# Browsing by Subject "Homotopy theory"

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Item A convenient category for geometric topology(2021-08-11) Clough, Christopher Adrian; Ben-Zvi, David, 1974-; Blumberg, Andrew J; Raskin, Samuel D; Freed, Daniel SShow more The ∞-topos Diff [superscript ∞] of of differentiable stacks, and its (ordinary) subcategory Diff [superscript ∞ over ≤ 0] of 0-truncated objects, the differential spaces, contain smooth manifolds as a full subcategory and have excellent formal properties: In both settings there is an intrinsic notion of underlying homotopy type of any object, as well as an intrinsic notion of what it means for an internal hom space to have the correct homotopy type. Extending and modernising work by Cisinski on (∞-)toposes and cofinality, we develop a suite of tools for constructing model structures and variants thereof in Diff [superscript ∞ over ≤ 0] and Diff [superscript ∞] which may be used to compare more classical constructions in geometric topology – for instance for computing underlying homotopy types – to the canonical constructions provided here, and thus to compare these classical notions with each other. Moreover, these tools are developed in a way so as to be highly customisable, with a view towards future applications. These model structures moreover allow Diff [superscript ∞ over ≤ 0] and Diff [superscript ∞] to adopt a second role as a model for the theory of homotopy types. In this latter capacity Diff [superscript ∞ over ≤ 0] may be favourably contrasted with quasi-topological spaces: Like the category of quasi-topological spaces, Diff [superscript ∞ over ≤ 0] is Cartesian closed and circumvents the construction of complicated topologies, but, additionally, we show that filtered colimits are homotopy colimits, and closed manifolds are compact in the categorical sense. This makes Diff [superscript ∞ over ≤ 0] a useful replacement for quasi-topological spaces in applications of the sheaf theoretic h-principle.Show more Item A weighty theorem of the heart for the algebraic K-theory of higher categories(2017-05-04) Fontes, Ernest Eugene; Blumberg, Andrew J.; Barwick, Clark; Ben-Zvi, David; Neitzke, AndrewShow more We introduce the notion of a bounded weight structure on a stable [infinity symbol]-category and prove a generalization of Waldhausen’s sphere theorem for the algebraic K-theory of higher categories. The algebraic K-theory of a stable [infinity symbol]-category with a bounded non-degenerate weight structure is proven to be equivalent to the algebraic K-theory of the heart of the weight structure. We relate this theorem to previous results as well as new applications.Show more Item Andre-Quillen (co)homology and equivariant stable homotopy theory(2019-07-30) Leeman, Ethan Jacob; Blumberg, Andrew J.; Allcock, Daniel; Ben-Zvi, David; Hill, MichaelShow more Andre and Quillen introduced a (co)homology theory for augmented commutative rings. Strickland [31] initially proposed some issues with the analogue of the abelianization functor in the equivariant setting. These were resolved by Hill [15] who further gave the notion of a genuine derivation and a module of Kähler differentials. We build on this endeavor by expanding to incomplete Tambara functors, introducing the cotangent complex and its various properties, and producing an analogue of the fundamental spectral sequence.Show more Item Equivariant aspects of topological Hochschild homology(2019-05-15) Sulyma, Yuri John Fraser; Blumberg, Andrew J.; Freed, Daniel S; Ben-Zvi, David D; Hill, Michael AShow more We study two invariants of topological Hochschild homology coming from equivariant homotopy theory: its RO(C [subscript p superscript n])-graded homotopy Mackey functors, and the regular slice filtration. In the case of RO(C [subscript p superscript n])-graded homotopy, we explain how to relate Angeltveit-Gerhardt's work to the gold elements, and in cases of interest give canonical identifications of the relevant groups in terms of the kernels of the Fontaine maps θ̃ [subscript r]. This is then used as input for studying the slice filtration on THH. When R is a torsionfree perfectoid ring, we show that the C [subscript p] -regular slice spectral sequence of THH(R; Z [subscript p]) collapses at E².Show more Item The moduli space of objects in differential graded categories glued along bimodules and a presentability result in the homotopy theory of commutative differential graded algebras(2019-09-23) Reyes, Nicolas Z.; Blumberg, Andrew J.; Ben-Zvi, David; Keel, Sean; Gepner, DavidShow more The moduli space of objects of a dg-category, T, is a derived stack introduced in (31) that paramatrizes "pseudo-perfect T [superscript op] -modules." This construction extends to a Morita invariant functor, [mathematical forumla], which is right adjoint to the functor that assigns to a derived stack it's dg-category of perfect complexes. In this thesis we are primarily concerned with the behavior of semi-orthogonal decompositions of dg categories under this functor. We show that when a dg category, C has a semi-orthogonal decomposition, H⁰ (C) =< H⁰(C₀), H⁰ (C₁) >, the moduli space of objects in C can be expressed as a certain pullback of stacks involving the moduli spaces of objects in C₀ and C₁. We also present a result on the cofibrant generation of a certain model category obtained as the total space of the Grothendieck fibration associated to the "module category" functor mapping a derived ring to its model category of modules.Show more