Related papers: Abstract Kleisli Structures on 2-categories
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
Kleisli simulation is a categorical notion introduced by Hasuo to verify finite trace inclusion. They allow us to give definitions of forward and backward simulation for various types of systems. A generic categorical theory behind Kleisli…
We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate…
The original purpose of component-based development was to provide techniques to master complex software, through composition, reuse and parametrisation. However, such systems are rapidly moving towards a level in which software becomes…
The study of categories abstracting the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. A previous paper offered a survey providing a modern and comprehensive…
We prove an equivalence between cocomplete Yoneda structures and certain proarrow equipments on a 2-category $\mathcal K$. In order to do this, we recognize the presheaf construction of a cocomplete Yoneda structure as a relative, lax…
Given a Hopf algebra in a symmetric monoidal category with duals, the category of modules inherits the structure of a monoidal category with duals. If the notion of algebra is replaced with that of monad on a monoidal category with duals…
These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians $Gr_2(\mathbb{C})$, and the appearance of…
The notion of pseudocategory, as considered in [11], is extended from the context of a 2-category to the more general one of a sesquicategory, which is considered as a category equipped with a 2-cell structure. Some particular examples of…
In this paper we redevelop the foundations of the category theory of quasi-categories (also called infinity-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among…
Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the…
We give a combinatorial characterization of amenability of monomial algebras and prove the existence of monomial Folner sequences, answering a question due to Ceccherini-Silberstein and Samet-Vaillant. We then use our characterization to…
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…
We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal…
With the increased interest in machine learning, and deep learning in particular, the use of automatic differentiation has become more wide-spread in computation. There have been two recent developments to provide the theoretical support…
We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman's neat…
Variations on the notions of Reedy model structures and projective model structures on categories of diagrams in a model category are introduced. These allow one to choose only a subset of the entries when defining weak equivalences, or to…
We show that some recent constructions in the literature, named `weak' generalizations, can be systematically treated by passing from 2-categories to categories enriched in the Cartesian monoidal category of Cauchy complete categories.
This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics. We review the following algebraic structures which appear in two-dimensional conformal field theory (CFT): The symmetries of two-dimensional…
P. Aluffi introduced in [1] a new graded algebra in order to conveniently express characteristic cycles in the theory of singular varieties. This algebra is attached to a surjective ring homomorphism $A\surjects B$ by taking a suitable…