Model Structures for the 𝔸1-homotopy Category

2019 ◽  
pp. 175-212
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Full Subcategory ◽  
Homotopy Category ◽  
Symmetric Power ◽  
Model Structure ◽  
Model Categories ◽  
Simplicial Presheaves ◽  
Basic Ideas ◽  
Quillen Model ◽  
Model Structures ◽  
Bousfield Localization

This chapter provides the 𝔸1-local projective model structure on the categories of simplicial presheaves and simplicial presheaves with transfers. These model categories, written as Δ‎opPshv(Sm)𝔸1 and Δ‎op PST(Sm)𝔸1, are first defined. Their respective homotopy categories are Ho(Sm) and the full subcategory DM eff nis ≤0 of DM eff nis. Afterward, this chapter introduces the notions of radditive presheaves and ̅Δ‎-closed classes, and develops their basic properties. The theory of ̅Δ‎-closed classes is needed because the extension of symmetric power functors to simplicial radditive presheaves is not a left adjoint. This chapter uses many of the basic ideas of Quillen model categories, which is a category equipped with three classes of morphisms satisfying five axioms. In addition, much of the material in this chapter is based upon the technique of Bousfield localization.

scholarly journals Morita homotopy theory for (∞,1)-categories and ∞-operads

2019 ◽  
Vol 31 (3) ◽  
pp. 661-684 ◽  
Author(s):  
Giovanni Caviglia ◽  
Javier J. Gutiérrez
Keyword(s):  
Homotopy Theory ◽  
Model Structure ◽  
Simplicial Sets ◽  
Simplicial Presheaves ◽  
Model Structures ◽  
Simplicial Algebras ◽  
Bousfield Localization

Abstract We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of {(\infty,1)} -categories and {\infty} -operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski–Moerdijk model structure on simplicial operads, respectively.

scholarly journals A homotopy-theoretic model of function extensionality in the effective topos

2018 ◽  
Vol 29 (4) ◽  
pp. 588-614 ◽  
Author(s):  
DAN FRUMIN ◽  
BENNO VAN DEN BERG
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Full Subcategory ◽  
Category Structure ◽  
Homotopy Category ◽  
Theoretic Model ◽  
Model Structure ◽  
Homotopy Equivalence ◽  
Homotopy Type Theory ◽  
Quillen Model

We present a way of constructing a Quillen model structure on a full subcategory of an elementary topos, starting with an interval object with connections and a certain dominance. The advantage of this method is that it does not require the underlying topos to be cocomplete. The resulting model category structure gives rise to a model of homotopy type theory with identity types, Σ- and Π-types, and functional extensionality. We apply the method to the effective topos with the interval object ∇2. In the resulting model structure we identify uniform inhabited objects as contractible objects, and show that discrete objects are fibrant. Moreover, we show that the unit of the discrete reflection is a homotopy equivalence and the homotopy category of fibrant assemblies is equivalent to the category of modest sets. We compare our work with the path object category construction on the effective topos by Jaap van Oosten.

Intermediate Model Structures for Simplicial Presheaves

2006 ◽  
Vol 49 (3) ◽  
pp. 407-413 ◽  
Author(s):  
J. F. Jardine
Keyword(s):  
Usual Sense ◽  
Model Structure ◽  
Closed Model ◽  
Intermediate Model ◽  
Standard Set ◽  
Simplicial Presheaves ◽  
Model Structures

AbstractThis note shows that any set of cofibrations containing the standard set of generating projective cofibrations determines a cofibrantly generated proper closed model structure on the category of simplicial presheaves on a small Grothendieck site, for which the weak equivalences are the local weak equivalences in the usual sense.

scholarly journals Homotopy limits for 2-categories

2008 ◽  
Vol 145 (1) ◽  
pp. 43-63 ◽  
Author(s):  
NICOLA GAMBINO
Keyword(s):  
Model Categories ◽  
Homotopy Limits ◽  
Quillen Model ◽  
Model Structures ◽  
Do So

AbstractWe study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

scholarly journals A Quillen model structure for Gray-categories

2010 ◽  
Vol 8 (2) ◽  
pp. 183-221 ◽  
Author(s):  
Stephen Lack
Keyword(s):  
Full Subcategory ◽  
Model Structure ◽  
Theoretic Proof ◽  
Quillen Model ◽  
Quillen Equivalent

AbstractA Quillen model structure on the category Gray-Cat of Gray-categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to provide a functorial and model-theoretic proof of the unpublished theorem of Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model structure on Gray-Cat is conjectured to be Quillen equivalent to a model structure on the category Tricat of tricategories and strict homomorphisms of tricategories.

