## Stable Homotopy of Algebraic Theories (2001)

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Venue: | Topology |

Citations: | 12 - 1 self |

### BibTeX

@ARTICLE{Schwede01stablehomotopy,

author = {Stefan Schwede},

title = {Stable Homotopy of Algebraic Theories},

journal = {Topology},

year = {2001},

volume = {40},

pages = {1--41}

}

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### Abstract

The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH}Quillen homology via certain spectral sequences. We show that the (co-)homology of an algebraic theory is isomorphic to the topological Hochschild (co-)homology of the parameterizing ring spectrum. # 2000 Elsevier Science Ltd. All rights reserved. MSC: 55U35; 18C10 Keywords: Algebraic theories; Ring spectra; AndreH}Quillen homology; #-spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For...

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Citation Context ...homology; �-spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For a map of commutative rings, Quille=-=n [31] de&q-=-uot;ned the cotangent complex as the left derived functor of abelianization; this construction is now referred to as AndreH}Quillen homology. We wanted to obtain a topological variant of the cotangent... |

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Citation Context ...ct notion of an algebraic theory might want to browse through the examples "rst, to get an idea of what the general theory is all about. We assume familiarity with the language of homotopical alg=-=ebra [16,32]. 1.-=- Review of �-spaces and Gamma-rings S. Schwede / Topology 40 (2001) 1}41 3 The category of �-spaces was introduced by Segal [38], who showed that it has a homotopy category equivalent to the stabl... |

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Citation Context ... over an FSP, hence we chose to work with a variant of this notion, which we call DFSPs. DFSPs are based on a symmetric monoidal smash product for \Gamma-spaces with good homotopical properties ([Se],=-=[BF]-=-,[Ly]). They are `FSPs defined on finite sets' and they represent all homotopy types of connective FSPs. We give model category structures for modules over a fixed DFSP, for the whole category of DFSP... |

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Citation Context ...r the p-local sphere spectrum, together with a `multiplicative "ltrationa.s34 S. Schwede / Topology 40 (2001) 1}41 This provides a di!erent view at the mod p-lower central series spectral sequenc=-=e of [8]. A nilp-=-otent group G is called p-local if for all primes qOp the set map x C x� is a bijection of G onto itself. On the category of nilpotent groups there exists a p-localization functor G C G ��� wh... |

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Citation Context ...se, the most convenient notion of ring spectrum is that of a Gamma-ring (see De"nition 1.12). Gamma-rings are based on a symmetric monoidal smash product for �-spaces with good homotopical prop=-=erties [9,25,38]-=-. The homotopy theory of Gamma-rings and their modules is developed in [37]. The generalization of the rational result [36, Theorem 3.2.3] then reads: Theorem. Let B be a commutative ring. Then the st... |

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Citation Context ...paces retractive over BG is equivalent to the homotopy theory of �[G]-modules, or spectra with an action of G. This is exploited by Klein and Rognes to prove a chain rule for the Calculus of Functor=-=s [22]-=-. 7.3. Monoids and groups. The theories of sets, monoids and groups have equivalent stable homotopy theories. This follows from the fact (see [29, Theorem 1]) that the free monoid and the free group g... |

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Citation Context ... commutative simplicial ring and consider the theory of augmented commutative B-algebras (alias commutative B-algebras without unit) Commutative simplicial algebras have been the object of much study =-=[14,17,18,31,36]-=-. The homology theory arising as the derived functor of abelianization in this case is known as AndreH}Quillen homology for commutative rings. We denote by DB the Gamma-ring arising from the theory of... |

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Citation Context ...motopy invariant. The Jibladze}Pirashvili cohomology plays the same role for algebraic theories that is played by Hochschild cohomology for algebras over a "eld, and it generalizes MacLane cohomo=-=logy [26]-=- for arbitrary rings. For example in [20, Section 4], Jibladze and Pirashvili give interpretations of the theory cohomology groups in dimensions 0, 1 and 2 as suitable `centera, `outer derivationa and... |

