Related papers: Infinity category theory from scratch

For a symplectic manifold satisfying some topological condition,we define a special class of modules over the deformation quantization algebra. For any two such modules we construct an infinity local system of morphisms. We construct such…

Coherence is here demonstrated for sesquicartesian categories, which are categories with nonempty finite products and arbitrary finite sums, including the empty sum, where moreover the first and the second projection from the product of the…

For every functor $\mathcal{F} : \mathcal{K} \to \mathbf{C}$, where $\mathcal{K}$ is a small category and $\mathbf{C}$ is a model category which satisfies some mild hypotheses, we define a model category $\mathbf{C}^m$ of…

We define and study notions of comprehension in $(\infty,1)$-category theory. In essence, we do so by implementing B\'{e}nabou's foundations of naive category theory in a univalent meta-theory. In particular, we develop natural…

In this paper, we justify and make precise an elementary approach that establishes the existence of (co)limits in $\mathbf{Cat}$. This approach, while conceptually evident, has not been made fully explicit or systematically described in the…

We introduce the notion of an accessible $\infty$-cosmos and prove that these include the basic examples of $\infty$-cosmoi and are stable under the main constructions. A consequence is that the vast majority of known examples of…

We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.

These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…

We show that both the $\infty$-category of $(\infty, \infty)$-categories with inductively defined equivalences, and with coinductively defined equivalences, satisfy universal properties with respect to weak enrichment in the sense of Gepner…

In part I, using the theory of $\infty$-categories, we constructed a natural ``continuous action'' of $\operatorname {Ham} (M, \omega) $ on the Fukaya category of a closed monotone symplectic manifold. Here we show that this action is…

We study exponentiable functors in the context of synthetic $\infty$-categories. We do this within the framework of simplicial Homotopy Type Theory of Riehl and Shulman. Our main result characterizes exponentiable functors. In order to…

In this paper we study a 2-dimensional version of Quillen's homotopy category construction. Given a category $\mathscr{A}$ and a class of morphisms $\Sigma \subset \mathscr{A}$ containing the identities, we construct a 2-category…

We develop a theory of categories which are simultaneously (1) indexed over a base category S with finite products, and (2) enriched over an S-indexed monoidal category V. This includes classical enriched categories, indexed and fibered…

We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is…

The goal of this work is to describe a categorical formalism for (Extended) Topological Quantum Field Theories (TQFTs) and present them as functors from a suitable category of cobordisms with corners to a linear category, generalizing 2d…

Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering…

In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterize its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed…

In this paper we propose a new treatment about infinite dimensional manifolds, using the language of category and functor. Our definition of infinite dimensional manifolds is a natural generalization of finite dimensional manifolds in the…

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having…

This paper provides a blueprint for the construction of a symplectic $(A_\infty,2)$-category, $\mathsf{Symp}$. We develop two ways of encoding the information in $\mathsf{Symp}$ -- one topological, one algebraic. The topological encoding is…