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In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally…

Let $\kappa$ be a regular cardinal. We study Gabriel-Ulmer duality when one restricts the 2-category of locally $\kappa$-presentable categories with $\kappa$-accessible right adjoints to its locally full sub-2-category of…

We propose a new model for multicategories with symmetries with respect to Zhang's group operads. The fully faithful embedding of the category of group operads into that of crossed interval groups is made use of, and it is shown that every…

We define and study opfibrations of $V$-enriched categories when $V$ is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with…

It is well-known that pseudo functors from bicategories of spans are equivalent to Beck-Chevalley bifibrations, and therefore capture the relationships underlying the adjunctions suitable as semantics for existential quantification. This…

We give a complete description of partially wrapped Fukaya categories of graded orbifold surfaces with stops. We show that a construction via global sections of a natural cosheaf of A$_\infty$ categories on a Lagrangian core of the surface…

We recapture Douglas' framework for twisted parametrized stable homotopy theory in the language of $\infty$- categories. A twisted spectrum is essentially a section of a bundle of presentable stable $\infty$-categories whose fiber is the…

We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying…

In this paper, we study equivariant Hurewicz fibrations, obtain their internal characteristics, and prove theorems on relationship between equivariant fibrations and fibrations generated by them. Local and global properties of equivariant…

In this paper we describe a general framework for constructing examples of locally linear semistrict monoidal 2-categories covering many examples appearing in link homology theory. The main input datum is a closed foam evaluation formula.…

Interest in weak cubical n-categories arises in various contexts, in particular in topological field theories. In this paper, we describe a concept of double bicategory, namely a strict model of the theory of bicategories in Bicat. We show…

This work presents an exposition of both the internal structure of derived category of an abelian category D*(A) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented…

This lecture series is based on joint work in progress with Shaul Barkan, as well as work in progress of the author. The five sections of these notes correspond to the five lectures, but more details have been added. $2$-dimensional…

Structured and decorated cospans are broadly applicable frameworks for building bicategories or double categories of open systems. We streamline and generalize these frameworks using central concepts of double category theory. We show that,…

We provide a calculus of mates for functors to the $\infty$-category of $\infty$-categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do…

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple…

In this paper I define the notion of a non-degenerate finitely semi-simple semi-strict spherical 2-category of non-zero dimension. Given such a 2-category I define a state-sum for any triangulated compact closed oriented 4-manifold and show…

We introduce the notion of a naive global 2-ring: a functor from the opposite of the $\infty$-category of global spaces to presentably symmetric monoidal stable $\infty$-categories. By passing to global sections, every naive global 2-ring…

Every Grothendieck fibration gives rise to a vertical/cartesian orthogonal factorization system on its domain. We define a cartesian factorization system to be an orthogonal factorization in which the left class satisfies 2-of-3 and is…

Classification questions are often about understanding components of a category. It is much more desirable however to be able to understand the entire homotopy type of this category and not just the set of its components. In this paper we…