Coherent Presentations of Monoidal Categories

Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations where the objects are considered modulo an equivalence relation, which is described by equational generators. When those form a convergent (abstract) rewriting system on objects, there are three very natural constructions that can be used to define the category which is described by the presentation: one consists in turning equational generators into identities (i.e. considering a quotient category), one consists in formally adding inverses to equational generators (i.e. localizing the category), and one consists in restricting to objects which are normal forms. We show that, under suitable coherence conditions on the presentation, the three constructions coincide, thus generalizing celebrated results on presentations of groups, and we extend those conditions to presentations of monoidal categories

Infinite root stacks and quasi-coherent sheaves on logarithmic schemes

We define and study infinite root stacks of fine and saturated logarithmic schemes, a limit version of the root stacks introduced by Niels Borne and the second author. We show in particular that the infinite root stack determines the logarithmic structure, and recovers the Kummer-flat topos of the logarithmic scheme. We also extend the correspondence between parabolic sheaves and quasi-coherent sheaves on root stacks to this new setting.Comment: v2: 61 pages. Final version, to appear in Proc. Lond. Math. So

Dualizability in Low-Dimensional Higher Category Theory

These lecture notes form an expanded account of a course given at the Summer School on Topology and Field Theories held at the Center for Mathematics at the University of Notre Dame, Indiana during the Summer of 2012. A similar lecture series was given in Hamburg in January 2013. The lecture notes are divided into two parts. The first part, consisting of the bulk of these notes, provides an expository account of the author's joint work with Christopher Douglas and Noah Snyder on dualizability in low-dimensional higher categories and the connection to low-dimensional topology. The cobordism hypothesis provides bridge between topology and algebra, establishing important connections between these two fields. One example of this is the prediction that the $n$-groupoid of so-called `fully-dualizable' objects in any symmetric monoidal $n$-category inherits an O(n)-action. However the proof of the cobordism hypothesis outlined by Lurie is elaborate and inductive. Many consequences of the cobordism hypothesis, such as the precise form of this O(n)-action, remain mysterious. The aim of these lectures is to explain how this O(n)-action emerges in a range of low category numbers ($n \leq 3$). The second part of these lecture notes focuses on the author's joint work with Clark Barwick on the Unicity Theorem, as presented in arXiv:1112.0040. This theorem and the accompanying machinery provide an axiomatization of the theory of $(\infty,n)$-categories and several tools for verifying these axioms. The aim of this portion of the lectures is to provide an introduction to this material.Comment: 65 pages, 8 figures. Lecture Note

Diagrammatics for Coxeter groups and their braid groups

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.Comment: Many figures, best viewed in color. Minor updates. This version agrees with the published versio

Strictification of etale stacky Lie groups

We define stacky Lie groups to be group objects in the 2-category of differentiable stacks. We show that every connected and etale stacky Lie group is equivalent to a crossed module of the form (H,G) where H is the fundamental group of the given stacky Lie group and G is the connected and simply connected Lie group integrating the Lie algebra of the stacky group. Our result is closely related to a strictification result of Baez and Lauda.Comment: 25 page

Loop group actions on categories and Whittaker invariants

We develop some aspects of the theory of $D$-modules on ind-schemes of pro-finite type. These notions are used to define $D$-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. Let $N$ be the maximal unipotent subgroup of a reductive group $G$. For a non-degenerate character $\chi: N(\!(t)\!) \to \mathbb{G}_a$ and a category $\mathcal{C}$ acted upon by $N(\!(t)\!)$, we define the category $\mathcal{C}^{N(\!(t)\!), \chi}$ of $(N(\!(t)\!), \chi)$-invariant objects, along with the coinvariant category $\mathcal{C}_{N(\!(t)\!), \chi}$. These are the Whittaker categories of $\mathcal{C}$, which are in general not equivalent. However, there is always a family of functors $\Theta_k: \mathcal{C}_{N(\!(t)\!), \chi} \to \mathcal{C}^{N(\!(t)\!), \chi}$, parametrized by $k \in \mathbb{Z}$. We conjecture that each $\Theta_k$ is an equivalence, provided that the $N(\!(t)\!)$-action on $\mathcal{C}$ extends to a $G(\!(t)\!)$-action. Using the Fourier-Deligne transform (adapted to Tate vector spaces), we prove this conjecture for $G= GL_n$ and show that the Whittaker categories can be obtained by taking invariants of $\mathcal{C}$ with respect to a very explicit pro-unipotent group subscheme (not ind-scheme) of $G(\!(t)\!)$