Universal Homotopy Theories

By Daniel Dugger. 2000

215 citations recorded by the source corpus.

Abstract Begin with a small category C . The goal of this short note is to point out that there is such a thing as a “universal model category built from C .” We describe applications of this to the study of homotopy colimits, the Dwyer–Kan theory of framings, sheaf theory, and the homotopy theory of schemes.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e17a68648bc292f18e1d41398ab8ffbee845b3af

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Differential cohomology theories as sheaves of spectra (2013, background): “The first statement is basically due to Dugger [Dug01] who shows it in a slightly different setting and only for C the (∞, 1)-category of spaces.”
Hypercovers and simplicial presheaves (2002, methodology): “on 2. We’d like to point out that the above theorem can be re-interpreted in terms of giving ‘generators’ and ‘relations’ for the homotopy theory of simplicial presheaves, in the manner introduced by [D]. Using the language of that paper, we prove Theorem 1.2. Jardine’s model category sPre(C)L is Qu”
Motivic cell structures (2003, background): “The idea is that the A-cellular objects are the ones that can, up to homotopy, be built out of objects in A. This definition is precisely the same as the usual notion of cellularity for the category of pointed topological spaces [DF, Ex. 2. D .2.1].”
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES (2005, methodology): “Using terminology of Dugger [6], this model category CSS is a presentation for the homotopy theory of homotopy theories, since it is a localization of a category of diagrams of spaces.”
Local Projective Model Structures on Simplicial Presheaves (2001, methodology): “Dan Dugger has also examined the local projective model structure for presheaves and his preprint [D] contains statements of many of his results, which overlap with some results in this note.”
Etale realization on the A1-homotopy theory of schemes (2001, background): “motopy theory. We assume familiarity with closed model structures. General references on this topic include [Hi], [Ho], or [Q1]. We conform to the conventions of [Hi] as closely as possible. See also [D] for more details on model structures as applied to simplicial presheaves. The ﬁrst major result ”
Quasi-categories vs Segal spaces (2006, background): “s of Stasheﬀ [MSS]. See also Batanin for A∞-categories [Ba]. A theory of homotopical categories was developed by Dwyer, Hirschhorn, Kan and Smith [DHKS]. Dugger studies universal homotopy theories in [D]. The model structure for quasi-categories belongs to a class of model structures in presheaf cat”
Generalized Brown representability in homotopy categories (2005, unknown): “Following [11], there is a left Quillen functor H : SSetD op → K such that HY = D where Y : D → SSetD is given by taking the discrete simplicial presheaves.”
Homotopical algebraic geometry. I. Topos theory. (2002, background): “to the set of morphisms deﬁned in the statement of Corollary 3.4.5). In the special case where (T,τ) is a usual Grothendieck site (i.e. when T is a category), the following corollary was announced in [Du] and proved in [DHI]. Corollary 3.4.7 An object F∈SPrτ(T) is ﬁbrant if and only if it is objectw”
Iterated wreath product of the simplex category and iterated loop spaces (2005, background): “Dugger [20] uses the more suggestive term homotopy-surjection for homotopy-reflections.”
Topological hypercovers and 1-realizations (2004, background): “Proof. By general nonsense from [ D2 ], to give a Quillen map Spc � (k)T → Top we just need to give a functor Sm/k → Top which respects the above relations.”
Manifold calculus and homotopy sheaves (2012, background): “rant replacement functor on PSh(C). Moreover, hMap(X;Y) ’Sing(RHom PSh(C)(X;Y)). A.2. Resolutions. In this section we discuss the construction of a resolution of a presheaf F2PSh(Disc k) (c.f. 2.6 in [Dug01]). More precisely, we wish to nd a cobrant presheaf Feand a weak equivalence Fe!F, where ever”
On the Unicity of the Homotopy Theory of Higher Categories (2011, unknown): “ement that these two classes of relations hold. In other words the homotopy theory of (∞,n)-categories is universal with respect to the ﬁrst three properties. Employing techniques discussed by Dugger [15] in the context of model categories, we are able to show that such a universal homotopy theory e”