Related papers: Flagged higher categories
Categorical spectra are spectrum objects in pointed $(\infty,\infty)$-categories: sequences $(X_n)$ equipped with equivalences $X_n\simeq \Omega X_{n+1}$. This thesis develops foundations for categorical spectra and constructs their tensor…
We describe various equivalent ways of associating to an orbifold, or more generally a higher \'etale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we…
We demonstrate that companionships and conjunctions in double $\infty$-categories -- and more generally, in double Segal spaces -- extend to functors out of the free-living companionship and conjunction respectively. Specifically, we prove…
A class of partially wrapped Fukaya categories in $T^* N$ are proven to be well defined and then studied. In the case of $N$ diffeomorphic to $\mathbb{R}^m \times \mathbb{T}^n$, it is shown that these categories provide homological mirrors…
We introduce the dendroidal analogs of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that…
We construct a monoidal category $\mathscr{C}_{w,v}$ which categorifies the doubly-invariant algebra $^{N'(w)}\mathbb{C}[N]^{N(v)}$ associated with Weyl group elements $w$ and $v$. It gives, after a localization, the coordinate algebra…
We identify natural symmetries of each rigid higher braided category. Specifically, we construct a functorial action by the continuous group $\Omega \mathsf{O}(n)$ on each $\mathcal{E}_{n-1}$-monoidal $(g,d)$-category $\mathcal{R}$ in which…
Stratifolds are considered from a categorical point of view. We show among others that the category of stratifolds fully faithfully embeds into the category of ${\mathbb R}$-algebras as does the category of smooth manifolds. We prove that a…
We show that a $\mathbb{P}$-object and simple configurations of $\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient…
We study the (\infty, 1)-category of autoequivalences of \infty-operads. Using techniques introduced by To\"en, Lurie, and Barwick and Schommer-Pries, we prove that this (\infty, 1)-category is a contractible \infty-groupoid. Our…
Category of pro-nilpotently extended differential graded commutative algebras is introduced. Chevalley-Eilenberg construction provides an equivalence between its certain full subcategory and the opposite to the full subcategory of strong…
We introduce enumerative invariants $F_{g,n}$ $(g\geq0$, $n \geq 1)$ associated to a cyclic $A_\infty$ algebra and a splitting of its non-commutative Hodge filtration. These invariants are defined by explicitly computable Feynman sums, and…
We extend Barwick's and Haugseng's construction of the double $\infty$-category of spans in a pullback-complete $\infty$-category $\mathfrak{C}$ to more general shapes: for a large class of algebraic patterns $\mathfrak{P}$, we define a…
We construct a weak representation of the category of framed affine tangles on a disjoint union of triangulated categories ${\mathcal D}_{2n}$. The categories we use are that of coherent sheaves on Springer fibers over a nilpotent element…
Evidence is given for the correctness of the Joyal-Riehl-Verity construction of the homotopy bicategory of the $(\infty, 2)$-category of $(\infty, 1)$-categories; in particular, it is shown that the analogous construction using complete…
In an enriched setting, we show that higher groupoids and higher categories form categories of fibrant objects. The nerve of a differential graded algebra is a higher category in the category of algebraic varieties, where covers are defined…
To any dg-category $T$ (over some base ring $k$), we define a $D^{-}$-stack $\mathcal{M}_{T}$ in the sense of \cite{hagII}, classifying certain $T^{op}$-dg-modules. When $T$ is saturated, $\mathcal{M}_{T}$ classifies compact objects in the…
Suppose $f,g$ are homogeneous polynomials of degree $d$ defining smooth hypersurfaces $X_f = V(f)\subset \mathbb{P}^{m-1}$ and $X_g = V(g)\subset\mathbb{P}^{n-1}$. Then the sum $f(x)+g(y)$ defines a smooth hypersurface…
We study the equivariant category associated to a finite group action on the derived category of coherent sheaves of a smooth projective variety. We discuss decompositions of the equivariant category and faithful actions, prove the…
Building on work by Fiore-Pronk-Paoli, we construct four model structures on the category of double categories, each modeling one of the following: simplicial spaces, Segal spaces, $(\infty,1)$-categories, and $\infty$-groupoids.…