RESOLUTION OF COLOURED OPERADS AND RECTIFICATION OF HOMOTOPY ALGEBRAS

By C. Berger, Ieke Moerdijk. 2005

192 citations recorded by the source corpus.

Dedicated to Ross Street on the occasion of his 60th birthday Abstract. We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen model struc- ture, and prove a Comparison Theorem to the effect that a weak equivalence between suitable such operads induces a Quillen equivalence between their cat- egories of algebras. We construct an explicit Boardman-Vogt style cofibrant resolution for coloured operads, thereby giving a uniform approach to algebraic structures up to homotopy over coloured operads. The Comparison Theorem implies that such structures can be rectified.

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

View Paper: https://www.semanticscholar.org/paper/629c2a76296e0951ae8b188df1aec2bfb1116e7a

How researchers see this paper

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In Algebra and Number Theory

1. Ieke Moerdijk
2. Joachim Kock
3. V. Hinich
4. Denis-Charles Cisinski
5. David White
6. Gijs Heuts
7. Markus Spitzweck
8. O. Rondigs

In Geometry and Topology

1. Denis-Charles Cisinski
2. Ieke Moerdijk
3. Markus Spitzweck
4. V. Hinich
5. David White
6. O. Rondigs
7. Gijs Heuts
8. Joachim Kock

In Mathematical Physics

1. Denis-Charles Cisinski
2. Ieke Moerdijk
3. Markus Spitzweck
4. Joachim Kock
5. V. Hinich
6. David White
7. O. Rondigs
8. Gijs Heuts

Citations in the corpus

• Motivic slices and coloured operads (2010, background): “For more recent accounts, we refer to [3, 5, 9, 16].”
• Model structures on commutative monoids in general model categories (2014, background): “4 of [4] does not apply here because the operads Com, ComR, and ComT are not Σcofibrant.”
• Homotopy theory for algebras over polynomial monads (2013, unknown): “the appendix of [13], where the category of I-coloured bouquets Bouq(I) is denoted F(I).”
• On the equivalence between Lurie's model and the dendroidal model for infinity-operads (2013, background): “where the right-hand corresponds to the notation of [7].”
• Polynomial Functors and Trees (2008, methodology): “(This viewpoint on Coll (I) was also used in the appendix of Berger-Moerdijk [4], except that they did not formulate it in terms of monomial functors.)”
• Dendroidal sets and simplicial operads (2011, methodology): “In Section 4, we take up the construction of a generalized Boardman– Vogt resolution of operads from [5, 6], and use it to construct the Quillen pair featuring in our main theorem.”