Related papers: Configurations spaces and Theta_n

Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as…

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is…

We study the construction of tensor products of representations up to homotopy, which are the A-infinity version of ordinary representations. We provide formulas for the construction of tensor products of representations up to homotopy and…

We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central…

We present the first definition of strictly associative and unital $\infty$-category. Our proposal takes the form of a type theory whose terms describe the operations of such structures, and whose definitional equality relation enforces…

We describe the relation of $r$-similarity and finite-order invariants on the homotopy set $[S^1,Y]=\pi_1(Y)$.

We give a new construction of the Joyal model structure on the category of simplicial sets, and we provide a simple characterization of the fibrations in it. We characterize the inner anodyne maps in terms of categorical equivalences and…

We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra…

These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the…

We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class…

We prove that the classifying space of a simplicial group is modeled by its homotopy coherent nerve.

Generalising Segal's approach to 1-fold loop spaces, the homotopy theory of $n$-fold loop spaces is shown to be equivalent to the homotopy theory of reduced $\Theta_n$-spaces, where $\Theta_n$ is an iterated wreath product of the simplex…

It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a large-cardinal axiom called Vopenka's principle.In this article we extend the…

We demonstrate that companionships and conjunctions in double $\infty$-categories -- and more generally, in double Segal spaces -- extend to functors out of the free-living companionship and conjunction respectively. Specifically, we prove…

A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of finiteness, countability and…

This paper is an expository account of the theory of stable infinity categories. We prove that the homotopy category of a stable infinity category is triangulated, and that the collection of stable infinity categories is closed under a…

Let $X$ be a compact toric variety. Let $Hol$ denote the space of based holomorphic maps from $CP^1$ to $X$ which lie in a fixed homotopy class. Let $Map$ denote the corresponding space of continuous maps. We show that $Hol$ has the same…

For $X$ a connected finite simplicial complex we consider $\Delta^d(X,n)$ the space of configurations of $n$ ordered points of $X$ such that no $d+1$ of them are equal, and $B^d(X,n)$ the analogous space of configurations of unordered…

Building on a previous definition of homotopy limit of model categories, we give a definition of homotopy colimit of model categories. Using the complete Segal space model for homotopy theories, we verify that this definition corresponds to…