Related papers: Higher Operads, Higher Categories
In [KW14], the new concept of Feynman categories was introduced to simplify the discussion of operad--like objects. In this present paper, we demonstrate the usefulness of this approach, by introducing the concept of decorated Feynman…
Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to…
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 recall several categories of graphs which are useful for describing homotopy-coherent versions of generalized operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms.…
We study the monoidal closed category of symmetric multicategories, especially in relation with its cartesian structure and with sequential multicategories (whose arrows are sequences of concurrent arrows in a given category). Then we…
An outline and summary of four new potential applications of category theory to OOP research are presented. These include (1) the use of operads to model Java subtyping, (2) the use of Yoneda's lemma and representable functors in the…
In recent years, there has been a surge of interest in higher-order topological phases (HOTPs) across various disciplines within the field of physics. These unique phases are characterized by their ability to harbor topological protected…
Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal…
In this survey, we provide an overview of category theory-derived machine learning from four mainstream perspectives: gradient-based learning, probability-based learning, invariance and equivalence-based learning, and topos-based learning.…
A brief overview of the recent developments of operadic and higher categorical techniques in algebraic quantum field theory is given. The relevance of such mathematical structures for the description of gauge theories is discussed.
We give a description of unital operads in a symmetric monoidal category as monoids in a monoidal category of unital $\Lambda$-sequences. This is a new variant of Kelly's old description of operads as monoids in the monoidal category of…
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced…
In this paper we introduce the notion of an operator category and two different models for homotopy theory of $\infty$-operads over an operator category -- one of which extends Lurie's theory of $\infty$-operads, the other of which is…
Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be…
We give a framework for comparing on the one hand theories of n-categories that are weakly enriched operadically, and on the other hand n-categories given as algebras for a contractible globular operad. Examples of the former are the…
It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are Z/2Z-orbifold versions of the little disks operads. We…
We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general…
An n-category is some sort of algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, together with various ways of composing them. We survey various concepts of…
We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we…
Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the "synthetic" development of homotopy…