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Morita homotopy theory for (∞,1)-categories and ∞-operads

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Published/Copyright: February 7, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 3

Abstract

We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of (∞,1)-categories and ∞-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski–Moerdijk model structure on simplicial operads, respectively.

Keywords: Morita model structure; simplicial category; quasicategory; simplicial operad; dendroidal set
MSC 2010: 55U35; 18G30; 18D50

Communicated by Frederick R. Cohen

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM2016-76453-C2-2-P

Funding statement: The second named author was supported by the Spanish Ministry of Economy under the grants MTM2016-76453-C2-2-P (AEI/FEDER, UE) and RYC-2014-15328 (Ramón y Cajal Program).

Acknowledgements

We would like to thank Joost Nuiten for several useful conversations related to the subject of this paper. We would also like to thank the referee for very useful comments and suggestions that helped improving the presentation of the paper.

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Received: 2018-02-06
Revised: 2018-08-29
Published Online: 2019-02-07
Published in Print: 2019-05-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
https://doi.org/10.1515/forum-2018-0033
Keywords for this article
Morita model structure; simplicial category; quasicategory; simplicial operad; dendroidal set

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
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