Cosimplicial objects and little n -cubes, I

McClure, James E.; Smith, Jeffrey H.

doi:10.1353/ajm.2004.0038

Cited by 62 publications

(124 citation statements)

References 19 publications

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“…which is associative and unital and satisfies The remainder of this section gives an outline of the proof of Theorem 4.3; for details see [28,Section 3].…”

Section: Provisional Definition 24 a Non-symmetric Operad O Is A Comentioning

confidence: 99%

“…These operations satisfy properties analogous to Propositions 7.4, 7.5, 7.6 and 7.8; the precise formulation is left to the reader (see [28,] for a hint).…”

Section: A Family Of Cochain Operationsmentioning

confidence: 99%

“…. , k} with complexity ≤ n. Suppose that the maps f are consistent with the cosimplicial operators (in the sense of [28,Definition 9.4]) and are commutative, associative and unital (in the sense of [28, Definitions 9.5, 9.6 and 9.7]). Then Tot(X • ) is an E n space.…”

Section: A Family Of Cochain Operationsmentioning

confidence: 99%

“…It remains to check that γ has the associativity, unitality and equivariance properties required by the definition of operad; for this we apply the coherence theorem one more time to see that Γ has associativity, unitality and equivariance properties (see [28,Section 4] for the explicit statements) from which those for γ can be deduced. This completes the proof of Remark 6.3(b).…”

Section: An Extension Of Remark 63(b)mentioning

confidence: 99%

“…For the proof see [28,Section 8]. Applying Proposition 11.1 we obtain an operad with k-th space (12.2) Hom…”

Section: An Extension Of Remark 63(b)mentioning

confidence: 99%

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Operads and Cosimplicial Objects: An Introduction

McClure

Smith

2004

Axiomatic, Enriched and Motivic Homotopy Theory

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“…which is associative and unital and satisfies The remainder of this section gives an outline of the proof of Theorem 4.3; for details see [28,Section 3].…”

Section: Provisional Definition 24 a Non-symmetric Operad O Is A Comentioning

confidence: 99%

“…These operations satisfy properties analogous to Propositions 7.4, 7.5, 7.6 and 7.8; the precise formulation is left to the reader (see [28,] for a hint).…”

Section: A Family Of Cochain Operationsmentioning

confidence: 99%

Section: A Family Of Cochain Operationsmentioning

confidence: 99%

Section: An Extension Of Remark 63(b)mentioning

confidence: 99%

“…For the proof see [28,Section 8]. Applying Proposition 11.1 we obtain an operad with k-th space (12.2) Hom…”

Section: An Extension Of Remark 63(b)mentioning

confidence: 99%

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Operads and Cosimplicial Objects: An Introduction

McClure

Smith

2004

Axiomatic, Enriched and Motivic Homotopy Theory

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A Mysterious Tensor Product in Topology

Moerdijk

2022

Lecture Notes in Mathematics

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The students of my generation had to survive without the internet and mobile phones, and depended on books and real paper to write on. As undergraduates in mathematics, we were always carrying yellow books around, and Springer-Verlag had a big part in our mathematical development. A little later, when I was a PhD student, two Springer Lecture Notes had a lasting influence on my own mathematical work: Homotopy Invariant Algebraic Structures on Topological Spaces by Boardman and Vogt [1] and The Geometry of Iterated Loop Spaces by Peter May [6]. These two books together shaped the foundation of the theory of operads about which I will write below.My contacts with Catriona go back to the preparation and publishing of "Sheaves in Geometry and Logic" with Saunders MacLane in the early 1990s. This period was also the beginning of the use of e-mail, and it is interesting and entertaining to see how e-mail customs and etiquette have changed over the years. I had my first e-mails to Catriona typed by a secretary, and Catriona had several people working for her who wrote on her behalf, all using an e-mail account and address under the name "Byrne". A few years after that, together with Albrecht Dold, Catriona helped me to get SLN 1616 into publishable shape. In more recent years, she remained most helpful in several matters, and I wish to thank her for that. Now I would like to come back to Boardman and Vogt, and May, and talk about some mathematics. To begin with, let me remind you that a (coloured) operad P consists of a set C = colours(P ) of colours, and for each sequence (c 1 , . . . , c n ; c) of elements of C (where n 0) a set of operations P (c 1 , . . . , c n ; c), to be thought of as taking inputs of "types" c 1 , . . . , c n , respectively, to an output of type c. Moreover, I. Moerdijk ( )

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Algebras of Higher Operads as Enriched Categories

Batanin

Weber

2008

Appl Categor Struct

101

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One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads (Batanin, Adv Math 136:39-103, 1998) to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n + 1)-operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)-category is something like a category enriched in weak n-categories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.

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Cosimplicial objects and little n -cubes, I

Abstract: In this paper we show that if a cosimplicial space has a certain kind of combinatorial structure then its total space has an action of an operad weakly equivalent to the little n-cubes operad. Our results are also valid for cosimplicial spectra.

Cited by 62 publications

References 19 publications

Operads and Cosimplicial Objects: An Introduction

Operads and Cosimplicial Objects: An Introduction

A Mysterious Tensor Product in Topology

Algebras of Higher Operads as Enriched Categories

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