scholarly journals STABLE LEFT AND RIGHT BOUSFIELD LOCALISATIONS

2013 ◽  
Vol 56 (1) ◽  
pp. 13-42 ◽  
Author(s):  
DAVID BARNES ◽  
CONSTANZE ROITZHEIM
Keyword(s):  
Homology Theory ◽  
Homotopy Category ◽  
Stable Model ◽  
Model Categories ◽  
Morita Theory ◽  
Left And Right ◽  
Torsion Modules ◽  
Model Structures ◽  
Insight Into ◽  
Complete Characterisation

AbstractWe study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra. We exploit stability to see that the resulting model structures are technically far better behaved than the general case. We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation.

Model structures for pro-simplicial presheaves

2011 ◽  
Vol 7 (3) ◽  
pp. 499-525 ◽  
Author(s):  
J.F. Jardine
Keyword(s):  
Type Structure ◽  
Model Structure ◽  
Simplicial Sets ◽  
Postnikov Tower ◽  
Simplicial Presheaves ◽  
Model Structures

AbstractThis paper displays model structures for the category of pro-objects in simplicial presheaves on an arbitrary small Grothendieck site. The first of these is an analogue of the Edwards-Hastings model structure for pro-simplicial sets, in which the cofibrations are monomorphisms and the weak equivalences are specified by comparisons of function complexes. Other model structures are built from the Edwards-Hastings structure by using Bousfield-Friedlander localization techniques. There is, in particular, an n-type structure for pro-simplicial presheaves, and also a model structure in which the map from a pro-object to its Postnikov tower is formally inverted.

scholarly journals A monoidal algebraic model for rational SO(2)-spectra

2016 ◽  
Vol 161 (1) ◽  
pp. 167-192 ◽  
Author(s):  
DAVID BARNES
Keyword(s):  
Abelian Category ◽  
Algebraic Model ◽  
Homotopy Category ◽  
Smash Product ◽  
Model Structure ◽  
Joint Work ◽  
Generating Sets ◽  
Stable Homotopy Category ◽  
Equivariant Spectra ◽  
Model Structures

AbstractThe category of rational SO(2)–equivariant spectra admits an algebraic model. That is, there is an abelian category ${\mathcal A}$(SO(2)) whose derived category is equivalent to the homotopy category of rational SO(2)–equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra?The category ${\mathcal A}$(SO(2)) is known as Greenlees' standard model, it is an abelian category that has no projective objects and is constructed from modules over a non–Noetherian ring. As a consequence, the standard techniques for constructing a monoidal model structure cannot be applied. In this paper a monoidal model structure on ${\mathcal A}$(SO(2)) is constructed and the derived tensor product on the homotopy category is shown to be compatible with the smash product of spectra. The method used is related to techniques developed by the author in earlier joint work with Roitzheim. That work constructed a monoidal model structure on Franke's exotic model for the K(p)–local stable homotopy category.A monoidal Quillen equivalence to a simpler monoidal model category R•-mod that has explicit generating sets is also given. Having monoidal model structures on ${\mathcal A}$(SO(2)) and R•-mod removes a serious obstruction to constructing a series of monoidal Quillen equivalences between the algebraic model and rational SO(2)–equivariant spectra.

Analysis and design of suitable model structures for activated sludge tanks with circulating flow

1999 ◽  
Vol 39 (4) ◽  
pp. 55-60 ◽  
Author(s):  
J. Alex ◽  
R. Tschepetzki ◽  
U. Jumar ◽  
F. Obenaus ◽  
K.-H. Rosenwinkel
Keyword(s):  
Activated Sludge ◽  
Large Scale ◽  
Wastewater Treatment Plants ◽  
Treatment Plant ◽  
High Sensitivity ◽  
Suitable Model ◽  
Model Structure ◽  
Analysis And Design ◽  
Hydraulic Behaviour ◽  
Model Structures

Activated sludge models are widely used for planning and optimisation of wastewater treatment plants and on line applications are under development to support the operation of complex treatment plants. A proper model is crucial for all of these applications. The task of parameter calibration is focused in several papers and applications. An essential precondition for this task is an appropriately defined model structure, which is often given much less attention. Different model structures for a large scale treatment plant with circulation flow are discussed in this paper. A more systematic method to derive a suitable model structure is applied to this case. Results of a numerical hydraulic model are used for this purpose. The importance of these efforts are proven by a high sensitivity of the simulation results with respect to the selection of the model structure and the hydraulic conditions. Finally it is shown, that model calibration was possible only by adjusting to the hydraulic behaviour and without any changes of biological parameters.

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