39 |
Through the looking glass: A dictionary between rational homotopy theory and local algebra, in: Algebra, algebraic topology, and their interactions
- Avramov, Halperin
- 1983
(Show Context)
Citation Context ...r to be more concrete and less formal. We repeat that many topologists have long known some of this material. For example, although the emphasis is quite different, our work overlaps that of [30] and =-=[2]-=-. On a technical note, we emphasize that, as in [30], k is an arbitrary commutative ring and we nowhere impose boundedness or flatness hypotheses. 1. Cell A-modules Motivated by the motivic context, w... |

38 |
Homotopy Lie algebras
- Hinich, Schechtman
- 1993
(Show Context)
Citation Context ...appear in print until the 1987 paper of Hinich and Schechtman [31]. Applications of such algebraic operads and their actions have appeared in a variety of contexts in other recent papers, for example =-=[27, 28, 29, 32, 34, 33, 56]-=-. In the algebraic setting, an operad C consists of suitably related chain complexes C (j) with actions by the symmetric groups Σj. An action of C on a chain complex A is specified by suitably related... |

34 |
Mixed Tate motives
- Bloch, Križ
- 1994
(Show Context)
Citation Context ...ndidate for the Abelian category M T M (F ) of mixed Tate motives over F . He (in [18]) and Bloch also proposed the category of finite dimensional comodules over χA as a candidate for M T M (F ), and =-=[6]-=- proves realization theorems in étale and Hodge theory starting from this definition. Our work shows that these two categories are equivalent, and it gives a fairly concrete and explicit description o... |

34 |
Abstract homotopy theory
- Brown
- 1965
(Show Context)
Citation Context ...i → ⊕Mi; equivalently, it is the cofiber of g : ⊕Mi → ⊕Mi, where g(m) = m − fi(m) for m ∈ Mi. We now have enough information to quote the categorical form of Brown’s representability theorem given in =-=[13]-=-, but we prefer to run through a quick concrete version of the proof.52 IGOR KRIZ AND J. P. MAY Theorem 3.2 (Brown). A contravariant functor J : DA → Sets is representable in the form J(M) ∼ = DA(M, ... |

31 |
On homotopy limit of homotopy algebras
- Hinich, Schechtman
- 1987
(Show Context)
Citation Context ...and Vogt [11], was a major step forwards. Perhaps for this reason, a chain level algebraic version of the definition of an operad did not appear in print until the 1987 paper of Hinich and Schechtman =-=[31]-=-. Applications of such algebraic operads and their actions have appeared in a variety of contexts in other recent papers, for example [27, 28, 29, 32, 34, 33, 56]. In the algebraic setting, an operad ... |

30 |
On the theory and applications of differential torsion products
- Gugenhein, May
- 1974
(Show Context)
Citation Context ... We prefer to be more concrete and less formal. We repeat that many topologists have long known some of this material. For example, although the emphasis is quite different, our work overlaps that of =-=[30]-=- and [2]. On a technical note, we emphasize that, as in [30], k is an arbitrary commutative ring and we nowhere impose boundedness or flatness hypotheses. 1. Cell A-modules Motivated by the motivic co... |

29 |
Operadic formulation of the notion of vertex operator algebra, in
- Huang, Lepowsky
- 1992
(Show Context)
Citation Context |

28 |
Infinitesimal structure of moduli spaces of G– bundles
- Beilinson, Ginzburg
- 1992
(Show Context)
Citation Context ...operads N and Hn−1(Cn), where the reduced sum is obtained from the direct sum by identifying the unit elements in N (1) and H0(Cn(1)). When char(k) = 0 and n = 1, the following result was implicit in =-=[4]-=- and was made explicit by Schechtman and Ginzburg. It was observed by Getzler and Jones [28] that the general case was already implicit in Cohen’s thesis [15]. Theorem 6.1. If char(k) ̸= 2 or 3, then,... |

28 | Vertex operator algebra and operads
- Huang, Lepwski
(Show Context)
Citation Context |

25 |
Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli spaces
- Stasheff
(Show Context)
Citation Context |

24 |
Bloch's higher Chow groups revisited
- Levine
- 1994
(Show Context)
Citation Context ... q) ⊗ Q ∼ = (Kq(X) ⊗ Q) (r) . Here the right side is the n r -eigenspace of the Adams operation ψ n for any n > 1 (which is independent of n); Kq(X) ⊗ Q is the direct sum of these eigenspaces. Levine =-=[39]-=- has recently given a different proof of this isomorphism that avoids Bloch’s hard moving lemma. As in Remark 6.2(ii), we choose an E∞ operad C of simplicial Abelian groups and regard our partial comm... |

