Related papers: Lax Distributive Laws for Topology, I
The monad of convex sets of probability distributions is a well-known tool for modelling the combination of nondeterministic and probabilistic computational effects. In this work we lift this monad from the category of sets to the category…
Linearly distributive categories (LDC), introduced by Cockett and Seely to model multiplicative linear logic, are categories equipped with two monoidal structures that interact via linear distributivities. A seminal result in monoidal…
In this paper we consider the conditions that need to be satisfied by two families of pseudofunctors with a common codomain for them to be collated into a bifunctor. We observe similarities between these conditions and distributive laws of…
Monads play an important role in both the syntax and semantics of modern functional programming languages. The problem of combining them has been of profound interest at least since the 90s, and different approaches have been employed to…
This dissertation is devoted to a study of adjunctions concerning categories enriched over a quantaloid Q (or Q-categories for short), with the following types of adjunctions involved: (1) adjoint functors between Q-categories; (2) adjoint…
We introduce notions of lax semiadditive and lax additive $(\infty,2)$-categories, categorifying the classical notions of semiadditive and additive 1-categories. To establish a well-behaved axiomatic framework, we develop a calculus of lax…
We develop a theory of weighted colimits in the framework of weakly bienriched $\infty$-categories, an extension of Lurie's notion of enriched $\infty$-categories. We prove an existence result for weighted colimits, study weighted colimits…
A basic theory of cowreath or extended distributive laws in the bicategory of unital bimodules, is deciphered. Precisely, we give in terms of tensor product over a scalar base ring, a simplest and equivalent definition for cowreath over…
The aim of this paper is to study categorified algebraic structures and their pseudo- and lax homomorphisms using the framework of Lawvere $2$-theories, and more generally, (enhanced) $2$-dimensional sketches. The key notion we focus on is…
Categories enriched over a commutative unital quantale can be studied as generalized, or many-valued, ordered structures. Because many concepts, such as complete distributivity, in lattice theory can be characterized by existence of certain…
We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg-Moore algebras for an oplax monoidal monad, we always have a natural…
We adapt the notion of an algebraic theory to work in the setting of quasicategories developed recently by Joyal and Lurie. We develop the general theory at some length. We study one extended example in detail: the theory of commutative…
This paper is part of a project that aims to give a homotopy cousin of Kelly's treatment of enriched category theory. After introducing unital co-Segal M-categories, we establish the unital version of a previous theorem that was proven for…
Generalizing the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC), a class of constraints involving a pair of operators was considered in \cite{ZGK2}. For a given constraint, the existences of extensions of…
We introduce a new type of weakly enriched categories over a given symmetric monoidal model category M; these are called Co-Segal categories. Their definition derives from the philosophy of classical (enriched) Segal categories. We study…
Our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored,…
In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result -- the lifting theorem for multitensors --…
We give a framework for combining $n$ monads on the same category via distributive laws satisfying Yang-Baxter equations, extending the classical result of Barr and Wells which combines two monads via one distributive law. We show that this…
The theory of commutative monads on cartesian closed categories provides a framework where aspects of the theory of distributions and other extensive quantities can be formulated and some results proved. We make explicit a link between our…
We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories…