范畴论

We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into $\mathbb{F}_1$-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids.…

We extend the standard construction of the adjoint representation of a Lie groupoid to the case of an arbitrary higher Lie groupoid. As for a Lie groupoid, the adjoint representation of a higher Lie groupoid turns out to be a representation…

The categorified theories known as "doctrines" specify a category equipped with extra structure, analogous to how ordinary theories specify a set with extra structure. We introduce a new framework for doctrines based on double category…

We prove an unstraightening result for lax transformations between functors from an arbitrary $(\infty,2)$-category to that of $(\infty,2)$-categories. We apply this to study partially (op)lax and weighted (co)limits, giving fibrational…

String diagrams are a graphical language used to represent processes that can be composed sequentially or in parallel, which correspond graphically to horizontal or vertical juxtaposition. In this paper we demonstrate how to compute the…

The Grothendieck construction is a fundamental link between indexed categories and opfibrations. This work is a detailed study of the Grothendieck construction over a small tight bipermutative category in the context of Cat-enriched…

In this article we develop formal category theory within augmented virtual double categories. Notably we formalise the classical notions of Kan extension, Yoneda embedding $\text y_A\colon A \to \hat A$, exact square, total category and…

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…

We provide a categorical interpretation for escrows, i.e. trading protocols in trustless environment, where the exchange between two agents is mediated by a third party where the buyer locks the money until they receive the goods they want…

Given an operad $\mathcal{O}$, we define a notion of weak $\mathcal{O}$-monoids -- which we term $\mathcal{O}$-pseudomonoids -- in a 2-category. In the special case with the 2-category in question is the 2-category $\mathsf{Cat}$ of…

We show how to treat families of $\infty$-categories fibered in categorical patterns (e.g., $\infty$-operads and monoidal $\infty$-categories) in terms of fibrations by relativizing the Grothendieck construction. As applications, we…

To every regular category $\mathcal{A}$ equipped with a degree function $\delta$ one can attach a pseudo-abelian tensor category $\mathcal{T}(\mathcal{A},\delta)$. We show that the generating objects of $\mathcal{T}$ decompose canonically…

In recent times, there has been a growing interest in a structuralist understanding of probability, measure and integration theory. The present thesis contributes to this programme in three ways. First, we construct a commutative…

Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$…

In the first part of this note we further the study of the interactions between Reedy and monoidal structures on a small category, building upon the work of Barwick. We define a Reedy monoidal category as a Reedy category $\mathcal{R}$…

In this article, we characterize convexity in terms of algebras over a PROP, and establish a tensor-product-like symmetric monoidal structure on the category of convex sets. Using these two structures, and the theory of $\scr{O}$-monoidal…

We give a categorical formulation of Kraus' "magic trick" for recovering information from truncated types. Rather than type theory, we work in Van den Berg-Moerdijk path categories with a univalent universe, and rather than propositional…

A neural code on $ n $ neurons is a collection of subsets of the set $ [n]=\{1,2,\dots,n\} $. Curto et al. \cite{curto2013neural} associated a ring $\mathcal{R}_{\mathcal{C}}$ (neural ring) to a neural code $\mathcal{C}$. A special class of…

We develop a self-dual, bivariant extension of the concept of an operadic category, its associated operads and their algebras. Our new theory covers, besides all classical subjects, also generalized traces and bivariant versions of…

We will show that the Morrison-Walker blob complex appearing in Topological Quantum Field Theory is an operadic bar resolution of a certain operad composed of fields and local relations. As a by-product we develop the theory of unary…