Relevant bibliographies by topics / Quillen equivalence

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Academic literature on the topic 'Quillen equivalence'

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Published: 4 June 2021
Last updated: 27 April 2022

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Journal articles on the topic "Quillen equivalence"

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Gaucher, Philippe. "Homotopy theory of Moore flows (II)." Extracta Mathematicae 36, no. 2 (December 1, 2021): 157–239. http://dx.doi.org/10.17398/2605-5686.36.2.157.

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This paper proves that the q-model structures of Moore flows and of multipointed d-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed d-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy of this functor using Moore flows.
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Pavlov, Dmitri, and Jakob Scholbach. "SYMMETRIC OPERADS IN ABSTRACT SYMMETRIC SPECTRA." Journal of the Institute of Mathematics of Jussieu 18, no. 4 (May 25, 2018): 707–58. http://dx.doi.org/10.1017/s1474748017000202.

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This paper sets up the foundations for derived algebraic geometry, Goerss–Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën–Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and $\text{E}_{\infty }$-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of $\text{E}_{\infty }$-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.
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Morrow, Matthew. "Pro unitality and pro excision in algebraic K-theory and cyclic homology." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 736 (March 1, 2018): 95–139. http://dx.doi.org/10.1515/crelle-2015-0007.

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AbstractThe purpose of this paper is to study pro excision in algebraicK-theory and cyclic homology, after Suslin–Wodzicki, Cuntz–Quillen, Cortiñas, and Geisser–Hesselholt, as well as continuity properties of André–Quillen and Hochschild homology. A key tool is first to establish the equivalence of various pro Tor vanishing conditions which appear in the literature.This allows us to prove that all ideals of commutative, Noetherian rings are pro unital in a suitable sense. We show moreover that such pro unital ideals satisfy pro excision in derived Hochschild and cyclic homology. It follows hence, and from the Suslin–Wodzicki criterion, that ideals of commutative, Noetherian rings satisfy pro excision in derived Hochschild and cyclic homology, and in algebraicK-theory.In addition, our techniques yield a strong form of the pro Hochschild–Kostant–Rosenberg theorem; an extension to general base rings of the Cuntz–Quillen excision theorem in periodic cyclic homology; a generalisation of the Feĭgin–Tsygan theorem; a short proof of pro excision in topological Hochschild and cyclic homology; and new Artin–Rees and continuity statements in André–Quillen and Hochschild homology.
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Moss, Sean. "Another approach to the Kan–Quillen model structure." Journal of Homotopy and Related Structures 15, no. 1 (September 24, 2019): 143–65. http://dx.doi.org/10.1007/s40062-019-00247-y.

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Abstract By careful analysis of the embedding of a simplicial set into its image under Kan’s $$\mathop {\mathop {\mathsf {Ex}}^\infty }$$Ex∞ functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we can quickly deduce some basic facts about $$\mathop {\mathop {\mathsf {Ex}}^\infty }$$Ex∞ and hence provide a new construction of the Kan–Quillen model structure on simplicial sets, one which avoids the use of topological spaces or minimal fibrations.
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BARNES, DAVID. "A monoidal algebraic model for rational SO(2)-spectra." Mathematical Proceedings of the Cambridge Philosophical Society 161, no. 1 (April 11, 2016): 167–92. http://dx.doi.org/10.1017/s0305004116000219.

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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.
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Piacenza, Robert J. "Homotopy Theory of Diagrams and CW-Complexes Over a Category." Canadian Journal of Mathematics 43, no. 4 (August 1, 1991): 814–24. http://dx.doi.org/10.4153/cjm-1991-046-3.

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The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.
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FRUMIN, DAN, and BENNO VAN DEN BERG. "A homotopy-theoretic model of function extensionality in the effective topos." Mathematical Structures in Computer Science 29, no. 4 (September 10, 2018): 588–614. http://dx.doi.org/10.1017/s0960129518000142.

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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.
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RIEHL, EMILY. "On the structure of simplicial categories associated to quasi-categories." Mathematical Proceedings of the Cambridge Philosophical Society 150, no. 3 (March 11, 2011): 489–504. http://dx.doi.org/10.1017/s0305004111000053.

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AbstractThe homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all Λ21 horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of X are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, X is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant “simplicial thickenings” of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.
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Dalezios, Georgios, Sergio Estrada, and Henrik Holm. "Quillen equivalences for stable categories." Journal of Algebra 501 (May 2018): 130–49. http://dx.doi.org/10.1016/j.jalgebra.2017.12.033.

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Lack, Stephen. "A Quillen model structure for Gray-categories." Journal of K-Theory 8, no. 2 (September 24, 2010): 183–221. http://dx.doi.org/10.1017/is010008014jkt127.

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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.
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Dissertations / Theses on the topic "Quillen equivalence"

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Ariotta, Stefano. "Model categories." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9151/.

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Slávik, Alexander. "Třídy modulů motivované algebraickou geometrií." Doctoral thesis, 2020. http://www.nusl.cz/ntk/nusl-436934.

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This thesis summarises the author's results in representation theory of rings and schemes, obtained with several collaborators. First, we show that for a quasicompact semiseparated scheme X, the derived category of very flat quasicoherent sheaves is equivalent to the derived category of flat quasicoherent sheaves, and if X is affine, this is further equivalent to the homotopy category of projectives. Next, we prove that if R is a commutative Noetherian ring, then every countably generated flat module is quite flat, i.e., a direct summand of a transfinite extension of localizations of R in countable multiplicative subsets. Further, we investigate the relations between the geometric and categorical purity in categories of sheaves; we give a characterization of indecomposable geometric pure-injectives in both the quasicoherent and non-quasicoherent case. In partic- ular, we describe the Ziegler spectrum and its geometric part for the category of quasicoherent sheaves on the projective line over a field. The final result is the equivalence of the following statements for a quasicompact quasiseparated scheme X: (1) the category QCoh(X) of all quasicoherent sheaves on X has a flat generator; (2) for every injective object E of QCoh(X), the internal Hom functor into E is exact; (3) for some injective...
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Helmstutler, Randall Douglas. "Quillen equivalent categories of functors /." 2004. http://wwwlib.umi.com/dissertations/fullcit/3149175.

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Book chapters on the topic "Quillen equivalence"

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Johnson, Niles, and Donald Yau. "The Whitehead Theorem for Bicategories." In 2-Dimensional Categories, 275–304. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198871378.003.0007.

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In this chapter, the Whitehead Theorem for bicategories is proved in detail. The Whitehead Theorem states that a pseudofunctor between bicategories is a biequivalence if and only if it is surjective up to adjoint equivalences on objects, surjective up to isomorphisms on 1-cells, and bijective on 2-cells. The chapter covers the lax slice bicategory, lax terminal objects, and the Quillen Theorem A for bicategories. A 2-categorical version of the Whitehead Theorem is also discussed.
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Journal articles
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