## Deo/nitions: operads, algebras and modules (1997)

Venue: | Contemporary Mathematics 202 |

Citations: | 24 - 3 self |

### BibTeX

@INPROCEEDINGS{May97deo/nitions:operads,,

author = {J. P. May},

title = {Deo/nitions: operads, algebras and modules},

booktitle = {Contemporary Mathematics 202},

year = {1997},

pages = {1--7}

}

### OpenURL

### Abstract

There are many different types of algebra: associative, associative and commutative, Lie, Poisson, etc., etc. Each comes with an appropriate notion of a module. As is becoming more and more important in a variety of fields, it is often necessary to deal with algebras and modules of these sorts “up to homotopy”. I shall give a very partial overview, concentrating on algebra, but saying a little about the original use of operads in topology. The development of abstract frameworks in which to study such algebras has a long history. As this conference attests, it now seems to be widely accepted that, for many purposes, the most convenient setting is that given by operads and their actions. While the notion was first written up in a purely topological framework [19], it was thoroughly understood by 1971 [12] that the basic definitions apply equally well in any underlying symmetric monoidal ( = tensor) category. The definitions and ideas had many precursors. I will indicate those that I was aware of at the time. • Algebraists such as Kaplansky, Herstein, and Jacobson systematically studied

### Citations

474 |
Categories for the working mathematician
- MacLane
- 1998
(Show Context)
Citation Context ...he name is meant to bring to mind both operations and monads. Incidentally, I persuaded MacLane to discard the term “triple” in favor of “monad” in his book “Categories for the working mathematician” =-=[17]-=- 1 , which was being written about the same time. I was convinced that the notion of an operad was an important one, and I wanted the names to mesh. What I did not foresee was just how flexible the no... |

319 |
Homotopy associativity of H-spaces
- Stasheff
- 1963
(Show Context)
Citation Context ...aic structures. • Adams and MacLane [16] developed certain chain level concepts, PROP’s and PACT’s, with a view towards understanding the algebraic structure of the singular chain complex. • Stasheff =-=[23]-=- introduced A∞ spaces and constructed their classifying spaces, using associahedra and what in retrospect was an example of an operad. • Milgram [22] proved an approximation theorem for iterated loop ... |

260 | The geometry of iterated loop spaces
- May
- 1972
(Show Context)
Citation Context ...ow seems to be widely accepted that, for many purposes, the most convenient setting is that given by operads and their actions. While the notion was first written up in a purely topological framework =-=[19]-=-, it was thoroughly understood by 1971 [12] that the basic definitions apply equally well in any underlying symmetric monoidal (= tensor) category. The definitions and ideas had many precursors. I wil... |

230 |
Koszul duality for operads
- Ginzburg, Kapranov
- 1994
(Show Context)
Citation Context ...1 ⊗ · · · ⊗ dk) is in I if either c or any of the ds is in I . There is then a quotient operad C /I with jth k-moduleOPERADS, ALGEBRAS AND MODULES 7 C (j)/I (j). As observed by Ginzburg and Kapranov =-=[9]-=-, one can construct the free operad F G generated by any sequence G = {G (j)} of κ[Σj]-modules, and one can then construct an operad that describes a particular type of algebra by quotienting out by t... |

166 |
Functorial Semantics of Algebraic Theories
- Lawvere
- 1963
(Show Context)
Citation Context ...ors. I will indicate those that I was aware of at the time. • Algebraists such as Kaplansky, Herstein, and Jacobson systematically studied algebras defined by different kinds of identities. • Lawvere =-=[15]-=- formalized algebraic theories as a way of codifying different kinds of algebraic structures. • Adams and MacLane [16] developed certain chain level concepts, PROP’s and PACT’s, with a view towards un... |

122 | Batalin-Vilkovisky algebras and two-dimensional topological field theories
- Getzler
- 1994
(Show Context)
Citation Context ...e Poisson formula asserts that the map dx = [x, ?]n is a graded derivation, in the sense that dx(yz) = dx(y)z + (−1) deg(y)deg(dx) ydx(z). Batalin-Vilkovisky algebras are examples of 1-braid algebras =-=[7]-=-, hence the general case, with non-zero differentials, is relevant to string theory. However, our concern here is with structures that have zero differential. Theorem 5.6. The homology H∗(X) is an n-b... |

94 |
The homology of iterated loop spaces
- Cohen, Lada, et al.
- 1976
(Show Context)
Citation Context ...-Lashof operations that are present in the homology of E∞ algebras. I will describe the characteristic zero information and a portion of the mod p information in Cohen’s exhaustive mod p calculations =-=[4, 5]-=-. (There were earlier partial calculations by Arnol’d [1] in the case n = 2.) Again, we take k to be a field. Cohen’s calculations have two starting points. One is his complete and explicit calculatio... |

34 |
Homotopy Lie algebras
- Hinich, Schechtman
- 1993
(Show Context)
Citation Context ...elated to Lie algebras as E∞ operads are related to commutative algebras, and there is a concomitant notion of a differential graded Lie algebra “up to homotopy,” or L∞ algebra. Hinich and Schechtman =-=[11]-=- called these “Lie May algebras”. An E∞ algebra A has a product for each degree zero element, necessarily a cycle, of C (2). Each such product is unital, associative, and commutative up to all possibl... |

