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We give describe several models for $(infty,n)$-categories, with an emphasis on models given by diagrams of sets and simplicial sets. We look most closely at the cases when $n leq 2$, then summarize methods of generalizing for all $n$.
In this paper we complete a chain of explicit Quillen equivalences between the model category for $Theta_{n+1}$-spaces and the model category of small categories enriched in $Theta_n$-spaces. The Quillen equivalences given here connect Segal category
We introduce a notion of bimodule in the setting of enriched $infty$-categories, and use this to construct a double $infty$-category of enriched $infty$-categories where the two kinds of 1-morphisms are functors and bimodules. We then consider a natu
Let $f:Gto mathrm{Pic}(R)$ be a map of $E_infty$-groups, where $mathrm{Pic}(R)$ denotes the Picard space of an $E_infty$-ring spectrum $R$. We determine the tensor $Xotimes_R Mf$ of the Thom $E_infty$-$R$-algebra $Mf$ with a space $X$; when $X$ is th
We show that a well behaved Noetherian, finite dimensional, stable, monoidal model category is equivalent to a model built from categories of modules over completed rings in an adelic fashion. For abelian groups this is based on the Hasse square, f
We propose foundations for a synthetic theory of $(infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of arbitrary t