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It is well known since Stasheffs work that 1-fold loop spaces can be described in terms of the existence of higher homotopies for associativity (coherence conditions) or equivalently as algebras of contractible non-symmetric operads. The combinatorics of these higher homotopies is well understood and is extremely useful. For $n ge 2$ the theory of symmetric operads encapsulated the corresponding higher homotopies, yet hid the combinatorics and it has remain a mystery for almost 40 years. However, the recent developments in many fields ranging from algebraic topology and algebraic geometry to mathematical physics and category theory show that this combinatorics in higher dimensions will be even more important than the one dimensional case. In this paper we are going to show that there exists a conceptual way to make these combinatorics explicit using the so called higher nonsymmetric $n$-operads.
This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoida
In this article we discuss a relation between the string topology and differential forms based on the theory of Chens iterated integrals and the cyclic bar complex.
Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in $(infty,1)$-category theory to study presheaves valued in $(infty,1)$-categories. In this work we define and study fibrations modeling p
We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets, on bisimplicial spaces, on bisimplicial sets, on marked simplicial spaces. The main way to pro
Mittag-Leffler condition ensures the exactness of the inverse limit of short exact sequences indexed on a partially ordered set $(I,leq)$ admitting a $countable$ cofinal subset. We extend Mittag-Leffler condition by relatively relaxing the countabili