# Browsing by Subject "Category 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 Autoequivalences, stability conditions, and n-gons : an example of how stability conditions illuminate the action of autoequivalences associated to derived categories(2010-05) Lowrey, Parker Eastin; Ben-Zvi, David, 1974-; Freed, Daniel; Uhlenbeck, Karen; Allcock, Daniel; Distler, JacquesShow more Understanding the action of an autoequivalence on a triangulated category is generally a very difficult problem. If one can find a stability condition for which the autoequivalence is "compatible", one can explicitly write down the action of this autoequivalence. In turn, the now understood autoequivalence can provide ways of extracting geometric information from the stability condition. In this thesis, we elaborate on what it means for an autoequivalence and stability condition to be "compatibile" and derive a sufficiency criterion.Show more Item Infinitesimal symmetries of Dixmier-Douady gerbes(2012-08) Collier, Braxton Livingston; Freed, Daniel S.; Allcock, Daniel; Ben-Zvi, David; Keel, Sean; Meinrenken, EckhardShow more This thesis introduces the infinitesimal symmetries of Dixmier-Douady gerbes over smooth manifolds. The collection of these symmetries are the counterpart for gerbes of the Lie algebra of circle invariant vector fields on principal circle bundles, and are intimately related to connective structures and curvings. We prove that these symmetries possess a Lie 2-algebra structure, and relate them to equivariant gerbes via a "differentiation functor". We also explain the relationship between the infinitesimal symmetries of gerbes and other mathematical structures including Courant algebroids and the String Lie 2-algebra.Show more