19 |
Homotopical Algebra, volume 43
- Quillen
- 1967
(Show Context)
Citation Context ...uences for their computation, drop out quite easily. Our methods can be abstracted and applied more generally, and some of what we do can be formalized in Quillen’s context of closed model categories =-=[54]-=-. We prefer to be more concrete and less formal. We repeat that many topologists have long known some of this material. For example, although the emphasis is quite different, our work overlaps that of... |

15 | Commutative algebra in stable homotopy theory
- Elmendorf, Kriz, et al.
- 1995
(Show Context)
Citation Context ...quences in Part V include Künneth and universal coefficient spectral sequences that are of clear utility in the study of generalized homology and cohomology. Some other applications were announced in =-=[24]-=-, and many more are now in place. An exposition of the analogy between the algebraic and topological theories is given in [51]. Parts II and V constitute a revision and expansion of material in the pr... |

15 | The uniqueness of infinite loop space machines
- May, Thomason
- 1978
(Show Context)
Citation Context ...ads [46]. This construction has also been used to prove topological analogs of many of the present algebraic results, along with various other results that are suggestive of further algebraic analogs =-=[47, 49, 26, 52]-=-. In particular, the proofs in Part II areOPERADS, ALGEBRAS, MODULES, AND MOTIVES 5 exactly analogous to a topological comparison between Segal’s Γ-spaces [55] and spaces with operad actions that is ... |

14 |
Homotopy Invariant Structures on Topological Spaces
- Boardman, Vogt
- 1973
(Show Context)
Citation Context ...OP’s and PACT’s of Adams and MacLane [42], were important precursors of operads. From a topological point of view, the switch from algebraic to topological PROP’s, which was made by Boardman and Vogt =-=[11]-=-, was a major step forwards. Perhaps for this reason, a chain level algebraic version of the definition of an operad did not appear in print until the 1987 paper of Hinich and Schechtman [31]. Applica... |

14 |
The geometry of iterated loop spaces, volume 271
- May
(Show Context)
Citation Context ...tract frameworks in which to study such algebras has a long history. It now seems to be widely accepted that, for most purposes, the most convenient setting is that given by operads and their actions =-=[46]-=-. While the notion was first written up in a purely topological framework, due in large part to the resistance of topologists to abstract nonsense at that period, it was already understood by 1971 tha... |

12 |
A quoi servent les motifs
- Deligne
- 1994
(Show Context)
Citation Context ... cohomologically connected. When the conjecture holds, the results just summarized apply to A = NQ. Assuming the Beilinson-Soulé conjecture (and assuming our construction of the DGA A), Deligne [20], =-=[17]-=-, proposed F HA as a candidate for the Abelian category M T M (F ) of mixed Tate motives over F . He (in [18]) and Bloch also proposed the category of finite dimensional comodules over χA as a candida... |

11 |
Higher Chow groups of affine varieties and étale cohomology, preprint
- Suslin
- 1995
(Show Context)
Citation Context ... Chow groups and there is only speculation as to their relationship to the higher algebraic K-groups of F . However, our results on derived categories work equally well if we reduce mod n, and Suslin =-=[59]-=- has recently proven that if F is an algebraically closed field of characteristic prime to n and X is a smooth affine variety over F , then, for r ≥ dim(X), CH r (X, q; Z/n) ∼ = H 2r−q ét (X, Z/n (r))... |

11 | The unstable Adams spectral sequence for free iterated loop spaces - Wellington - 1982 |

8 |
On the operads of
- Kelly
- 2005
(Show Context)
Citation Context ...resistance of topologists to abstract nonsense at that period, it was already understood by 1971 that the basic definitions apply equally well in any underlying symmetric monoidal (= tensor) category =-=[35]-=-. In fact, certain chain level concepts, the PROP’s and PACT’s of Adams and MacLane [42], were important precursors of operads. From a topological point of view, the switch from algebraic to topologic... |

6 |
A variant of E. H. Brown’s representability theorem
- Adams
- 1971
(Show Context)
Citation Context ...s): C (k) ⊗ C (j1) ⊗ X j1 ⊗ · · · ⊗ C (jk−1) ⊗ X jk−1 ⊗ C (jk) ⊗ X jk−1 ⊗ Y shuffle �� C (k) ⊗ C (j1) ⊗ · · · ⊗ C (jk) ⊗ Xj−1 ⊗ Y γ⊗Id � � C (j) ⊗ X j−1 ⊗ Y18 IGOR KRIZ AND J. P. MAY Define a monad C=-=[1]-=- in the category of pairs (X; Y ) by letting C[1](X; Y ) = (CX; C(X; Y )). The unit η and product µ of C[1] are given by the evident pairs (η; η) and (µ; µ). Proposition 4.4. A C -algebra structure on... |

6 |
Homology operations and loop spaces
- Browder
- 1960
(Show Context)
Citation Context ...or any Cn+1-space X and any field of coefficients. The n-bracket is denoted λn and called a Browder operation in [45, §6], and [16, II], where the theorem is proven. The first appearance of λn was in =-=[12]-=-, in characteristic 2. We have displayed (iii) since that is the version of the Poisson formula given in [16] (where signs are garbled on page 216 but correct on page 317). Identity (iii) of Definitio... |

