Homotopical Adjoint Lifting Theorem

This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. We work in the setting of colored operads, and we do not require them to be $\Sigma$-cofibrant. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as numerous new results. In particular, we recover a recent result of Richter-Shipley about a zig-zag of Quillen equivalences between commutative $H\mathbb{Q}$-algebra spectra and commutative differential graded $\mathbb{Q}$-algebras, but our version involves only three Quillen equivalences instead of six. We also work out the theory of how to lift Quillen equivalences to categories of colored operad algebras after a left Bousfield localization.Comment: This is the final, journal versio

Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups

Let $W$ be an affine Weyl group, and let $\Bbbk$ be a field of characteristic $p>0$. The diagrammatic Hecke category $\mathcal{D}$ for $W$ over $\Bbbk$ is a categorification of the Hecke algebra for $W$ with rich connections to modular representation theory. We explicitly construct a functor from $\mathcal{D}$ to a matrix category which categorifies a recursive representation $\xi : \mathbb{Z}W \rightarrow M_{p^r}(\mathbb{Z}W)$, where $r$ is the rank of the underlying finite root system. This functor gives a method for understanding diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules which are "smaller" by a factor of $p$. It also explains the presence of self-similarity in the $p$-canonical basis, which has been observed in small examples. By decategorifying we obtain a new lower bound on the $p$-canonical basis, which corresponds to new lower bounds on the characters of the indecomposable tilting modules by the recent $p$-canonical tilting character formula due to Achar-Makisumi-Riche-Williamson.Comment: 62 pages, many figures, best viewed in colo

Do-It-Yourself Single Camera 3D Pointer Input Device

We present a new algorithm for single camera 3D reconstruction, or 3D input for human-computer interfaces, based on precise tracking of an elongated object, such as a pen, having a pattern of colored bands. To configure the system, the user provides no more than one labelled image of a handmade pointer, measurements of its colored bands, and the camera's pinhole projection matrix. Other systems are of much higher cost and complexity, requiring combinations of multiple cameras, stereocameras, and pointers with sensors and lights. Instead of relying on information from multiple devices, we examine our single view more closely, integrating geometric and appearance constraints to robustly track the pointer in the presence of occlusion and distractor objects. By probing objects of known geometry with the pointer, we demonstrate acceptable accuracy of 3D localization.Comment: 8 pages, 6 figures, 2018 15th Conference on Computer and Robot Visio

Specialization orders on atom spectra of Grothendieck categories

We introduce systematic methods to construct Grothendieck categories from colored quivers and develop a theory of the specialization orders on the atom spectra of Grothendieck categories. We show that any partially ordered set is realized as the atom spectrum of some Grothendieck category, which is an analog of Hochster's result in commutative ring theory. We also show that there exists a Grothendieck category which has empty atom spectrum but has nonempty injective spectrum.Comment: 39 page