arXiv subjects
Publications and source records attributed to Marcy Robertson.
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.
A theory of $\infty$-properads is developed, extending both the Joyal-Lurie $\infty$-categories and the Cisinski-Moerdijk-Weiss $\infty$-operads. Every connected wheel-free graph generates a properad, giving rise to the graphical category $Γ$ of properads. Using graphical analogs of coface maps and the properadic nerve functor, an $\infty$-properad is defined as an object in the graphical set category $Set^{Γ^{op}}$ that satisfies some inner horn extension property. Symmetric monoidal closed structures are constructed in the categories of properads and of graphical sets. Strict $\infty$-properads, in which inner horns have unique fillers, are given two alternative characterizations, one in terms of graphical analogs of the Segal maps, and the other as images of the properadic nerve. The fundamental properad of an $\infty$-properad is characterized in terms of homotopy classes of $1$-dimensional elements. Using all connected graphs instead of connected wheel-free graphs, a parallel theory of $\infty$-wheeled properads is also developed.
We consider triangulated orbit categories, with the motivating example of cluster categories, in their usual context of algebraic triangulated categories, then present them from another perspective in the framework of topological triangulated categories.
The notion of a derived A-infinity algebra arose in the work of Sagave as a natural generalisation of the classical A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We develop some of the basic operadic theory of derived A-infinity algebras, building on work of Livernet-Roitzheim-Whitehouse. In particular, we study the coalgebras over the Koszul dual cooperad of the operad dAs, and provide a simple description of these. We study representations of derived A-infinity algebras and explain how these are a two-sided version of Sagave's modules over derived A-infinity algebras. We also give a new explicit example of a derived A-infinity algebra.
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. The titular category has nice formal properties: it is bicomplete and is a symmetric monoidal category, with monoidal product closely related to the Boardman-Vogt tensor product of operads. Tools developed in this article, which is the first part of a larger work, include a generalized version of multilinearity of functors, a free prop construction defined on certain "generalized graphs", and the relationship between the category of props and the categories of permutative categories and of operads.
In this paper we construct a cofibrantly generated model category structure on the category of all small symmetric multicategories enriched in simplicial sets.
The purpose of this paper is to study the derived category of simplicial multicategories with arbitrary sets of objects (also known as, colored operads in simplicial sets). Our main result is a derived Morita theory for operads-where we describe the derived mapping spaces between two multicategories P and Q in terms of the nerve of a certain category of P-Q-bimodules. As an application, we show that the derived category possesses internal Hom-objects.