# Research

## Description

A central or recurring theme in my work is Category Theory, in particular its applications to Algebraic Geometry and Homotopy Theory.

In more details, I am interested in higher algebra, which is, in a broad sense, the study of familiar algebraic structures—groups, rings, and others—suitably translated to (higher) categories.

Some instances where these gadgets concretely manifest themselves concern Intersection Theory and $K$-Theory, and the geometry of homotopy types. Here are some of the main themes:

• The correspondence between (Cartier) divisors and line bundles is well known. Its analog in codimension 2 (think about a point on a surface) is that to such a cycle corresponds a gerbe bound1 by the sheaf $\mathcal{K}_{2,X}$, whose stalks are the Quillen $K_2(\mathcal{O}_X)$-groups. The construction of this gerbe, as well as the construction of the cup-product corresponding to the intersection product via liftings to a central extension, is joint work with Niranjan Ramachandran.

• Categorical rings are Picard groupoids2 equipped with a second monoidal structure turning them into the categorical analog of a ring. (Picard groupoids can conveniently model the 1-type of connective spectra, such as the ones arising in $K$-Theory from, say, exact categories. A monoidal structure at this level turns the corresponding 1-type into a categorical ring.) In general, Picard stacks have very convenient algebraic models—or presentations, as they are called. Some facts we can prove about them are: (1) for presented categorical rings the structure is encoded by a biextension; (2) for presented categorical rings whose underlying Picard stack is strict, the biextension is trivial and the presentation is a crossed bimodule—these categorical rings represent classes in André-Quillen cohomology; (3) Picard groupoids comprise a multi-category in which categorical rings are the weak monoids, and this correspondence is transported to the presentations by way of a multi-functor.

• Loosely speaking, a Homotopy Type classifies the connectivity of a space up to a certain degree, so you say $n$-Homotopy Type to consider everything up to the $n^\mathrm{th}$ homotopy group. The interesting bit is to compute the space of morphisms between two homotopy types. For $n=1$ we are just talking about groups and the homomorphisms between them, but for $n>1$ resorting to (higher) categories and homotopy theory is essential. For $n=2,3$ we obtained relatively simpler answers by way of diagrams called Butterflies, with neat applications to nonabelian cohomology, with Behrang Noohi.

## Projects

• (With U. Bruzzo and V. Rubtsov) Extensions and cohomology of Lie Algebroids on a scheme $X$.

• (With Niranjan Ramachandran) Extend the correspondence between codimension 2 cycles and gerbes, as well as the intersection product via extensions, to any codimension. A second strand is to understand the infinitesimal theory, where the base scheme $X$ is replaced by its thickening $\mathbf{X}=X\otimes_k \mathbf{A}$, $\mathbf{A}$ being an Artinian simplicial ring, in the sense of derived geometry.

Part of this project requires generalizing the Heisenberg central extension and its relation to the cup product to simplicial presheaves of abelian groups and presheaves of spectra. This is developed in part with my student Michael Niemeier.

• Extend the functor $D$ from Waldhausen categories to the (homotopy) category of 1-types, which assigns to a Waldhausen category $\mathcal{C}$ a stable crossed module $D(\mathcal{C})$, a presentation for the 1-type of $\mathcal{C}$, to a multi-functor.3 With my student Yaineli Valdes.

• (With Niranjan Ramachandran and Denis Eriksson) Universal 2-determinants, 2-types, and $K_2$.

## Papers

1. We should say ‘‘banded,’’ using a rather ugly neologism.

2. Or stacks, if we work over a site.

3. It is known Waldhausen categories form a multi-category. So do Picard groupoids and their presentations.