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We introduce two-sorted theories in the style of [CN10] for the complexity classes \oplusL and DET, whose complete problems include determinants over Z2 and Z, respectively. We then describe interpretations of Soltys' linear algebra theory…

The Kalton-Peck $Z_2$ space is the derived space obtained from the scale of $\ell_p$ spaces by complex interpolation at $1/2$. If we denote by by $Z_2^{real}$ the derived space obtained from the scale of $\ell_p$ spaces by real…

We functorially associate to each relative $\infty$-category $(R,W)$ a simplicial space $N^R_\infty(R,W)$, called its Rezk nerve (a straightforward generalization of Rezk's "classification diagram" construction for relative categories). We…

We prove that the category of rational SO(2)-equivariant spectra has a simple algebraic model. Furthermore, all of our model categories and Quillen equivalences are monoidal, so we can use this classification to understand ring spectra and…

In this paper, we address the construction of homotopy bicategories of $(\infty,2)$-categories, which we take as being modeled by 2-fold Segal spaces. Our main result is the concrete construction of a functor $h_2$ from the category of…

We relate completions of infinitesimal Hecke algebras of sl2 to noncommutative deformations of Kleinian singularities of type D of Crawley-Boevey and Holland. As a consequence, we show an analogue of the inequality of Bernstein and…

Let $\theta$ and $\theta'$ be a pair of exceptional representations in the sense of Kazhdan and Patterson [KP], of a metaplectic double cover of $GL_n$. The tensor $\theta\otimes\theta'$ is a (very large) representation of $GL_n$. We…

We prove a Tannaka duality theorem for $(\infty,1)$-categories. This is a duality between certain derived group stacks, or more generally certain derived gerbes, and symmetric monoidal $(\infty,1)$-categories endowed with particular…

The subject of this paper is a nerve construction for bicategories introduced by Leinster, which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over Joyal's…

We classify all the \emph{$\Delta$-}coherent pairs of measures of the second kind on the real line. We obtain $5$ cases, corresponding to all the families of discrete semiclassical orthogonal polynomials of class $s\leq1.$

We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category with finite limits and colimits has a…

In this paper we construct the category of birational spaces as the category in which Temkin's relative Riemann-Zariski spaces are naturally included. Furthermore we develop an analogue of Raynaud's theory. We prove that the category of…

We construct a 2-equivalence $\mathfrak{CohTheory}^\text{op} \simeq \mathfrak{TypeSpaceFunc}$. Here $\mathfrak{CohTheory}$ is the 2-category of positive theories and $\mathfrak{TypeSpaceFunc}$ is the 2-category of type space functors. We…

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a "tessellation" if there is a discrete subgroup D of G, such that D acts properly discontinuously on G/H, and the…

J. Lurie proved in Higher Topos Theory that for $K$ a simplicial set, $\mathcal{C}$ a simplicial category, $f: \mathfrak{C}[K] \rightarrow \mathcal{C}^{\text{op}}$ an equivalence of simplicial categories, we have a Quillen equivalence…

In this paper, we consider an equivalence relation within the class of finitely presented discrete groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, we say that two finitely presented…

We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's…

In a previous paper I gave a presentation for the Quillen higher algebraic K-groups of an exact category in terms of "acyclic binary multicomplexes". In this paper I take that presentation as a definition of the higher K-groups, generalize…

We apply the Dwyer-Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [DS1], we give a streamlined proof of the Quillen…

In this article, we construct a cofibrantly generated Quillen model structure on the category of small topological categories $\mathbf{Cat}_{\mathbf{Top}}$. It is Quillen equivalent to the Joyal model structure of $(\infty,1)$-categories…