Geometry & Topology

Commutative ring objects in pro-categories and generalized Moore spectra

Daniel G Davis and Tyler Lawson

Full-text: Open access


We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M J Hopkins that certain towers of generalized Moore spectra, closely related to the K(n)–local sphere, are E–algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have homotopically tractable endomorphism operads.

Article information

Geom. Topol., Volume 18, Number 1 (2014), 103-140.

Received: 22 August 2012
Revised: 7 April 2013
Accepted: 13 June 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55U35: Abstract and axiomatic homotopy theory
Secondary: 18D20: Enriched categories (over closed or monoidal categories) 18D50: Operads [See also 55P48] 18G55: Homotopical algebra

Moore spectra pro-objects structured ring spectra endomorphism operad


Davis, Daniel G; Lawson, Tyler. Commutative ring objects in pro-categories and generalized Moore spectra. Geom. Topol. 18 (2014), no. 1, 103--140. doi:10.2140/gt.2014.18.103.

Export citation


  • V Angeltveit, Enriched Reedy categories, Proc. Amer. Math. Soc. 136 (2008) 2323–2332
  • M Artin, B Mazur, Etale homotopy, Lecture Notes in Mathematics 100, Springer, Berlin (1969)
  • C Ausoni, J Rognes, Algebraic $K$–theory of the fraction field of topological $K$–theory, unpublished (2008)
  • A Baker, A Lazarev, On the Adams spectral sequence for $R$–modules, Algebr. Geom. Topol. 1 (2001) 173–199
  • T Bauer, Formal plethories
  • M Behrens, The tower of generalized Moore spectra is $H_\infty$, unpublished (2004)
  • M Behrens, D G Davis, The homotopy fixed point spectra of profinite Galois extensions, Trans. Amer. Math. Soc. 362 (2010) 4983–5042
  • C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805–831
  • A K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257–281
  • G Carlsson, Derived completions in stable homotopy theory, J. Pure Appl. Algebra 212 (2008) 550–577
  • B I Dundas, Localization of $V$–categories, Theory Appl. Categ. 8 (2001) 284–312
  • W Dwyer, K Hess, Long knots and maps between operads, Geom. Topol. 16 (2012) 919–955
  • W G Dwyer, P S Hirschhorn, D M Kan, J H Smith, Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs 113, Amer. Math. Soc. (2004)
  • W G Dwyer, D M Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980) 267–284
  • D A Edwards, H M Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics 542, Springer, Berlin (1976)
  • H Fausk, Equivariant homotopy theory for pro-spectra, Geom. Topol. 12 (2008) 103–176
  • H Fausk, D C Isaksen, Model structures on pro-categories, Homology Homotopy Appl. 9 (2007) 367–398
  • H Fausk, D C Isaksen, t–model structures, Homology Homotopy Appl. 9 (2007) 399–438
  • P G Goerss, (Pre-)sheaves of ring spectra over the moduli stack of formal group laws, from: “Axiomatic, enriched and motivic homotopy theory”, (J P C Greenlees, editor), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer Acad. Publ., Dordrecht, Netherlands (2004) 101–131
  • P Goerss, H-W Henn, M Mahowald, C Rezk, A resolution of the $K(2)$–local sphere at the prime $3$, Ann. of Math. 162 (2005) 777–822
  • P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: “Structured ring spectra”, (A Baker, B Richter, editors), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151–200
  • M Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture, from: “The Čech centennial 1993”, (M Cenkl, H Miller, editors), Contemp. Math. 181, Amer. Math. Soc. (1995) 225–250
  • M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc. (1999)
  • M Hovey, Morava $E$–theory of filtered colimits, Trans. Amer. Math. Soc. 360 (2008) 369–382
  • M Hovey, J H Palmieri, N P Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997) 114
  • M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
  • M Hovey, N P Strickland, Morava $K$–theories and localisation, Mem. Amer. Math. Soc. 139 (1999) viii+100
  • D C Isaksen, Strict model structures for pro-categories, from: “Categorical decomposition techniques in algebraic topology”, (G Arone, J Hubbuck, R Levi, M Weiss, editors), Progr. Math. 215, Birkhäuser, Basel (2004) 179–198
  • M Markl, S Shnider, J Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96, Amer. Math. Soc. (2002)
  • S Oka, Ring spectra with few cells, Japan. J. Math. 5 (1979) 81–100
  • C W Rezk, Spaces of algebra structures and cohomology of operads, PhD thesis, Massachusetts Institute of Technology (1996)
  • C Rezk, Notes on the Hopkins–Miller theorem, from: “Homotopy theory via algebraic geometry and group representations”, (M Mahowald, S Priddy, editors), Contemp. Math. 220, Amer. Math. Soc. (1998) 313–366
  • C Rezk, Every homotopy theory of simplicial algebras admits a proper model, Topology Appl. 119 (2002) 65–94
  • C Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006) 969–1014
  • J Rognes, Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc. 192 (2008) 137
  • S Schwede, Stable homotopical algebra and $\Gamma$–spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999) 329–356
  • S Schwede, The stable homotopy category is rigid, Ann. of Math. 166 (2007) 837–863
  • S Schwede, An untitled book project about symmetric spectra (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
  • S Schwede, B Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003) 287–334
  • B Shipley, A convenient model category for commutative ring spectra, from: “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$–theory”, (P Goerss, S Priddy, editors), Contemp. Math. 346, Amer. Math. Soc. (2004) 473–483
  • R W Thomason, Algebraic $K$–theory and étale cohomology, Ann. Sci. École Norm. Sup. 18 (1985) 437–552