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The homotopy theory of infinity-operads is defined by extending Joyal's homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets is endowed with a model category structure…
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…
The paper studies complementary choice functions, i.e. monotonic and consistent choice functions. Such choice functions were introduced and used in the work \cite{RY} for investigation of matchings with complementary contracts. Three…
The notions and certain fundamental characteristics of the proximal and limiting normal cones with respect to a set are first presented in this paper. We present the ideas of the limiting coderivative and subdifferential with respect to a…
This paper examines the structure of poset matrices by formulating a set of new construction rules for this purpose. In this direction, the technique of partial composition operation will be introduced as the basis for the construction of…
Lenses are a well-established structure for modelling bidirectional transformations, such as the interactions between a database and a view of it. Lenses may be symmetric or asymmetric, and may be composed, forming the morphisms of a…
We introduce a monoid structure on a certain set of labelled binary trees, by a process similar to the construction of the plactic monoid. This leads to a new interpretation of the algebra of planar binary trees of Loday-Ronco.
The main objective of this paper is to construct a symmetric monoidal closed model category of coherently commutative monoidal quasi-categories. We construct another model category structure whose fibrant objects are (essentially) those…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
Inspired by the classical theory of modules over a monoid, we give a first account of the natural notion of module over a monad. The associated notion of morphism of left modules ("Linear" natural transformations) captures an important…
We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the…
By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with…
This work addresses the homotopical analysis of enveloping operads in a general cofibrantly generated symmetric monoidal model category. We show the potential of this analysis by obtaining, in a uniform way, several central results…
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic…
An important goal in studying the relations between unitary VOAs and conformal nets is to prove the equivalence of their ribbon categories. In this article, we prove this conjecture for many familiar examples. Our main idea is to construct…
We introduce the condensed fiber product of two $G$-crossed braided fusion categories, generalizing existing constructions in the literature. We show that this product is closely related to the cohomological construction known as zesting.…
We develop a theory of weak omega categories that will be accessible to anyone who is familiar with the language of categories and functors and who has encountered the definition of a strict 2-category. The most remarkable feature of this…
Web Services provide interoperable mechanisms for describing, locating and invoking services over the Internet; composition further enables to build complex services out of simpler ones for complex B2B applications. While current studies on…
Reasoning about weak higher categorical structures constitutes a challenging task, even to the experts. One principal reason is that the language of set theory is not invariant under the weaker notions of equivalence at play, such as…
One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce "structured cospans" as a way to study networks with inputs and outputs. Given a functor $L \colon…