## 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}

}

### OpenURL

### Abstract

this paper came from the attempt to understand how a rational result about the stable homotopy theory of commutative rings might generalize to arbitrary characteristic. We showed in [Sch, 3.2] that for a commutative

### Citations

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Citation Context ... to are DFSPs, i.e., the monoids in the category of \Gamma-spaces with respect to the smash product, and `FSPs on spheres', which are the monoids in the category of symmetric spectra, investigated in =-=[HSS]-=-. So in this section we consider a cofibrantly generated closed model category C which is also a symmetric monoidal category ([MacL, VII.7], [Bor, 6.1]) under a product denotedswith unit I. The model ... |

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Citation Context ...table equivalence. There are certain standard Tor spectral sequences converging to the homotopy groups of M�� � N, see [37, Lemma 3.1]. 2. Algebraic theories Algebraic theories, introduced by La=-=wvere [23], -=-formalize the concept of an algebraic object as a set together with n-ary operations for various n3� and equational relations. A detailed exposition of algebraic theories can be found in [6, Section... |

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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 ... theories to be simplicial sets with actions of the simplicial loop groups of the respective spaces involved. We assume that the reader is familiar with the language of homotopical algebra (cf. [Q1], =-=[DS]-=-) and with the basic ideas concerning monoidal and symmetric monoidal categories (cf. [MacL, VII], [Bor, 6]) and triples (also called monads, cf. [MacL, VI.1], [Bor, 4]). This paper is a slightly modi... |

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Citation Context ...tabilized to an FSP QF defined via QF (K) = hocolim I Sing\Omega n jF (S nsK)j where I is the category of finite sets and injective maps (details of this construction can be found in [Bo], [PW, 2] or =-=[BHM, 3]-=-). There is a map F \Gamma! QF of FSPs which is a weak equivalence. The spectrum underlying QF is an\Omega\Gamma/3 ectrum in the sense that the maps jQF (K)j \Gamma!\Omega jQF (\SigmaK)j adjoint to \S... |

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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 ...llen homology. We wanted to obtain a topological variant of the cotangent complex by replacing `abelianizationa by `stabilizationa, i.e., passage to spectra in the sense of stable homotopy theory. In =-=[36]-=- we made this precise by introducing a model category of spectra for simplicial commutative algebras. At the same time we showed that over the rational numbers, nothing really new is happening. More p... |

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Citation Context ... than simplicial rings, i.e., if one works with a suitable notion of homotopy ring. It turned out that the right notion to consider here is that of a `Functor with Smash Product' (FSP), introduced in =-=[Bo]-=-. There is currently no Quillen model category structure available for modules over an FSP, hence we chose to work with a variant of this notion, which we call DFSPs. DFSPs are based on a symmetric mo... |

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Citation Context ..."brations, co"brations and weak equivalences make the category of spectra into a closed simplicial model category. We show in Lemma A.3 that this model category structure is coxbrantly gener=-=ated (see [15], [37, De"n-=-ition A.2]). 1.7. Quillen equivalences. An adjoint functor pair between model categories is called a Quillen pair if the left adjoint ¸ preserves co"brations and acyclic co"brations. An equ... |

30 | T.: Cohomology of algebraic theories - Jibladze, Pirashvili - 1991 |

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Citation Context ...homology of with coe$cients in a non-additive functor can di!er from the topological Hochschild cohomology, see Remark 6.15. To prove the homological part of Theorem 6.7 we use the same strategy as [=-=30]-=-: we show that topological Hochschild homology has the universal properties of the derived functors of Q. The cohomological part follows from a comparison of the derived category of the abelian catego... |

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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 ...lgebras and the category of pointed simplicial sets. The category of -algebras is locally "nitely presentable, see Lemma A.1. As the "brant replacement functor Q we can take either Kan's fun=-=ctor Ex� [21]-=- or the composition of the singular complex and geometric realization functor (we then have to work in the category of compactly generated topological spaces). Each of these functors is simplicial and... |

18 | Some relations between homotopy and homology - Curtis - 1965 |

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Homotopy theory of \Gamma-spaces, spectra, and bisimplicial sets
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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... |

17 | Symmetric spectra and topological Hochschild homology, K-Theory 19(2 - Shipley - 2000 |

16 |
The mod-p lower central series and the Adams spectral sequence, Topology 5
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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... |

16 | Homotopy limits - Bousfield, Kan - 1972 |

14 | A chain rule in the calculus of homotopy functors
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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... |

13 | Homotopy operations for simplicial commutative algebras
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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... |

13 | New model categories from old - Blanc - 1996 |

12 | Coequalizers in categories of algebras - Linton - 1969 |

10 | Closed model structures for algebraic models of n-types - Cabello, Garzón - 1995 |

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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... |

7 |
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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... |

6 |
Smash products and -spaces
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- 1991
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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... |

6 | Γ + -structures. II. A recognition principle for infinite loop spaces - Barratt, Eccles |

6 | On H-spaces and infinite loop spaces - Beck - 1969 |

5 |
Stable homotopical algebra and #-spaces
- Schwede
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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... |

4 |
A Hilton-Milnor theorem for categories of simplicial algebras
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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... |

4 | Lawvere Functorial semantics of algebraic theories - W - 1963 |

2 | The homotopical foundations of algebraic K-theory. In Algebraic K-theory and algebraic number theory - Jardine - 1987 |

2 |
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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... |

2 |
Goodwillie: “Calculus I, The first derivative of pseudoisotopy theory, K-Theory 4
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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... |

2 | The categories of A1 - and E1 -monoids and ring spaces as closed simplicial and topological model categories - Schwanzl - 1991 |

1 |
structures II: a recognition principle for in"nite loop spaces, Topology 13
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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 ... |

1 | On H-spaces and in"nite loop spaces, in Category Theory, Homology Theory and Their - Beck - 1969 |

1 |
kstedt, Topological Hochschild homology
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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... |

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... |

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... |