30 |
Homology of iterated loop spaces
- Dyer, Lashoff
- 1962
(Show Context)
Citation Context ...c to topological PROP’s and used PROP’s to prove a recognition principle in infinite loop space theory. • Beck [2] pointed out the relevance of monads to infinite loop space theory. • Dyer and Lashof =-=[6]-=- systematized homology operations for iterated loop spaces as analogs of Steenrod operations in the cohomology of spaces. It was my extraordinary good fortune to have had close mathematical contact wi... |

28 |
Homotopy limits of homotopy algebras
- Hinich, Schechtman
(Show Context)
Citation Context ...nd Σ-free; C (j) is then a k[Σj]-free resolution of k. By an E∞ algebra, we mean a C -algebra for any E∞ operad C . These were called “May algebras” when they were introduced by Hinich and Schechtman =-=[10]-=-. If we ignore symmetric groups, we obtain the notion of an A∞ algebra. These are commutative and non-commutative differential graded algebras up to homotopy. Similarly, there is a class of operads th... |

18 |
E∞ ring spaces and E∞ ring spectra
- May
- 1977
(Show Context)
Citation Context ... which they are applied. As an aside, since in the absence of diagonals it is unclear that there is a workable algebraic analog, we note that a topological theory of E∞ ring spaces has been developed =-=[20, 21]-=-. The sum and product, with the appropriate version of the distributive law, are codified in actions by two suitably interrelated operads. I may say a bit more about this in my second talk. Fix an ope... |

14 |
Homotopy Invariant Structures on Topological Spaces
- Boardman, Vogt
- 1973
(Show Context)
Citation Context ...associahedra and what in retrospect was an example of an operad. • Milgram [22] proved an approximation theorem for iterated loop spaces, using what are now known as permutahedra. • Boardman and Vogt =-=[3]-=- switched from algebraic to topological PROP’s and used PROP’s to prove a recognition principle in infinite loop space theory. • Beck [2] pointed out the relevance of monads to infinite loop space the... |

9 |
On the operads of
- Kelly
(Show Context)
Citation Context ...ny purposes, the most convenient setting is that given by operads and their actions. While the notion was first written up in a purely topological framework [19], it was thoroughly understood by 1971 =-=[12]-=- that the basic definitions apply equally well in any underlying symmetric monoidal (= tensor) category. The definitions and ideas had many precursors. I will indicate those that I was aware of at the... |

6 |
On H-spaces and infinite loop spaces
- Beck
- 1969
(Show Context)
Citation Context ... using what are now known as permutahedra. • Boardman and Vogt [3] switched from algebraic to topological PROP’s and used PROP’s to prove a recognition principle in infinite loop space theory. • Beck =-=[2]-=- pointed out the relevance of monads to infinite loop space theory. • Dyer and Lashof [6] systematized homology operations for iterated loop spaces as analogs of Steenrod operations in the cohomology ... |

6 |
n-Algebras and Batalin-Vilkovisky algebras
- Getzler, Jones
- 1992
(Show Context)
Citation Context ...ent literature of string theory. Although the theorems I’m about to state were proven in the early 1970’s, their statements came much later, in work of Ginzberg and Kapranov [9] and Getzler and Jones =-=[8]-=-. For each n > 0, there is a little n-cubes operad Cn. It was invented, before the introduction of operads, by Boardman and Vogt [3]. Its jth space Cn(j) consists of j-tuples of little n-cubes embedde... |

5 |
Derived categories and motives
- Kriz, May
- 1994
(Show Context)
Citation Context ...ds, in any characteristic. This has real force. Using it, Kriz and I were able to carry out a suggestion of Deligne for the construction of categories of both integral and rational mixed Tate motives =-=[13, 14]-=-. However, that is a subject for another talk. I hope that this has given a bit of a feel for some of the ways that operads work, and I look forward to learning more from all of you during the confere... |

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 =-=[24]-=-, 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... |

3 |
Multiplicative ininite loop space theory
- May
- 1982
(Show Context)
Citation Context ... which they are applied. As an aside, since in the absence of diagonals it is unclear that there is a workable algebraic analog, we note that a topological theory of E∞ ring spaces has been developed =-=[20, 21]-=-. The sum and product, with the appropriate version of the distributive law, are codified in actions by two suitably interrelated operads. I may say a bit more about this in my second talk. Fix an ope... |

3 |
Milgram Iterated loop spaces
- J
(Show Context)
Citation Context ...tructure of the singular chain complex. • Stasheff [23] introduced A∞ spaces and constructed their classifying spaces, using associahedra and what in retrospect was an example of an operad. • Milgram =-=[22]-=- proved an approximation theorem for iterated loop spaces, using what are now known as permutahedra. • Boardman and Vogt [3] switched from algebraic to topological PROP’s and used PROP’s to prove a re... |

2 |
A general approach to Steenrod operations
- May
- 1970
(Show Context)
Citation Context ...re at Chicago, or were there when the relevant work was done, or were regular visitors there. I wanted a notion that carried the combinatorial structure familiar to me from a paper that I had written =-=[18]-=- that gave a general algebraic approach to Steenrod operations. The diagrams in the definition of an operad action are generalizations of diagrams that were used there to prove the Cartan formula and ... |

1 |
Cohomology of the group of dyed braids
- Arnol’d
(Show Context)
Citation Context ...works and there is a huge difference between the two constructions, with the reduced construction being by far the more important one. Modules also admit a monadic reinterpretation: there is a monad C=-=[1]-=- in the category S 2 such that a C[1]-algebra (A, M) is a C-algebra A together with an6 J. P. MAY A-module M. There is a free A-module functor FA for any C -algebra A, and C[1](X; Y ) is the pair (CX... |