Axiomatic homotopy theory for operads

By C. Berger, Ieke Moerdijk. 2002

357 citations recorded by the source corpus.

Abstract We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ed194a1efe6084e4d751980d8a15e0f33e74dbe1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Martin Markl
2. Benoit Fresse
3. John E. Harper
4. David White
5. L. Pereira
6. P. Bonventre
7. C. Berger
8. Michael Shulman

In Geometry and Topology

1. Michael Shulman
2. David White
3. John E. Harper
4. L. Pereira
5. P. Bonventre
6. Martin Markl
7. Benoit Fresse
8. Michael Batanin

In Mathematical Physics

1. Denis-Charles Cisinski
2. John E. Harper
3. Benoit Fresse
4. Michael Shulman
5. Martin Markl
6. David White
7. L. Pereira
8. P. Bonventre

Citations in the corpus

• Homotopy theory of modules over operads in symmetric spectra (2007, methodology): “The reader may contrast this with a path object approach explored in [2], which amounts to verifying the conditions of [38, Lemma 2.”
• Model Categories and Simplicial Methods (2006, background): “6, also due to Quillen in his original work [30] and recently highlighted in [2].”
• Symmetrisation of n-operads and compactification of real configuration spaces (2006, methodology): “1 [7]) There exists a transferred model structure on the category of reduced symmetric operads in a monoidal model category V if”
• Homotopy-theoretic aspects of 2-monads (2006, background): “In [1] operadic analogues are established to our lifted model structures on T -Algs and on Mndf (K ), generalizing earlier work by various authors.”
• Model structures on commutative monoids in general model categories (2014, background): “This semi-model structure is often a full model structure if D satisfies stronger conditions ([5] Theorem 3.”
• Props in model categories and homotopy invariance of structures (2008, methodology): “mapping spaces in the category of operads (see [38]) and such equivalences can be used to transport structures in the context of operads (see [1, 38]).”
• Mixed Weil cohomologies (2007, background): “As we are working with rational coefficients, we can see that there is a commutative monoid structure15 on the derived tensor product E ⊗Q HQ (see [7,29]).”
• On equivariant homotopy theory for model categories (2013, methodology): “Recall that the normalized chain complex C(∆[1];R) contains R in degree 1, two copies of R in degree 0 and is zero in the other degrees.”
• Operads and PROPs (2006, background): “ructures of the categories of dg commutative associative algebras and dg Lie algebras [95, 107]. Operads, in a OPERADS AND PROPS 3 reasonable monoidal model category, themselves form a model category [7, 31] such that algebras over cofibrant operads are homotopy invariant, see also [104]. Minimal ope”
• Modules over Operads and Functors (2007, methodology): “We refer also to [3] for the definition of a model category of operads in a general setting, to [22] for the semi-model category of operads in dg-module, and to [30] for the case of operads in orthogonal spectra.”
• Coloured Koszul duality and strongly homotopy operads (2003, methodology): “Let i : V −→ W and r : W −→ V be dg linear maps, and H : W −→ W [1] a chain homotopy between i ◦ r and idW .”
• Genuine equivariant operads (2017, methodology): “Furthermore, the arrows of ∣I●∣ are generated by the image of the arrows in I0 ≃ I0 × [0] and the image of the arrows in [1] ×Ob(I1).”
• Pre-Lie deformation theory (2015, background): “The first part, which recalls the general notion of cofibrant replacement and a proof of the homotopy transfer theorem from [BM03, Fre10], is not new.”