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Citation Context ... an FSP, hence we chose to work with a variant of this notion, which we call DFSPs. DFSPs are based on a symmetric monoidal smash product for \Gamma-spaces with good homotopical properties ([Se],[BF],=-=[Ly]-=-). They are `FSPs defined on finite sets' and they represent all homotopy types of connective FSPs. We give model category structures for modules over a fixed DFSP, for the whole category of DFSPs and... |

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Citation Context ...se, the most convenient notion of ring spectrum is that of a Gamma-ring (see De"nition 1.12). Gamma-rings are based on a symmetric monoidal smash product for �-spaces with good homotopical prop=-=erties [9,25,38]-=-. The homotopy theory of Gamma-rings and their modules is developed in [37]. The generalization of the rational result [36, Theorem 3.2.3] then reads: Theorem. Let B be a commutative ring. Then the st... |

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Citation Context ...nition 1.12). Gamma-rings are based on a symmetric monoidal smash product for �-spaces with good homotopical properties [9,25,38]. The homotopy theory of Gamma-rings and their modules is developed i=-=n [37]-=-. The generalization of the rational result [36, Theorem 3.2.3] then reads: Theorem. Let B be a commutative ring. Then the stable homotopy theory of augmented commutative simplicial B-algebras is equi... |

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Citation Context ... commutative simplicial ring and consider the theory of augmented commutative B-algebras (alias commutative B-algebras without unit) Commutative simplicial algebras have been the object of much study =-=[14,17,18,31,36]-=-. The homology theory arising as the derived functor of abelianization in this case is known as AndreH}Quillen homology for commutative rings. We denote by DB the Gamma-ring arising from the theory of... |

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Citation Context ...ories can be found in [6, Section 3]. To do homotopy theory, we use algebraic theories which are enriched over the category of simplicial sets; these simplicial theories have been considered by Reedy =-=[33]-=-. The version of algebraic theories enriched over topological spaces can be found in [3, Eq. (6); 4, Chapter II] or [34,35]. For many purposes, topological and simplicial theories can be used intercha... |

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Citation Context ...erivatives of the constituent functors. We recall the notions of stable excision and derivative for a homotopy functor. These were introduced by T. Goodwillie in the framework of Calculus of functors =-=[Gw1,2]-=-. The extension of the definitions from topological spaces to algebras over a simplicial theory is straightforward: Definition 5.2.1 (cf. [Gw1, 1.8, 1.13]) Let S; T be simplicial theories and F; H : S... |

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Citation Context ...p lower central series spectral sequence of [8]. From the E�-term on this spectral sequence is the Adams spectral sequence. 7.6. In5nite loop spaces. In our simplicial setup, the Barratt}Eccles mode=-=l [1,2] gives an algebraic-=- theory modeling in"nite loop spaces. Barratt and Eccles de"ne a functor �� from the category of pointed simplicial sets to itself [1, De"nition 3.1]. To avoid notational confusion ... |

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Citation Context ... homotopy groups of the derived smash product of S and M as cS-S-bimodules, THH (S; M)"� (S �� � � ���� ��M). This is not the original de"nition of topological Hochschi=-=ld homology given by BoK kstedt [5]. How-=-ever Shipley [39, Section 4] shows in the context of symmetric spectra that the two de"nitions are equivalent; a proof of the analogous statements in the context of Gamma-rings is similar, but ea... |

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Citation Context ...tions of H commutative simplicial B-algebras. These operations are also referred to as the stable Cartan}Bous"eld}Dwyer algebra (since these authors calculated the unstable operations for B"=-=� , � see [11,7,14]).-=- Additively, � DB is the direct sum of the stable derived functors, in the sense of Dold H and Puppe [13, Section 8.3], of the symmetric power functors on the category of B-modules. In [7, Section 1... |

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Citation Context ...tions of H commutative simplicial B-algebras. These operations are also referred to as the stable Cartan}Bous"eld}Dwyer algebra (since these authors calculated the unstable operations for B"=-=� , � see [11,7,14]).-=- Additively, � DB is the direct sum of the stable derived functors, in the sense of Dold H and Puppe [13, Section 8.3], of the symmetric power functors on the category of B-modules. In [7, Section 1... |