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25
A model category structure on the category of simplicial categories
, 2005
"... In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories. ..."
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Cited by 49 (6 self)
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In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.
Three models for the homotopy theory of homotopy theories
, 2005
"... its simplicial localization yields a “homotopy theory of homotopy theories. ” In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent ..."
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Cited by 43 (8 self)
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its simplicial localization yields a “homotopy theory of homotopy theories. ” In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with their respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory. 1.
HZalgebra spectra are differential graded algebras
 Amer. Jour. Math
, 2004
"... Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Qu ..."
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Cited by 32 (10 self)
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Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Qalgebra (with many objects). 1.
Iterated wreath product of the simplex category and iterated loop spaces
 Adv. Math
"... Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternat ..."
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Cited by 11 (4 self)
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Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternative description of the Segal spectrum associated to a Γspace. In particular, each EilenbergMacLane space has a canonical reduced Θnset model. The number of (n + d)dimensional cells of the resulting CWcomplex of type K(Z/2Z, n) is a generalised Fibonacci number.
Quasicategories vs Segal spaces
 IN CATEGORIES IN ALGEBRA, GEOMETRY AND MATHEMATICAL
, 2006
"... We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories. ..."
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Cited by 11 (0 self)
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We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories.
Simplicial monoids and Segal categories
 Contemp. Math
"... Abstract. Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids. ..."
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Cited by 4 (3 self)
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Abstract. Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids.
ADDING INVERSES TO DIAGRAMS ENCODING ALGEBRAIC STRUCTURES
, 2008
"... Abstract. We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segal’s mod ..."
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Cited by 3 (2 self)
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Abstract. We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segal’s model for simplicial abelian monoids in such a way that it becomes a model for simplicial abelian groups. 1.
Adding inverses to diagrams II: Invertible homotopy theories are spaces, preprint available at math.AT/0710.2254
"... Abstract. In previous work, we showed that there are appropriate model ..."
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Cited by 3 (1 self)
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Abstract. In previous work, we showed that there are appropriate model