相关论文: Higher Operads, Higher Categories

Expansion of the categorical point of view on many areas of the mathematics and mathematical physics will cause to deeper understanding of genuine features of these problems. New applications of categorical methods are connected with new…

Operads may be represented as symmetric monoidal functors on a small symmetric monoidal category. We discuss the axioms which must be imposed on a symmetric monoidal functor in order that it give rise to a theory similar to the theory of…

There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict n-categories and…

This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions…

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2-categories. In this paper we continue the developments of [3] and [2] by understanding the…

We write down a series of basic laws for (strict) higher-order circuit diagrams. More precisely, we define higher-order circuit theories in terms of: (a) nesting, (b) temporal and spatial composition, and (c) equivalence between lower-order…

Layered monoidal theories provide a categorical framework for studying scientific theories at different levels of abstraction, via string diagrammatic algebra. We introduce models for three closely related classes of layered monoidal…

Polygraphs are a higher-dimensional generalization of the notion of directed graph. Based on those as unifying concept, this monograph on polygraphs revisits the theory of rewriting in the context of strict higher categories, adopting the…

We study the $2$-categories BIon, of (generalized) bounded ionads, and $\text{Acc}_\omega$, of accessible categories with directed colimits, as an abstract framework to approach formal model theory. We relate them to topoi and (lex)…

Over suitable monoidal model categories, we construct a Dwyer-Kan model category structure on the category of algebras over an augmented operadic collection. As examples we obtain Dwyer-Kan model category structure on the categories of…

Multimodal normal incestual systems are investigated in terms of multiple categories. The different sorted composition of operators are exhibited as 2-cells in multiple categories built up from 2-categories giving rise to different axioms.…

I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an…

A new generalisation of the notion of space, called "vectoid", is suggested in this work. Basic definitions, examples and properties are presented, as well as a construction of direct product of vectoids. Proofs of more complicated…

Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also…

We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial…

This introduction to higher category theory is intended to a give the reader an intuition for what $(\infty,1)$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from…

We introduce a category of locally constant $n$-operads which can be considered as the category of higher braided operads. For $n=1,2,\infty$ the homotopy category of locally constant $n$-operads is equivalent to the homotopy category of…

In this paper we discuss various philosophical aspects of the hyperstructure concept extending networks and higher categories. By this discussion we hope to pave the way for applications and further developments of the mathematical theory…

In these lecture notes, we give a brief introduction to some elements of category theory. The choice of topics is guided by applications to functional programming. Firstly, we study initial algebras, which provide a mathematical…

We introduce the notion of a braiding on a skew monoidal category, whose curious feature is that the defining isomorphisms involve three objects rather than two. These braidings are shown to arise from, and classify, cobraidings (also known…