6 |
n-Algebras and Batalin-Vilkovisky algebras
- Getzler, Jones
- 1992
(Show Context)
Citation Context ...g the unit elements in N (1) and H0(Cn(1)). When char(k) = 0 and n = 1, the following result was implicit in [4] and was made explicit by Schechtman and Ginzburg. It was observed by Getzler and Jones =-=[28]-=- that the general case was already implicit in Cohen’s thesis [15]. Theorem 6.1. If char(k) ̸= 2 or 3, then, for all n ≥ 1, the degree zero translate of the operad Hn(Cn+1) is isomorphic to the operad... |

5 |
Derived categories and motives
- Kriz, May
- 1994
(Show Context)
Citation Context ...lose analogy between the two theories and the resulting expository duplication, the technical differences dictate separate and self-contained treatments. Some of the present results were announced in =-=[38]-=-. Each part has its own introduction, and we have tried to make the parts readable independently of one another. Part III has nothing whatever to do with operads and is wholly independent of Parts I a... |

5 |
A canonical operad
- Steiner
(Show Context)
Citation Context ...radial contraction and translation. These are better suited to considerations of group actions and of geometry, but they do not stabilize over n. There is a more sophisticated variant, due to Steiner =-=[57]-=-, that enjoys the good properties of both the little n-cubes and the little n-disks operads. Each of these operads comes with a canonical equivalence from its jth space to the configuration space F (R... |

4 | Homologie nicht-additiver Functoren - Dold, Puppe - 1961 |

4 | The spectra associated to I -monoids
- May
(Show Context)
Citation Context ...metric intuition, think of intersection products that are only defined between elements that are in general position. Such partial algebras have been used in topology since the 1970’s, for example in =-=[48]-=- and in unpublished work of Boardman and Segal. In Part II, we shall generalize the notions of algebras over operads and of modules over algebras over operads to the context of partially defined struc... |

3 |
to Bloch and May dated March 19
- Letters
- 1990
(Show Context)
Citation Context ...er tensoring with the rationals, we obtain a commutative differential graded algebra (DGA) NQ(X) that is quasi-isomorphic to N (X) ⊗ Q. The construction of these algebras answers questions of Deligne =-=[20]-=- that were the starting point of the present work. His motivation was the intuition that, when X = Spec(F ) for a field F , the associated derived categories of modules ought to be the appropriate hom... |

3 | Homology operations revisited - Fiedorowicz, May - 1982 |

3 |
Multiplicative ininite loop space theory
- May
- 1982
(Show Context)
Citation Context ...ads [46]. This construction has also been used to prove topological analogs of many of the present algebraic results, along with various other results that are suggestive of further algebraic analogs =-=[47, 49, 26, 52]-=-. In particular, the proofs in Part II areOPERADS, ALGEBRAS, MODULES, AND MOTIVES 5 exactly analogous to a topological comparison between Segal’s Γ-spaces [55] and spaces with operad actions that is ... |

3 |
The dual Whitehead theorems
- May
- 1983
(Show Context)
Citation Context ...is the cycle d(j q (r)) ⊗ j s (t) + (−1) q (j q (r)) ⊗ d(j s (t)). □ 2. Whitehead’s theorem and the derived category A quick space level version of some of the results of this section may be found in =-=[50]-=-, and the spectrum level model is given in [40, I §5]. We construct the derived category explicitly in terms of cell modules. As in topology, the “homotopy extension and lifting property” is pivotal. ... |

2 |
Letter to Kriz and
- Bloch
- 1992
(Show Context)
Citation Context ...(X, q) spanned by those j-tuples of simplices all intersections of subsets of which meet all faces properly. An “easy” moving lemma of Bloch, implicit in [7] and proven in detail when X = Spec(F ) in =-=[8]-=-, gives that the inclusion Z(X)j → Z(X) j is a quasi-isomorphism. It is evident that the intersection product is commutative, associative, and unital. If π : X → Y is a flat map, we obtain a map π ∗ :... |

2 |
The Homology of Iterated Loop Spaces. Volume 533
- Cohen, Lada, et al.
- 1976
(Show Context)
Citation Context ...lgebra generated by the free n-Lie algebra generated by V . Actually, the results just summarized are the easy characteristic zero case of Cohen’s much deeper calculations in arbitrary characteristic =-=[15, 16]-=-, now over twenty years old. Operads and their actions are specified in terms of maps that are defined on tensor products of chain complexes. In practice, one often encounters structures that behave m... |

2 | Tannakian categories, volume 900 - Deligne, Milne - 1982 |

2 | E∞ ring spaces and E∞ ring spectra, volume 577 - May - 1977 |