## Algebraic & Geometric Topology

### Homotopy theory of non-symmetric operads, II: Change of base category and left properness

Fernando Muro

#### Abstract

We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 1 (2014), 229-281.

Dates
Revised: 9 August 2013
Accepted: 9 August 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715800

Digital Object Identifier
doi:10.2140/agt.2014.14.229

Mathematical Reviews number (MathSciNet)
MR3158759

Zentralblatt MATH identifier
1281.18001

Subjects
Secondary: 18G55: Homotopical algebra

#### Citation

Muro, Fernando. Homotopy theory of non-symmetric operads, II: Change of base category and left properness. Algebr. Geom. Topol. 14 (2014), no. 1, 229--281. doi:10.2140/agt.2014.14.229. https://projecteuclid.org/euclid.agt/1513715800

#### References

• M Aguiar, S Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph Series 29, Amer. Math. Soc. (2010)
• M A Batanin, Homotopy coherent category theory and $A\sb \infty$–structures in monoidal categories, J. Pure Appl. Algebra 123 (1998) 67–103
• H J Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge Univ. Press (1989)
• C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805–831
• F Borceux, Handbook of categorical algebra, 2: Categories and structures, Encyclopedia of Mathematics and its Applications 51, Cambridge Univ. Press (1994)
• P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc. (2003)
• M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc. (1999)
• A Joyal, R Street, Tortile Yang–Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991) 43–51
• S Mac Lane, Categories for the working mathematician, 2nd edition, Graduate Texts in Mathematics 5, Springer, New York (1998)
• M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
• F Muro, Homotopy theory of nonsymmetric operads, Algebr. Geom. Topol. 11 (2011) 1541–1599
• F Muro, Homotopy units in ${A}$–infinity algebras (2011)
• F Muro, Moduli spaces of algebras over non-symmetric operads (2011)
• S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
• S Schwede, B E Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003) 287–334
• B E Shipley, $H\Bbb Z$–algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351–379
• B Toën, G Vezzosi, Homotopical algebraic geometry, II: Geometric stacks and applications, Mem. Amer. Math. Soc. 902, AMS (2008)